Daily Papers

Differentiable volumetric rendering-based methods made significant progress in novel view synthesis. On one hand, innovative methods have replaced the Neural Radiance Fields (NeRF) network with locally parameterized structures, enabling high-quality renderings in a reasonable time. On the other hand, approaches have used differentiable splatting instead of NeRF's ray casting to optimize radiance fields rapidly using Gaussian kernels, allowing for fine adaptation to the scene. However, differentiable ray casting of irregularly spaced kernels has been scarcely explored, while splatting, despite enabling fast rendering times, is susceptible to clearly visible artifacts. Our work closes this gap by providing a physically consistent formulation of the emitted radiance c and density {\sigma}, decomposed with Gaussian functions associated with Spherical Gaussians/Harmonics for all-frequency colorimetric representation. We also introduce a method enabling differentiable ray casting of irregularly distributed Gaussians using an algorithm that integrates radiance fields slab by slab and leverages a BVH structure. This allows our approach to finely adapt to the scene while avoiding splatting artifacts. As a result, we achieve superior rendering quality compared to the state-of-the-art while maintaining reasonable training times and achieving inference speeds of 25 FPS on the Blender dataset. Project page with videos and code: https://raygauss.github.io/

Structural optimization is a popular method for designing objects such as bridge trusses, airplane wings, and optical devices. Unfortunately, the quality of solutions depends heavily on how the problem is parameterized. In this paper, we propose using the implicit bias over functions induced by neural networks to improve the parameterization of structural optimization. Rather than directly optimizing densities on a grid, we instead optimize the parameters of a neural network which outputs those densities. This reparameterization leads to different and often better solutions. On a selection of 116 structural optimization tasks, our approach produces the best design 50% more often than the best baseline method.

Sequential learning paradigms pose challenges for gradient-based deep learning due to difficulties incorporating new data and retaining prior knowledge. While Gaussian processes elegantly tackle these problems, they struggle with scalability and handling rich inputs, such as images. To address these issues, we introduce a technique that converts neural networks from weight space to function space, through a dual parameterization. Our parameterization offers: (i) a way to scale function-space methods to large data sets via sparsification, (ii) retention of prior knowledge when access to past data is limited, and (iii) a mechanism to incorporate new data without retraining. Our experiments demonstrate that we can retain knowledge in continual learning and incorporate new data efficiently. We further show its strengths in uncertainty quantification and guiding exploration in model-based RL. Further information and code is available on the project website.

This paper introduces a new parameterization of deep neural networks (both fully-connected and convolutional) with guaranteed ell^2 Lipschitz bounds, i.e. limited sensitivity to input perturbations. The Lipschitz guarantees are equivalent to the tightest-known bounds based on certification via a semidefinite program (SDP). We provide a ``direct'' parameterization, i.e., a smooth mapping from mathbb R^N onto the set of weights satisfying the SDP-based bound. Moreover, our parameterization is complete, i.e. a neural network satisfies the SDP bound if and only if it can be represented via our parameterization. This enables training using standard gradient methods, without any inner approximation or computationally intensive tasks (e.g. projections or barrier terms) for the SDP constraint. The new parameterization can equivalently be thought of as either a new layer type (the sandwich layer), or a novel parameterization of standard feedforward networks with parameter sharing between neighbouring layers. A comprehensive set of experiments on image classification shows that sandwich layers outperform previous approaches on both empirical and certified robust accuracy. Code is available at https://github.com/acfr/LBDN.

The cost of hyperparameter tuning in deep learning has been rising with model sizes, prompting practitioners to find new tuning methods using a proxy of smaller networks. One such proposal uses muP parameterized networks, where the optimal hyperparameters for small width networks transfer to networks with arbitrarily large width. However, in this scheme, hyperparameters do not transfer across depths. As a remedy, we study residual networks with a residual branch scale of 1/text{depth} in combination with the muP parameterization. We provide experiments demonstrating that residual architectures including convolutional ResNets and Vision Transformers trained with this parameterization exhibit transfer of optimal hyperparameters across width and depth on CIFAR-10 and ImageNet. Furthermore, our empirical findings are supported and motivated by theory. Using recent developments in the dynamical mean field theory (DMFT) description of neural network learning dynamics, we show that this parameterization of ResNets admits a well-defined feature learning joint infinite-width and infinite-depth limit and show convergence of finite-size network dynamics towards this limit.

In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.

The creation of photorealistic virtual worlds requires the accurate modeling of 3D surface geometry for a wide range of objects. For this, meshes are appealing since they 1) enable fast physics-based rendering with realistic material and lighting, 2) support physical simulation, and 3) are memory-efficient for modern graphics pipelines. Recent work on reconstructing and statistically modeling 3D shape, however, has critiqued meshes as being topologically inflexible. To capture a wide range of object shapes, any 3D representation must be able to model solid, watertight, shapes as well as thin, open, surfaces. Recent work has focused on the former, and methods for reconstructing open surfaces do not support fast reconstruction with material and lighting or unconditional generative modelling. Inspired by the observation that open surfaces can be seen as islands floating on watertight surfaces, we parameterize open surfaces by defining a manifold signed distance field on watertight templates. With this parameterization, we further develop a grid-based and differentiable representation that parameterizes both watertight and non-watertight meshes of arbitrary topology. Our new representation, called Ghost-on-the-Shell (G-Shell), enables two important applications: differentiable rasterization-based reconstruction from multiview images and generative modelling of non-watertight meshes. We empirically demonstrate that G-Shell achieves state-of-the-art performance on non-watertight mesh reconstruction and generation tasks, while also performing effectively for watertight meshes.

Geometric Deep Learning has recently made striking progress with the advent of continuous Deep Implicit Fields. They allow for detailed modeling of watertight surfaces of arbitrary topology while not relying on a 3D Euclidean grid, resulting in a learnable parameterization that is not limited in resolution. Unfortunately, these methods are often not suitable for applications that require an explicit mesh-based surface representation because converting an implicit field to such a representation relies on the Marching Cubes algorithm, which cannot be differentiated with respect to the underlying implicit field. In this work, we remove this limitation and introduce a differentiable way to produce explicit surface mesh representations from Deep Signed Distance Functions. Our key insight is that by reasoning on how implicit field perturbations impact local surface geometry, one can ultimately differentiate the 3D location of surface samples with respect to the underlying deep implicit field. We exploit this to define MeshSDF, an end-to-end differentiable mesh representation which can vary its topology. We use two different applications to validate our theoretical insight: Single-View Reconstruction via Differentiable Rendering and Physically-Driven Shape Optimization. In both cases our differentiable parameterization gives us an edge over state-of-the-art algorithms.

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s geq 3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young et al., Phys. Rev. E 96, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.

The structure learning problem consists of fitting data generated by a Directed Acyclic Graph (DAG) to correctly reconstruct its arcs. In this context, differentiable approaches constrain or regularize the optimization problem using a continuous relaxation of the acyclicity property. The computational cost of evaluating graph acyclicity is cubic on the number of nodes and significantly affects scalability. In this paper we introduce COSMO, a constraint-free continuous optimization scheme for acyclic structure learning. At the core of our method, we define a differentiable approximation of an orientation matrix parameterized by a single priority vector. Differently from previous work, our parameterization fits a smooth orientation matrix and the resulting acyclic adjacency matrix without evaluating acyclicity at any step. Despite the absence of explicit constraints, we prove that COSMO always converges to an acyclic solution. In addition to being asymptotically faster, our empirical analysis highlights how COSMO performance on graph reconstruction compares favorably with competing structure learning methods.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood

Tensor network (TN) is a powerful framework in machine learning, but selecting a good TN model, known as TN structure search (TN-SS), is a challenging and computationally intensive task. The recent approach TNLS~li2022permutation showed promising results for this task, however, its computational efficiency is still unaffordable, requiring too many evaluations of the objective function. We propose TnALE, a new algorithm that updates each structure-related variable alternately by local enumeration, greatly reducing the number of evaluations compared to TNLS. We theoretically investigate the descent steps for TNLS and TnALE, proving that both algorithms can achieve linear convergence up to a constant if a sufficient reduction of the objective is reached in each neighborhood. We also compare the evaluation efficiency of TNLS and TnALE, revealing that Omega(2^N) evaluations are typically required in TNLS for reaching the objective reduction in the neighborhood, while ideally O(N^2R) evaluations are sufficient in TnALE, where N denotes the tensor order and R reflects the ``low-rankness'' of the neighborhood. Experimental results verify that TnALE can find practically good TN-ranks and permutations with vastly fewer evaluations than the state-of-the-art algorithms.

Automatic analysis of the enormous sets of images is a critical task in life sciences. This faces many challenges such as: algorithms are highly parameterized, significant human input is intertwined, and lacking a standard meta-visualization approach. This paper proposes an alternative iterative approach for optimizing input parameters, saving time by minimizing the user involvement, and allowing for understanding the workflow of algorithms and discovering new ones. The main focus is on developing an interactive visualization technique that enables users to analyze the relationships between sampled input parameters and corresponding output. This technique is implemented as a prototype called Veni Vidi Vici, or "I came, I saw, I conquered." This strategy is inspired by the mathematical formulas of numbering computable functions and is developed atop ImageJ, a scientific image processing program. A case study is presented to investigate the proposed framework. Finally, the paper explores some potential future issues in the application of the proposed approach in parameter space analysis in visualization.

We define the bicategory of Graph Convolutional Neural Networks GCNN_n for an arbitrary graph with n nodes. We show it can be factored through the already existing categorical constructions for deep learning called Para and Lens with the base category set to the CoKleisli category of the product comonad. We prove that there exists an injective-on-objects, faithful 2-functor GCNN_n to Para(CoKl(R^{n times n} times -)). We show that this construction allows us to treat the adjacency matrix of a GCNN as a global parameter instead of a a local, layer-wise one. This gives us a high-level categorical characterisation of a particular kind of inductive bias GCNNs possess. Lastly, we hypothesize about possible generalisations of GCNNs to general message-passing graph neural networks, connections to equivariant learning, and the (lack of) functoriality of activation functions.

Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.

In this work, we propose a communication-efficient parameterization, FedPara, for federated learning (FL) to overcome the burdens on frequent model uploads and downloads. Our method re-parameterizes weight parameters of layers using low-rank weights followed by the Hadamard product. Compared to the conventional low-rank parameterization, our FedPara method is not restricted to low-rank constraints, and thereby it has a far larger capacity. This property enables to achieve comparable performance while requiring 3 to 10 times lower communication costs than the model with the original layers, which is not achievable by the traditional low-rank methods. The efficiency of our method can be further improved by combining with other efficient FL optimizers. In addition, we extend our method to a personalized FL application, pFedPara, which separates parameters into global and local ones. We show that pFedPara outperforms competing personalized FL methods with more than three times fewer parameters.

In "Backprop as functor", the authors show that the fundamental elements of deep learning -- gradient descent and backpropagation -- can be conceptualized as a strong monoidal functor Para(Euc)toLearn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism LearncongPara(Slens), where Slens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that Slens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor Amapsto Ay^A. Using the fact that (Poly,otimes) is monoidal closed, we show that a map Ato B in Para(Slens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type [Ay^A,By^B]. Finally, we review the fact that the category p-Coalg of dynamical systems on any p in Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.

In this article, we study the parameterized complexity of the Set Cover problem restricted to semi-ladder-free hypergraphs, a class defined by Fabianski et al. [Proceedings of STACS 2019]. We observe that two algorithms introduced by Langerman and Morin [Discrete & Computational Geometry 2005] in the context of geometric covering problems can be adapted to this setting, yielding simple FPT and kernelization algorithms for Set Cover in semi-ladder-free hypergraphs. We complement our algorithmic results with a compression lower bound for the problem, which proves the tightness of our kernelization under standard complexity-theoretic assumptions.

Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph G=(V, D,B) is parameterized by a rational function with parameters for each vertex and edge in G. This rational parametrization naturally allows for the study of these models from an algebraic and combinatorial point of view. Indeed, this point of view has led to a collection of results in the literature, mainly focusing on questions related to identifiability and determining relationships between covariances (i.e., finding polynomials in the Gaussian vanishing ideal). So far, a large proportion of these results has focused on the case when D, the directed part of the mixed graph G, is acyclic. This is due to the fact that in the acyclic case, the parametrization becomes polynomial and there is a description of the entries of the covariance matrices in terms of a finite sum. We move beyond the acyclic case and give a closed form expression for the entries of the covariance matrices in terms of the one-connections in a graph obtained from D through some small operations. This closed form expression then allows us to show that if G is simple, then the parametrization map is generically finite-to-one. Finally, having a closed form expression for the covariance matrices allows for the development of an algorithm for systematically exploring possible polynomials in the Gaussian vanishing ideal.

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.

In the realm of generative models for graphs, extensive research has been conducted. However, most existing methods struggle with large graphs due to the complexity of representing the entire joint distribution across all node pairs and capturing both global and local graph structures simultaneously. To overcome these issues, we introduce a method that generates a graph by progressively expanding a single node to a target graph. In each step, nodes and edges are added in a localized manner through denoising diffusion, building first the global structure, and then refining the local details. The local generation avoids modeling the entire joint distribution over all node pairs, achieving substantial computational savings with subquadratic runtime relative to node count while maintaining high expressivity through multiscale generation. Our experiments show that our model achieves state-of-the-art performance on well-established benchmark datasets while successfully scaling to graphs with at least 5000 nodes. Our method is also the first to successfully extrapolate to graphs outside of the training distribution, showcasing a much better generalization capability over existing methods.

Directed acyclic graphs (DAGs) encode a lot of information about a particular distribution in their structure. However, compute required to infer these structures is typically super-exponential in the number of variables, as inference requires a sweep of a combinatorially large space of potential structures. That is, until recent advances made it possible to search this space using a differentiable metric, drastically reducing search time. While this technique -- named NOTEARS -- is widely considered a seminal work in DAG-discovery, it concedes an important property in favour of differentiability: transportability. To be transportable, the structures discovered on one dataset must apply to another dataset from the same domain. We introduce D-Struct which recovers transportability in the discovered structures through a novel architecture and loss function while remaining fully differentiable. Because D-Struct remains differentiable, our method can be easily adopted in existing differentiable architectures, as was previously done with NOTEARS. In our experiments, we empirically validate D-Struct with respect to edge accuracy and structural Hamming distance in a variety of settings.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f =(f_1, ldots, f_N)in C[x_1, ldots, x_n]^N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes the multiplicity structure at the root. We use $alpha$-theory test to certify the quadratic convergence, and togive bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.

Mechanistic Interpretability aims to reverse engineer the algorithms implemented by neural networks by studying their weights and activations. An obstacle to reverse engineering neural networks is that many of the parameters inside a network are not involved in the computation being implemented by the network. These degenerate parameters may obfuscate internal structure. Singular learning theory teaches us that neural network parameterizations are biased towards being more degenerate, and parameterizations with more degeneracy are likely to generalize further. We identify 3 ways that network parameters can be degenerate: linear dependence between activations in a layer; linear dependence between gradients passed back to a layer; ReLUs which fire on the same subset of datapoints. We also present a heuristic argument that modular networks are likely to be more degenerate, and we develop a metric for identifying modules in a network that is based on this argument. We propose that if we can represent a neural network in a way that is invariant to reparameterizations that exploit the degeneracies, then this representation is likely to be more interpretable, and we provide some evidence that such a representation is likely to have sparser interactions. We introduce the Interaction Basis, a tractable technique to obtain a representation that is invariant to degeneracies from linear dependence of activations or Jacobians.

This paper studies the online node classification problem under a transductive learning setting. Current methods either invert a graph kernel matrix with O(n^3) runtime and O(n^2) space complexity or sample a large volume of random spanning trees, thus are difficult to scale to large graphs. In this work, we propose an improvement based on the online relaxation technique introduced by a series of works (Rakhlin et al.,2012; Rakhlin and Sridharan, 2015; 2017). We first prove an effective regret O(n^{1+gamma}) when suitable parameterized graph kernels are chosen, then propose an approximate algorithm FastONL enjoying O(kn^{1+gamma}) regret based on this relaxation. The key of FastONL is a generalized local push method that effectively approximates inverse matrix columns and applies to a series of popular kernels. Furthermore, the per-prediction cost is O(vol({S})log 1/epsilon) locally dependent on the graph with linear memory cost. Experiments show that our scalable method enjoys a better tradeoff between local and global consistency.

Large artificial neural networks have become a mainstay of language, vision, and audio processing and synthesis, yet their initializations and learning rates are often set in an unsophisticated fashion, due to the high cost of hyperparameter sweeps at scale. The mu-Parameterization (muP) offers a potential solution to this challenge, yielding scaling rules for model initialization and learning rates while reportedly enabling zero-shot hyperparameter transfer from small to large models. Despite its evident promise, the muP method is not yet widely adopted, perhaps due to higher implementation complexity, many variations, or complex theoretical background. This work investigates muP empirically, focusing on the ubiquitous transformer architecture, and aims to answer a simple question: does mu-Transfer yield optimal learning rates in practice? Studying models of up to 10B parameters and training budgets of up to 190B tokens, we find mu-Transfer works as intended for the majority of important cases, yet also identify a few cases where it may not.

When learning simulations for modeling physical phenomena in industrial designs, geometrical variabilities are of prime interest. While classical regression techniques prove effective for parameterized geometries, practical scenarios often involve the absence of shape parametrization during the inference stage, leaving us with only mesh discretizations as available data. Learning simulations from such mesh-based representations poses significant challenges, with recent advances relying heavily on deep graph neural networks to overcome the limitations of conventional machine learning approaches. Despite their promising results, graph neural networks exhibit certain drawbacks, including their dependency on extensive datasets and limitations in providing built-in predictive uncertainties or handling large meshes. In this work, we propose a machine learning method that do not rely on graph neural networks. Complex geometrical shapes and variations with fixed topology are dealt with using well-known mesh morphing onto a common support, combined with classical dimensionality reduction techniques and Gaussian processes. The proposed methodology can easily deal with large meshes without the need for explicit shape parameterization and provides crucial predictive uncertainties, which are essential for informed decision-making. In the considered numerical experiments, the proposed method is competitive with respect to existing graph neural networks, regarding training efficiency and accuracy of the predictions.

The advancements in neural rendering have increased the need for techniques that enable intuitive editing of 3D objects represented as neural implicit surfaces. This paper introduces a novel neural algorithm for parameterizing neural implicit surfaces to simple parametric domains like spheres and polycubes. Our method allows users to specify the number of cubes in the parametric domain, learning a configuration that closely resembles the target 3D object's geometry. It computes bi-directional deformation between the object and the domain using a forward mapping from the object's zero level set and an inverse deformation for backward mapping. We ensure nearly bijective mapping with a cycle loss and optimize deformation smoothness. The parameterization quality, assessed by angle and area distortions, is guaranteed using a Laplacian regularizer and an optimized learned parametric domain. Our framework integrates with existing neural rendering pipelines, using multi-view images of a single object or multiple objects of similar geometries to reconstruct 3D geometry and compute texture maps automatically, eliminating the need for any prior information. We demonstrate the method's effectiveness on images of human heads and man-made objects.

Robust and effective scaling of models from small to large width typically requires the precise adjustment of many algorithmic and architectural details, such as parameterization and optimizer choices. In this work, we propose a new perspective on parameterization by investigating a key assumption in prior work about the alignment between parameters and data and derive new theoretical results under weaker assumptions and a broader set of optimizers. Our extensive empirical investigation includes tens of thousands of models trained with all combinations of three optimizers, four parameterizations, several alignment assumptions, more than a dozen learning rates, and fourteen model sizes up to 26.8B parameters. We find that the best learning rate scaling prescription would often have been excluded by the assumptions in prior work. Our results show that all parameterizations, not just maximal update parameterization (muP), can achieve hyperparameter transfer; moreover, our novel per-layer learning rate prescription for standard parameterization outperforms muP. Finally, we demonstrate that an overlooked aspect of parameterization, the epsilon parameter in Adam, must be scaled correctly to avoid gradient underflow and propose Adam-atan2, a new numerically stable, scale-invariant version of Adam that eliminates the epsilon hyperparameter entirely.

We present Generalized LoRA (GLoRA), an advanced approach for universal parameter-efficient fine-tuning tasks. Enhancing Low-Rank Adaptation (LoRA), GLoRA employs a generalized prompt module to optimize pre-trained model weights and adjust intermediate activations, providing more flexibility and capability across diverse tasks and datasets. Moreover, GLoRA facilitates efficient parameter adaptation by employing a scalable, modular, layer-wise structure search that learns individual adapter of each layer. Originating from a unified mathematical formulation, GLoRA exhibits strong transfer learning, few-shot learning and domain generalization abilities, as it adjusts to new tasks through additional dimensions on weights and activations. Comprehensive experiments demonstrate that GLoRA outperforms all previous methods in natural, specialized, and structured benchmarks, achieving superior accuracy with fewer parameters and computations on various datasets. Furthermore, our structural re-parameterization design ensures that GLoRA incurs no extra inference cost, rendering it a practical solution for resource-limited applications. Code is available at: https://github.com/Arnav0400/ViT-Slim/tree/master/GLoRA.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

A neural network with the widely-used ReLU activation has been shown to partition the sample space into many convex polytopes for prediction. However, the parameterized way a neural network and other machine learning models use to partition the space has imperfections, e.g., the compromised interpretability for complex models, the inflexibility in decision boundary construction due to the generic character of the model, and the risk of being trapped into shortcut solutions. In contrast, although the non-parameterized models can adorably avoid or downplay these issues, they are usually insufficiently powerful either due to over-simplification or the failure to accommodate the manifold structures of data. In this context, we first propose a new type of machine learning models referred to as Manifoldron that directly derives decision boundaries from data and partitions the space via manifold structure discovery. Then, we systematically analyze the key characteristics of the Manifoldron such as manifold characterization capability and its link to neural networks. The experimental results on 4 synthetic examples, 20 public benchmark datasets, and 1 real-world application demonstrate that the proposed Manifoldron performs competitively compared to the mainstream machine learning models. We have shared our code in https://github.com/wdayang/Manifoldron for free download and evaluation.

A key requirement for graph neural networks is that they must process a graph in a way that does not depend on how the graph is described. Traditionally this has been taken to mean that a graph network must be equivariant to node permutations. Here we show that instead of equivariance, the more general concept of naturality is sufficient for a graph network to be well-defined, opening up a larger class of graph networks. We define global and local natural graph networks, the latter of which are as scalable as conventional message passing graph neural networks while being more flexible. We give one practical instantiation of a natural network on graphs which uses an equivariant message network parameterization, yielding good performance on several benchmarks.

We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor graph associated with the emission distribution. This allows the exponential-in-dimension cost of exact filtering and smoothing to be avoided. We prove that the approximation accuracy, measured in a local total variation norm, is "dimension-free" in the sense that as the overall dimension of the model increases the error bounds we derive do not necessarily degrade. A key step in the analysis is to quantify the error introduced by localizing the likelihood function in a Bayes' rule update. The factorial structure of the likelihood function which we exploit arises naturally when data have known spatial or network structure. We demonstrate the new algorithms on synthetic examples and a London Underground passenger flow problem, where the factor graph is effectively given by the train network.

In Multi-Task Learning (MTL), tasks may compete and limit the performance achieved on each other, rather than guiding the optimization to a solution, superior to all its single-task trained counterparts. Since there is often not a unique solution optimal for all tasks, practitioners have to balance tradeoffs between tasks' performance, and resort to optimality in the Pareto sense. Most MTL methodologies either completely neglect this aspect, and instead of aiming at learning a Pareto Front, produce one solution predefined by their optimization schemes, or produce diverse but discrete solutions. Recent approaches parameterize the Pareto Front via neural networks, leading to complex mappings from tradeoff to objective space. In this paper, we conjecture that the Pareto Front admits a linear parameterization in parameter space, which leads us to propose Pareto Manifold Learning, an ensembling method in weight space. Our approach produces a continuous Pareto Front in a single training run, that allows to modulate the performance on each task during inference. Experiments on multi-task learning benchmarks, ranging from image classification to tabular datasets and scene understanding, show that Pareto Manifold Learning outperforms state-of-the-art single-point algorithms, while learning a better Pareto parameterization than multi-point baselines.

Popular parameter-efficient fine-tuning (PEFT) methods, such as LoRA and its variants, freeze pre-trained model weights \(W\) and inject learnable matrices \(\Delta W\). These \(\Delta W\) matrices are structured for efficient parameterization, often using techniques like low-rank approximations or scaling vectors. However, these methods typically show a performance gap compared to full fine-tuning. Although recent PEFT methods have narrowed this gap, they do so at the cost of additional learnable parameters. We propose SVFT, a simple approach that fundamentally differs from existing methods: the structure imposed on \(\Delta W\) depends on the specific weight matrix \(W\). Specifically, SVFT updates \(W\) as a sparse combination of outer products of its singular vectors, training only the coefficients (scales) of these sparse combinations. This approach allows fine-grained control over expressivity through the number of coefficients. Extensive experiments on language and vision benchmarks show that SVFT recovers up to 96% of full fine-tuning performance while training only 0.006 to 0.25% of parameters, outperforming existing methods that only recover up to 85% performance using 0.03 to 0.8% of the trainable parameter budget.

Developing robust representations of chemical structures that enable models to learn topological inductive biases is challenging. In this manuscript, we present a representation of atomistic systems. We begin by proving that our representation satisfies all structural, geometric, efficiency, and generalizability constraints. Afterward, we provide a general algorithm to encode any atomistic system. Finally, we report performance comparable to state-of-the-art methods on numerous tasks. We open-source all code and datasets. The code and data are available at https://github.com/rahulkhorana/PolyatomicComplexes.

We consider a family of linear systems A_mu alpha=C with system matrix A_mu depending on a parameter mu and for simplicity parameter-independent right-hand side C. These linear systems typically result from the finite-dimensional approximation of a parameter-dependent boundary-value problem. We derive a procedure based on the Empirical Interpolation Method to obtain a separated representation of the system matrix in the form A_muapproxsum_{m}beta_m(mu)A_{mu_m} for some selected values of the parameter. Such a separated representation is in particular useful in the Reduced Basis Method. The procedure is called nonintrusive since it only requires to access the matrices A_{mu_m}. As such, it offers a crucial advantage over existing approaches that instead derive separated representations requiring to enter the code at the level of assembly. Numerical examples illustrate the performance of our new procedure on a simple one-dimensional boundary-value problem and on three-dimensional acoustic scattering problems solved by a boundary element method.

Given a complex of groups, we construct a new class of complex of groups that records its local data and offer a functorial perspective on the statement that complexes of groups are locally developable. We also construct a new notion of an immersion of complexes of groups and establish that a locally isometric immersion of a complex of groups into a non-positively curved complex of groups is pi_1-injective. Furthermore, the domain complex of groups is developable and the induced map on geometric realizations of developments is an isometric embedding.

We study compute efficiency of LLM training when using different parameterizations, i.e., rules for adjusting model and optimizer hyperparameters (HPs) as model size changes. Some parameterizations fail to transfer optimal base HPs (such as learning rate) across changes in model depth, requiring practitioners to either re-tune these HPs as they scale up (expensive), or accept sub-optimal training when re-tuning is prohibitive. Even when they achieve HP transfer, we develop theory to show parameterizations may still exist in the lazy learning regime where layers learn only features close to their linearization, preventing effective use of depth and nonlinearity. Finally, we identify and adopt the parameterization we call CompleteP that achieves both depth-wise HP transfer and non-lazy learning in all layers. CompleteP enables a wider range of model width/depth ratios to remain compute-efficient, unlocking shapes better suited for different hardware settings and operational contexts. Moreover, CompleteP enables 12-34% compute efficiency improvements over the prior state-of-the-art.

The performance of a language model has been shown to be effectively modeled as a power-law in its parameter count. Here we study the scaling behaviors of Routing Networks: architectures that conditionally use only a subset of their parameters while processing an input. For these models, parameter count and computational requirement form two independent axes along which an increase leads to better performance. In this work we derive and justify scaling laws defined on these two variables which generalize those known for standard language models and describe the performance of a wide range of routing architectures trained via three different techniques. Afterwards we provide two applications of these laws: first deriving an Effective Parameter Count along which all models scale at the same rate, and then using the scaling coefficients to give a quantitative comparison of the three routing techniques considered. Our analysis derives from an extensive evaluation of Routing Networks across five orders of magnitude of size, including models with hundreds of experts and hundreds of billions of parameters.

We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings. Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement. We extend Baudot and Bennequin's definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature. Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions. The only 1-cocycles are Shannon entropy or Tsallis alpha-entropy, depending on the value of the parameter.

Graph foundation models represent a transformative paradigm for learning transferable representations across diverse graph domains. Recent methods leverage large language models to unify graph and text modalities into a shared representation space using contrastive learning. However, systematic evaluations reveal significant performance degradation at structural boundaries where distinct topological patterns converge, with accuracy losses exceeding 20 percentage points. This issue arises from a key limitation: current methods assume all graph structures can be encoded within a single Euclidean space. In reality, tree structures require hyperbolic geometry to preserve hierarchical branching, while cyclic patterns depend on spherical geometry for closure properties. At structural boundaries, nodes experience conflicting geometric constraints that uniform encoding spaces cannot resolve. This raises a crucial challenge: Can alignment frameworks be designed to respect the intrinsic geometric diversity of graph structures? We introduce GraphShaper, a geometry-aware framework that enhances graph encoding through multi-geometric specialization. Our approach employs expert networks tailored to different geometric spaces, dynamically computing fusion weights to adaptively integrate geometric properties based on local structural characteristics. This adaptive fusion preserves structural integrity before alignment with text embeddings. Extensive experiments demonstrate that GraphShaper achieves 9.47\% accuracy improvements on citation networks and 7.63\% on social networks in zero-shot settings.

We enhance the calculus of string diagrams for monoidal categories with hierarchical features in order to capture closed monoidal (and cartesian closed) structure. Using this new syntax we formulate an automatic differentiation algorithm for (applied) simply typed lambda calculus in the style of [Pearlmutter and Siskind 2008] and we prove for the first time its soundness. To give an efficient yet principled implementation of the AD algorithm we define a sound and complete representation of hierarchical string diagrams as a class of hierarchical hypergraphs we call hypernets.

Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single real-valued function (the one-parameter setting), there is a canonical choice of descriptor for persistent homology: the barcode. The operation mapping a real-valued function to its barcode is differentiable almost everywhere, and the convergence of gradient descent for losses using barcodes is relatively well understood. When optimizing a vector-valued function (the multiparameter setting), there is no unique choice of descriptor for multiparameter persistent homology, and many distinct descriptors have been proposed. This calls for the development of a general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors. In this article, we develop such a framework and show that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape. We complement the theory with numerical experiments supporting the idea that optimizing multiparameter homological descriptors can lead to improved performances compared to optimizing one-parameter descriptors, even when using the simplest and most efficiently computable multiparameter descriptors.

We propose a novel random walk-based algorithm for unbiased estimation of arbitrary functions of a weighted adjacency matrix, coined universal graph random features (u-GRFs). This includes many of the most popular examples of kernels defined on the nodes of a graph. Our algorithm enjoys subquadratic time complexity with respect to the number of nodes, overcoming the notoriously prohibitive cubic scaling of exact graph kernel evaluation. It can also be trivially distributed across machines, permitting learning on much larger networks. At the heart of the algorithm is a modulation function which upweights or downweights the contribution from different random walks depending on their lengths. We show that by parameterising it with a neural network we can obtain u-GRFs that give higher-quality kernel estimates or perform efficient, scalable kernel learning. We provide robust theoretical analysis and support our findings with experiments including pointwise estimation of fixed graph kernels, solving non-homogeneous graph ordinary differential equations, node clustering and kernel regression on triangular meshes.

Geometric Deep Learning has recently made striking progress with the advent of continuous deep implicit fields. They allow for detailed modeling of watertight surfaces of arbitrary topology while not relying on a 3D Euclidean grid, resulting in a learnable parameterization that is unlimited in resolution. Unfortunately, these methods are often unsuitable for applications that require an explicit mesh-based surface representation because converting an implicit field to such a representation relies on the Marching Cubes algorithm, which cannot be differentiated with respect to the underlying implicit field. In this work, we remove this limitation and introduce a differentiable way to produce explicit surface mesh representations from Deep Implicit Fields. Our key insight is that by reasoning on how implicit field perturbations impact local surface geometry, one can ultimately differentiate the 3D location of surface samples with respect to the underlying deep implicit field. We exploit this to define DeepMesh - an end-to-end differentiable mesh representation that can vary its topology. We validate our theoretical insight through several applications: Single view 3D Reconstruction via Differentiable Rendering, Physically-Driven Shape Optimization, Full Scene 3D Reconstruction from Scans and End-to-End Training. In all cases our end-to-end differentiable parameterization gives us an edge over state-of-the-art algorithms.

Grid-based structures are commonly used to encode explicit features for graphics primitives such as images, signed distance functions (SDF), and neural radiance fields (NeRF) due to their simple implementation. However, in n-dimensional space, calculating the value of a sampled point requires interpolating the values of its 2^n neighboring vertices. The exponential scaling with dimension leads to significant computational overheads. To address this issue, we propose a simplex-based approach for encoding graphics primitives. The number of vertices in a simplex-based structure increases linearly with dimension, making it a more efficient and generalizable alternative to grid-based representations. Using the non-axis-aligned simplicial structure property, we derive and prove a coordinate transformation, simplicial subdivision, and barycentric interpolation scheme for efficient sampling, which resembles transformation procedures in the simplex noise algorithm. Finally, we use hash tables to store multiresolution features of all interest points in the simplicial grid, which are passed into a tiny fully connected neural network to parameterize graphics primitives. We implemented a detailed simplex-based structure encoding algorithm in C++ and CUDA using the methods outlined in our approach. In the 2D image fitting task, the proposed method is capable of fitting a giga-pixel image with 9.4% less time compared to the baseline method proposed by instant-ngp, while maintaining the same quality and compression rate. In the volumetric rendering setup, we observe a maximum 41.2% speedup when the samples are dense enough.

Modularization is a cornerstone of computer science, abstracting complex functions into atomic building blocks. In this paper, we introduce a new level of modularization by abstracting generative models into atomic generative modules. Analogous to fractals in mathematics, our method constructs a new type of generative model by recursively invoking atomic generative modules, resulting in self-similar fractal architectures that we call fractal generative models. As a running example, we instantiate our fractal framework using autoregressive models as the atomic generative modules and examine it on the challenging task of pixel-by-pixel image generation, demonstrating strong performance in both likelihood estimation and generation quality. We hope this work could open a new paradigm in generative modeling and provide a fertile ground for future research. Code is available at https://github.com/LTH14/fractalgen.

We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are assumed to be linear subspaces, this reduces to the classical problem of subspace clustering, which has been studied extensively over the past two decades. Unfortunately, many real-world datasets such as natural images can not be well approximated by linear subspaces. On the other hand, numerous works have attempted to learn an appropriate transformation of the data, such that data is mapped from a union of general non-linear manifolds to a union of linear subspaces (with points from the same manifold being mapped to the same subspace). However, many existing works have limitations such as assuming knowledge of the membership of samples to clusters, requiring high sampling density, or being shown theoretically to learn trivial representations. In this paper, we propose to optimize the Maximal Coding Rate Reduction metric with respect to both the data representation and a novel doubly stochastic cluster membership, inspired by state-of-the-art subspace clustering results. We give a parameterization of such a representation and membership, allowing efficient mini-batching and one-shot initialization. Experiments on CIFAR-10, -20, -100, and TinyImageNet-200 datasets show that the proposed method is much more accurate and scalable than state-of-the-art deep clustering methods, and further learns a latent linear representation of the data.

We present two new classes of algorithms for efficient field integration on graphs encoding point clouds. The first class, SeparatorFactorization(SF), leverages the bounded genus of point cloud mesh graphs, while the second class, RFDiffusion(RFD), uses popular epsilon-nearest-neighbor graph representations for point clouds. Both can be viewed as providing the functionality of Fast Multipole Methods (FMMs), which have had a tremendous impact on efficient integration, but for non-Euclidean spaces. We focus on geometries induced by distributions of walk lengths between points (e.g., shortest-path distance). We provide an extensive theoretical analysis of our algorithms, obtaining new results in structural graph theory as a byproduct. We also perform exhaustive empirical evaluation, including on-surface interpolation for rigid and deformable objects (particularly for mesh-dynamics modeling), Wasserstein distance computations for point clouds, and the Gromov-Wasserstein variant.

Global climate models typically operate at a grid resolution of hundreds of kilometers and fail to resolve atmospheric mesoscale processes, e.g., clouds, precipitation, and gravity waves (GWs). Model representation of these processes and their sources is essential to the global circulation and planetary energy budget, but subgrid scale contributions from these processes are often only approximately represented in models using parameterizations. These parameterizations are subject to approximations and idealizations, which limit their capability and accuracy. The most drastic of these approximations is the "single-column approximation" which completely neglects the horizontal evolution of these processes, resulting in key biases in current climate models. With a focus on atmospheric GWs, we present the first-ever global simulation of atmospheric GW fluxes using machine learning (ML) models trained on the WINDSET dataset to emulate global GW emulation in the atmosphere, as an alternative to traditional single-column parameterizations. Using an Attention U-Net-based architecture trained on globally resolved GW momentum fluxes, we illustrate the importance and effectiveness of global nonlocality, when simulating GWs using data-driven schemes.

We investigate novel random graph embeddings that can be computed in expected polynomial time and that are able to distinguish all non-isomorphic graphs in expectation. Previous graph embeddings have limited expressiveness and either cannot distinguish all graphs or cannot be computed efficiently for every graph. To be able to approximate arbitrary functions on graphs, we are interested in efficient alternatives that become arbitrarily expressive with increasing resources. Our approach is based on Lov\'asz' characterisation of graph isomorphism through an infinite dimensional vector of homomorphism counts. Our empirical evaluation shows competitive results on several benchmark graph learning tasks.

Local graph clustering methods aim to detect small clusters in very large graphs without the need to process the whole graph. They are fundamental and scalable tools for a wide range of tasks such as local community detection, node ranking and node embedding. While prior work on local graph clustering mainly focuses on graphs without node attributes, modern real-world graph datasets typically come with node attributes that provide valuable additional information. We present a simple local graph clustering algorithm for graphs with node attributes, based on the idea of diffusing mass locally in the graph while accounting for both structural and attribute proximities. Using high-dimensional concentration results, we provide statistical guarantees on the performance of the algorithm for the recovery of a target cluster with a single seed node. We give conditions under which a target cluster generated from a fairly general contextual random graph model, which includes both the stochastic block model and the planted cluster model as special cases, can be fully recovered with bounded false positives. Empirically, we validate all theoretical claims using synthetic data, and we show that incorporating node attributes leads to superior local clustering performances using real-world graph datasets.

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and that factor through hierarchical clustering functors. We then use this characterization to prove refinement bounds on manifold learning loss functions and construct a hierarchy of manifold learning algorithms based on their equivariants. We express several popular manifold learning algorithms as functors at different levels of this hierarchy, including Metric Multidimensional Scaling, IsoMap, and UMAP. Next, we use interleaving distance to study the stability of a broad class of manifold learning algorithms. We present bounds on how closely the embeddings these algorithms produce from noisy data approximate the embeddings they would learn from noiseless data. Finally, we use our framework to derive a set of novel manifold learning algorithms, which we experimentally demonstrate are competitive with the state of the art.

We propose a new localized inference algorithm for answering marginalization queries in large graphical models with the correlation decay property. Given a query variable and a large graphical model, we define a much smaller model in a local region around the query variable in the target model so that the marginal distribution of the query variable can be accurately approximated. We introduce two approximation error bounds based on the Dobrushin's comparison theorem and apply our bounds to derive a greedy expansion algorithm that efficiently guides the selection of neighbor nodes for localized inference. We verify our theoretical bounds on various datasets and demonstrate that our localized inference algorithm can provide fast and accurate approximation for large graphical models.

Random walks are widely used for mining networks due to the computational efficiency of computing them. For instance, graph representation learning learns a d-dimensional embedding space, so that the nodes that tend to co-occur on random walks (a proxy of being in the same network neighborhood) are close in the embedding space. Specific local network topology (i.e., structure) influences the co-occurrence of nodes on random walks, so random walks of limited length capture only partial topological information, hence diminishing the performance of downstream methods. We explicitly capture all topological neighborhood information and improve performance by introducing orbit adjacencies that quantify the adjacencies of two nodes as co-occurring on a given pair of graphlet orbits, which are symmetric positions on graphlets (small, connected, non-isomorphic, induced subgraphs of a large network). Importantly, we mathematically prove that random walks on up to k nodes capture only a subset of all the possible orbit adjacencies for up to k-node graphlets. Furthermore, we enable orbit adjacency-based analysis of networks by developing an efficient GRaphlet-orbit ADjacency COunter (GRADCO), which exhaustively computes all 28 orbit adjacency matrices for up to four-node graphlets. Note that four-node graphlets suffice, because real networks are usually small-world. In large networks on around 20,000 nodes, GRADCOcomputesthe28matricesinminutes. Onsixrealnetworksfromvarious domains, we compare the performance of node-label predictors obtained by using the network embeddings based on our orbit adjacencies to those based on random walks. We find that orbit adjacencies, which include those unseen by random walks, outperform random walk-based adjacencies, demonstrating the importance of the inclusion of the topological neighborhood information that is unseen by random walks.

In this note we establish a 1-to-1 correspondence between the class of generalized quadrangles with ovoids and the class of balanced incomplete block designs that posses a non-triangular local resolution system and have the appropriate parameters. We present a non-triangular local resolution system for a difference family BIBD construction of Sprott.

A supervised learning algorithm searches over a set of functions A to B parametrised by a space P to find the best approximation to some ideal function fcolon A to B. It does this by taking examples (a,f(a)) in Atimes B, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent---with respect to a fixed step size and an error function satisfying a certain property---defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.

Adapting pre-trained foundation models for various downstream tasks has been prevalent in artificial intelligence. Due to the vast number of tasks and high costs, adjusting all parameters becomes unfeasible. To mitigate this, several fine-tuning techniques have been developed to update the pre-trained model weights in a more resource-efficient manner, such as through low-rank adjustments. Yet, almost all of these methods focus on linear weights, neglecting the intricacies of parameter spaces in higher dimensions like 4D. Alternatively, some methods can be adapted for high-dimensional parameter space by compressing changes in the original space into two dimensions and then employing low-rank matrix decomposition. However, these approaches destructs the structural integrity of the involved high-dimensional spaces. To tackle the diversity of dimensional spaces across different foundation models and provide a more precise representation of the changes within these spaces, this paper introduces a generalized parameter-efficient fine-tuning framework, FLoRA, designed for various dimensional parameter space. Specifically, utilizing Tucker decomposition, FLoRA asserts that changes in each dimensional parameter space are based on a low-rank core space which maintains the consistent topological structure with the original space. It then models the changes through this core space alongside corresponding weights to reconstruct alterations in the original space. FLoRA effectively preserves the structural integrity of the change of original N-dimensional parameter space, meanwhile decomposes it via low-rank tensor decomposition. Extensive experiments on computer vision, natural language processing and multi-modal tasks validate FLoRA's effectiveness. Codes are available at https://github.com/SJTU-DeepVisionLab/FLoRA.

Neural networks are famously nonlinear. However, linearity is defined relative to a pair of vector spaces, f:XtoY. Is it possible to identify a pair of non-standard vector spaces for which a conventionally nonlinear function is, in fact, linear? This paper introduces a method that makes such vector spaces explicit by construction. We find that if we sandwich a linear operator A between two invertible neural networks, f(x)=g_y^{-1}(A g_x(x)), then the corresponding vector spaces X and Y are induced by newly defined addition and scaling actions derived from g_x and g_y. We term this kind of architecture a Linearizer. This framework makes the entire arsenal of linear algebra, including SVD, pseudo-inverse, orthogonal projection and more, applicable to nonlinear mappings. Furthermore, we show that the composition of two Linearizers that share a neural network is also a Linearizer. We leverage this property and demonstrate that training diffusion models using our architecture makes the hundreds of sampling steps collapse into a single step. We further utilize our framework to enforce idempotency (i.e. f(f(x))=f(x)) on networks leading to a globally projective generative model and to demonstrate modular style transfer.

Multi-task learning (MTL) is an inductive transfer mechanism designed to leverage useful information from multiple tasks to improve generalization performance compared to single-task learning. It has been extensively explored in traditional machine learning to address issues such as data sparsity and overfitting in neural networks. In this work, we apply MTL to problems in science and engineering governed by partial differential equations (PDEs). However, implementing MTL in this context is complex, as it requires task-specific modifications to accommodate various scenarios representing different physical processes. To this end, we present a multi-task deep operator network (MT-DeepONet) to learn solutions across various functional forms of source terms in a PDE and multiple geometries in a single concurrent training session. We introduce modifications in the branch network of the vanilla DeepONet to account for various functional forms of a parameterized coefficient in a PDE. Additionally, we handle parameterized geometries by introducing a binary mask in the branch network and incorporating it into the loss term to improve convergence and generalization to new geometry tasks. Our approach is demonstrated on three benchmark problems: (1) learning different functional forms of the source term in the Fisher equation; (2) learning multiple geometries in a 2D Darcy Flow problem and showcasing better transfer learning capabilities to new geometries; and (3) learning 3D parameterized geometries for a heat transfer problem and demonstrate the ability to predict on new but similar geometries. Our MT-DeepONet framework offers a novel approach to solving PDE problems in engineering and science under a unified umbrella based on synergistic learning that reduces the overall training cost for neural operators.

Generalized Tur\'an problems ask for the maximum number of copies of a graph H in an n-vertex, F-free graph, denoted by ex(n,H,F). We show how to extend the new, localized approach of Bradac, Malec, and Tompkins to generalized Tur\'{a}n problems. We weight the copies of H (typically taking H=K_t), instead of the edges, based on the size of the largest clique, path, or star containing the vertices of the copy of H, and in each case prove a tight upper bound on the sum of the weights. A consequence of our new localized theorems is an asymptotic determination of ex(n,H,K_{1,r}) for every H having at least one dominating vertex and mex(m,H,K_{1,r}) for every H having at least two dominating vertices.

We present CHARM, a novel parametric representation and generative framework for anime hairstyle modeling. While traditional hair modeling methods focus on realistic hair using strand-based or volumetric representations, anime hairstyle exhibits highly stylized, piecewise-structured geometry that challenges existing techniques. Existing works often rely on dense mesh modeling or hand-crafted spline curves, making them inefficient for editing and unsuitable for scalable learning. CHARM introduces a compact, invertible control-point-based parameterization, where a sequence of control points represents each hair card, and each point is encoded with only five geometric parameters. This efficient and accurate representation supports both artist-friendly design and learning-based generation. Built upon this representation, CHARM introduces an autoregressive generative framework that effectively generates anime hairstyles from input images or point clouds. By interpreting anime hairstyles as a sequential "hair language", our autoregressive transformer captures both local geometry and global hairstyle topology, resulting in high-fidelity anime hairstyle creation. To facilitate both training and evaluation of anime hairstyle generation, we construct AnimeHair, a large-scale dataset of 37K high-quality anime hairstyles with separated hair cards and processed mesh data. Extensive experiments demonstrate state-of-the-art performance of CHARM in both reconstruction accuracy and generation quality, offering an expressive and scalable solution for anime hairstyle modeling. Project page: https://hyzcluster.github.io/charm/

Structural and Positional Encodings can significantly improve the performance of Graph Neural Networks in downstream tasks. Recent literature has begun to systematically investigate differences in the structural properties that these approaches encode, as well as performance trade-offs between them. However, the question of which structural properties yield the most effective encoding remains open. In this paper, we investigate this question from a geometric perspective. We propose a novel structural encoding based on discrete Ricci curvature (Local Curvature Profiles, short LCP) and show that it significantly outperforms existing encoding approaches. We further show that combining local structural encodings, such as LCP, with global positional encodings improves downstream performance, suggesting that they capture complementary geometric information. Finally, we compare different encoding types with (curvature-based) rewiring techniques. Rewiring has recently received a surge of interest due to its ability to improve the performance of Graph Neural Networks by mitigating over-smoothing and over-squashing effects. Our results suggest that utilizing curvature information for structural encodings delivers significantly larger performance increases than rewiring.

Structural pruning enables model acceleration by removing structurally-grouped parameters from neural networks. However, the parameter-grouping patterns vary widely across different models, making architecture-specific pruners, which rely on manually-designed grouping schemes, non-generalizable to new architectures. In this work, we study a highly-challenging yet barely-explored task, any structural pruning, to tackle general structural pruning of arbitrary architecture like CNNs, RNNs, GNNs and Transformers. The most prominent obstacle towards this goal lies in the structural coupling, which not only forces different layers to be pruned simultaneously, but also expects all removed parameters to be consistently unimportant, thereby avoiding structural issues and significant performance degradation after pruning. To address this problem, we propose a general and {fully automatic} method, Dependency Graph (DepGraph), to explicitly model the dependency between layers and comprehensively group coupled parameters for pruning. In this work, we extensively evaluate our method on several architectures and tasks, including ResNe(X)t, DenseNet, MobileNet and Vision transformer for images, GAT for graph, DGCNN for 3D point cloud, alongside LSTM for language, and demonstrate that, even with a simple norm-based criterion, the proposed method consistently yields gratifying performances.

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

Graph Neural Networks (GNNs) had been demonstrated to be inherently susceptible to the problems of over-smoothing and over-squashing. These issues prohibit the ability of GNNs to model complex graph interactions by limiting their effectiveness in taking into account distant information. Our study reveals the key connection between the local graph geometry and the occurrence of both of these issues, thereby providing a unified framework for studying them at a local scale using the Ollivier-Ricci curvature. Specifically, we demonstrate that over-smoothing is linked to positive graph curvature while over-squashing is linked to negative graph curvature. Based on our theory, we propose the Batch Ollivier-Ricci Flow, a novel rewiring algorithm capable of simultaneously addressing both over-smoothing and over-squashing.

Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer this question by training networks not in their native parameter space, but instead in a smaller, randomly oriented subspace. We slowly increase the dimension of this subspace, note at which dimension solutions first appear, and define this to be the intrinsic dimension of the objective landscape. The approach is simple to implement, computationally tractable, and produces several suggestive conclusions. Many problems have smaller intrinsic dimensions than one might suspect, and the intrinsic dimension for a given dataset varies little across a family of models with vastly different sizes. This latter result has the profound implication that once a parameter space is large enough to solve a problem, extra parameters serve directly to increase the dimensionality of the solution manifold. Intrinsic dimension allows some quantitative comparison of problem difficulty across supervised, reinforcement, and other types of learning where we conclude, for example, that solving the inverted pendulum problem is 100 times easier than classifying digits from MNIST, and playing Atari Pong from pixels is about as hard as classifying CIFAR-10. In addition to providing new cartography of the objective landscapes wandered by parameterized models, the method is a simple technique for constructively obtaining an upper bound on the minimum description length of a solution. A byproduct of this construction is a simple approach for compressing networks, in some cases by more than 100 times.

This paper investigates scaling laws for local SGD in LLM training, a distributed optimization algorithm that facilitates training on loosely connected devices. Through extensive experiments, we show that local SGD achieves competitive results compared to conventional methods, given equivalent model parameters, datasets, and computational resources. Furthermore, we explore the application of local SGD in various practical scenarios, including multi-cluster setups and edge computing environments. Our findings elucidate the necessary conditions for effective multi-cluster LLM training and examine the potential and limitations of leveraging edge computing resources in the LLM training process. This demonstrates its viability as an alternative to single large-cluster training.

State space models (SSM) have recently been shown to be very effective as a deep learning layer as a promising alternative to sequence models such as RNNs, CNNs, or Transformers. The first version to show this potential was the S4 model, which is particularly effective on tasks involving long-range dependencies by using a prescribed state matrix called the HiPPO matrix. While this has an interpretable mathematical mechanism for modeling long dependencies, it introduces a custom representation and algorithm that can be difficult to implement. On the other hand, a recent variant of S4 called DSS showed that restricting the state matrix to be fully diagonal can still preserve the performance of the original model when using a specific initialization based on approximating S4's matrix. This work seeks to systematically understand how to parameterize and initialize such diagonal state space models. While it follows from classical results that almost all SSMs have an equivalent diagonal form, we show that the initialization is critical for performance. We explain why DSS works mathematically, by showing that the diagonal restriction of S4's matrix surprisingly recovers the same kernel in the limit of infinite state dimension. We also systematically describe various design choices in parameterizing and computing diagonal SSMs, and perform a controlled empirical study ablating the effects of these choices. Our final model S4D is a simple diagonal version of S4 whose kernel computation requires just 2 lines of code and performs comparably to S4 in almost all settings, with state-of-the-art results for image, audio, and medical time-series domains, and averaging 85\% on the Long Range Arena benchmark.

Recent years have seen a growing interest and adoption of LLMs, with muTransfer becoming a key technique for tuning hyperparameters in large-scale training. Meanwhile, Mixture-of-Experts (MoE) has emerged as a leading architecture in extremely large models. However, the intersection of these two advancements has remained unexplored. In this work, we derive a mu-Parameterization (muP) for MoE, providing theoretical guarantees for feature learning across model widths in both the router and experts. We empirically validate our parameterization and further investigate how scaling the number of experts and granularity affects the optimal learning rate.

Machine learning models are often tuned by nesting optimization of model weights inside the optimization of hyperparameters. We give a method to collapse this nested optimization into joint stochastic optimization of weights and hyperparameters. Our process trains a neural network to output approximately optimal weights as a function of hyperparameters. We show that our technique converges to locally optimal weights and hyperparameters for sufficiently large hypernetworks. We compare this method to standard hyperparameter optimization strategies and demonstrate its effectiveness for tuning thousands of hyperparameters.

The growing interest in machine learning problems over graphs with additional node information such as texts, images, or labels has popularized methods that require the costly operation of processing the entire graph. Yet, little effort has been made to the development of fast local methods (i.e. without accessing the entire graph) that extract useful information from such data. To that end, we propose a study of local graph clustering using noisy node labels as a proxy for additional node information. In this setting, nodes receive initial binary labels based on cluster affiliation: 1 if they belong to the target cluster and 0 otherwise. Subsequently, a fraction of these labels is flipped. We investigate the benefits of incorporating noisy labels for local graph clustering. By constructing a weighted graph with such labels, we study the performance of graph diffusion-based local clustering method on both the original and the weighted graphs. From a theoretical perspective, we consider recovering an unknown target cluster with a single seed node in a random graph with independent noisy node labels. We provide sufficient conditions on the label noise under which, with high probability, using diffusion in the weighted graph yields a more accurate recovery of the target cluster. This approach proves more effective than using the given labels alone or using diffusion in the label-free original graph. Empirically, we show that reliable node labels can be obtained with just a few samples from an attributed graph. Moreover, utilizing these labels via diffusion in the weighted graph leads to significantly better local clustering performance across several real-world datasets, improving F1 scores by up to 13%.

The Maximal Update Parametrization (muP) aims to make the optimal hyperparameters (HPs) of a model independent of its size, allowing them to be swept using a cheap proxy model rather than the full-size target model. We present a new scheme, u-muP, which improves upon muP by combining it with Unit Scaling, a method for designing models that makes them easy to train in low-precision. The two techniques have a natural affinity: muP ensures that the scale of activations is independent of model size, and Unit Scaling ensures that activations, weights and gradients begin training with a scale of one. This synthesis opens the door to a simpler scheme, whose default values are near-optimal. This in turn facilitates a more efficient sweeping strategy, with u-muP models reaching a lower loss than comparable muP models and working out-of-the-box in FP8.

The notion of shortcut partition, introduced recently by Chang, Conroy, Le, Milenkovi\'c, Solomon, and Than [CCLMST23], is a new type of graph partition into low-diameter clusters. Roughly speaking, the shortcut partition guarantees that for every two vertices u and v in the graph, there exists a path between u and v that intersects only a few clusters. They proved that any planar graph admits a shortcut partition and gave several applications, including a construction of tree cover for arbitrary planar graphs with stretch 1+varepsilon and O(1) many trees for any fixed varepsilon in (0,1). However, the construction heavily exploits planarity in multiple steps, and is thus inherently limited to planar graphs. In this work, we breach the "planarity barrier" to construct a shortcut partition for K_r-minor-free graphs for any r. To this end, we take a completely different approach -- our key contribution is a novel deterministic variant of the cop decomposition in minor-free graphs [And86, AGG14]. Our shortcut partition for K_r-minor-free graphs yields several direct applications. Most notably, we construct the first optimal distance oracle for K_r-minor-free graphs, with 1+varepsilon stretch, linear space, and constant query time for any fixed varepsilon in (0,1). The previous best distance oracle [AG06] uses O(nlog n) space and O(log n) query time, and its construction relies on Robertson-Seymour structural theorem and other sophisticated tools. We also obtain the first tree cover of O(1) size for minor-free graphs with stretch 1+varepsilon, while the previous best (1+varepsilon)-tree cover has size O(log^2 n) [BFN19].

One method for obtaining generalizable solutions to machine learning tasks when presented with diverse training environments is to find invariant representations of the data. These are representations of the covariates such that the best model on top of the representation is invariant across training environments. In the context of linear Structural Equation Models (SEMs), invariant representations might allow us to learn models with out-of-distribution guarantees, i.e., models that are robust to interventions in the SEM. To address the invariant representation problem in a {\em finite sample} setting, we consider the notion of epsilon-approximate invariance. We study the following question: If a representation is approximately invariant with respect to a given number of training interventions, will it continue to be approximately invariant on a larger collection of unseen SEMs? This larger collection of SEMs is generated through a parameterized family of interventions. Inspired by PAC learning, we obtain finite-sample out-of-distribution generalization guarantees for approximate invariance that holds probabilistically over a family of linear SEMs without faithfulness assumptions. Our results show bounds that do not scale in ambient dimension when intervention sites are restricted to lie in a constant size subset of in-degree bounded nodes. We also show how to extend our results to a linear indirect observation model that incorporates latent variables.

We introduce a novel approach for analyzing the training dynamics of ReLU networks by examining the characteristic activation boundaries of individual ReLU neurons. Our proposed analysis reveals a critical instability in common neural network parameterizations and normalizations during stochastic optimization, which impedes fast convergence and hurts generalization performance. Addressing this, we propose Geometric Parameterization (GmP), a novel neural network parameterization technique that effectively separates the radial and angular components of weights in the hyperspherical coordinate system. We show theoretically that GmP resolves the aforementioned instability issue. We report empirical results on various models and benchmarks to verify GmP's theoretical advantages of optimization stability, convergence speed and generalization performance.

Many methods in differentially private model training rely on computing the similarity between a query point (such as public or synthetic data) and private data. We abstract out this common subroutine and study the following fundamental algorithmic problem: Given a similarity function f and a large high-dimensional private dataset X subset R^d, output a differentially private (DP) data structure which approximates sum_{x in X} f(x,y) for any query y. We consider the cases where f is a kernel function, such as f(x,y) = e^{-|x-y|_2^2/sigma^2} (also known as DP kernel density estimation), or a distance function such as f(x,y) = |x-y|_2, among others. Our theoretical results improve upon prior work and give better privacy-utility trade-offs as well as faster query times for a wide range of kernels and distance functions. The unifying approach behind our results is leveraging `low-dimensional structures' present in the specific functions f that we study, using tools such as provable dimensionality reduction, approximation theory, and one-dimensional decomposition of the functions. Our algorithms empirically exhibit improved query times and accuracy over prior state of the art. We also present an application to DP classification. Our experiments demonstrate that the simple methodology of classifying based on average similarity is orders of magnitude faster than prior DP-SGD based approaches for comparable accuracy.

This paper investigates efficient deep neural networks (DNNs) to replace dense unstructured weight matrices with structured ones that possess desired properties. The challenge arises because the optimal weight matrix structure in popular neural network models is obscure in most cases and may vary from layer to layer even in the same network. Prior structured matrices proposed for efficient DNNs were mostly hand-crafted without a generalized framework to systematically learn them. To address this issue, we propose a generalized and differentiable framework to learn efficient structures of weight matrices by gradient descent. We first define a new class of structured matrices that covers a wide range of structured matrices in the literature by adjusting the structural parameters. Then, the frequency-domain differentiable parameterization scheme based on the Gaussian-Dirichlet kernel is adopted to learn the structural parameters by proximal gradient descent. On the image and language tasks, our method learns efficient DNNs with structured matrices, achieving lower complexity and/or higher performance than prior approaches that employ low-rank, block-sparse, or block-low-rank matrices.

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

Estimating the structure of directed acyclic graphs (DAGs, also known as Bayesian networks) is a challenging problem since the search space of DAGs is combinatorial and scales superexponentially with the number of nodes. Existing approaches rely on various local heuristics for enforcing the acyclicity constraint. In this paper, we introduce a fundamentally different strategy: We formulate the structure learning problem as a purely continuous optimization problem over real matrices that avoids this combinatorial constraint entirely. This is achieved by a novel characterization of acyclicity that is not only smooth but also exact. The resulting problem can be efficiently solved by standard numerical algorithms, which also makes implementation effortless. The proposed method outperforms existing ones, without imposing any structural assumptions on the graph such as bounded treewidth or in-degree. Code implementing the proposed algorithm is open-source and publicly available at https://github.com/xunzheng/notears.

We propose RepMLP, a multi-layer-perceptron-style neural network building block for image recognition, which is composed of a series of fully-connected (FC) layers. Compared to convolutional layers, FC layers are more efficient, better at modeling the long-range dependencies and positional patterns, but worse at capturing the local structures, hence usually less favored for image recognition. We propose a structural re-parameterization technique that adds local prior into an FC to make it powerful for image recognition. Specifically, we construct convolutional layers inside a RepMLP during training and merge them into the FC for inference. On CIFAR, a simple pure-MLP model shows performance very close to CNN. By inserting RepMLP in traditional CNN, we improve ResNets by 1.8% accuracy on ImageNet, 2.9% for face recognition, and 2.3% mIoU on Cityscapes with lower FLOPs. Our intriguing findings highlight that combining the global representational capacity and positional perception of FC with the local prior of convolution can improve the performance of neural network with faster speed on both the tasks with translation invariance (e.g., semantic segmentation) and those with aligned images and positional patterns (e.g., face recognition). The code and models are available at https://github.com/DingXiaoH/RepMLP.

We study Helly graphs of finite combinatorial dimension, i.e. whose injective hull is finite-dimensional. We describe very simple fine simplicial subdivisions of the injective hull of a Helly graph, following work of Lang. We also give a very explicit simplicial model of the injective hull of a Helly graphs, in terms of cliques which are intersections of balls. We use these subdivisions to prove that any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic. Moreover, every such hyperbolic automorphism has an axis in an appropriate Helly subdivision, and its translation length is rational with uniformly bounded denominator.

We present a new approach for the approximate K-nearest neighbor search based on navigable small world graphs with controllable hierarchy (Hierarchical NSW, HNSW). The proposed solution is fully graph-based, without any need for additional search structures, which are typically used at the coarse search stage of the most proximity graph techniques. Hierarchical NSW incrementally builds a multi-layer structure consisting from hierarchical set of proximity graphs (layers) for nested subsets of the stored elements. The maximum layer in which an element is present is selected randomly with an exponentially decaying probability distribution. This allows producing graphs similar to the previously studied Navigable Small World (NSW) structures while additionally having the links separated by their characteristic distance scales. Starting search from the upper layer together with utilizing the scale separation boosts the performance compared to NSW and allows a logarithmic complexity scaling. Additional employment of a heuristic for selecting proximity graph neighbors significantly increases performance at high recall and in case of highly clustered data. Performance evaluation has demonstrated that the proposed general metric space search index is able to strongly outperform previous opensource state-of-the-art vector-only approaches. Similarity of the algorithm to the skip list structure allows straightforward balanced distributed implementation.

Modeling and designing urban building layouts is of significant interest in computer vision, computer graphics, and urban applications. A building layout consists of a set of buildings in city blocks defined by a network of roads. We observe that building layouts are discrete structures, consisting of multiple rows of buildings of various shapes, and are amenable to skeletonization for mapping arbitrary city block shapes to a canonical form. Hence, we propose a fully automatic approach to building layout generation using graph attention networks. Our method generates realistic urban layouts given arbitrary road networks, and enables conditional generation based on learned priors. Our results, including user study, demonstrate superior performance as compared to prior layout generation networks, support arbitrary city block and varying building shapes as demonstrated by generating layouts for 28 large cities.

Transfer learning has recently become the dominant paradigm of machine learning. Pre-trained models fine-tuned for downstream tasks achieve better performance with fewer labelled examples. Nonetheless, it remains unclear how to develop models that specialise towards multiple tasks without incurring negative interference and that generalise systematically to non-identically distributed tasks. Modular deep learning has emerged as a promising solution to these challenges. In this framework, units of computation are often implemented as autonomous parameter-efficient modules. Information is conditionally routed to a subset of modules and subsequently aggregated. These properties enable positive transfer and systematic generalisation by separating computation from routing and updating modules locally. We offer a survey of modular architectures, providing a unified view over several threads of research that evolved independently in the scientific literature. Moreover, we explore various additional purposes of modularity, including scaling language models, causal inference, programme induction, and planning in reinforcement learning. Finally, we report various concrete applications where modularity has been successfully deployed such as cross-lingual and cross-modal knowledge transfer. Related talks and projects to this survey, are available at https://www.modulardeeplearning.com/.

We study the problem of visualizing large-scale and high-dimensional data in a low-dimensional (typically 2D or 3D) space. Much success has been reported recently by techniques that first compute a similarity structure of the data points and then project them into a low-dimensional space with the structure preserved. These two steps suffer from considerable computational costs, preventing the state-of-the-art methods such as the t-SNE from scaling to large-scale and high-dimensional data (e.g., millions of data points and hundreds of dimensions). We propose the LargeVis, a technique that first constructs an accurately approximated K-nearest neighbor graph from the data and then layouts the graph in the low-dimensional space. Comparing to t-SNE, LargeVis significantly reduces the computational cost of the graph construction step and employs a principled probabilistic model for the visualization step, the objective of which can be effectively optimized through asynchronous stochastic gradient descent with a linear time complexity. The whole procedure thus easily scales to millions of high-dimensional data points. Experimental results on real-world data sets demonstrate that the LargeVis outperforms the state-of-the-art methods in both efficiency and effectiveness. The hyper-parameters of LargeVis are also much more stable over different data sets.

This thesis introduces quantum natural language processing (QNLP) models based on a simple yet powerful analogy between computational linguistics and quantum mechanics: grammar as entanglement. The grammatical structure of text and sentences connects the meaning of words in the same way that entanglement structure connects the states of quantum systems. Category theory allows to make this language-to-qubit analogy formal: it is a monoidal functor from grammar to vector spaces. We turn this abstract analogy into a concrete algorithm that translates the grammatical structure onto the architecture of parameterised quantum circuits. We then use a hybrid classical-quantum algorithm to train the model so that evaluating the circuits computes the meaning of sentences in data-driven tasks. The implementation of QNLP models motivated the development of DisCoPy (Distributional Compositional Python), the toolkit for applied category theory of which the first chapter gives a comprehensive overview. String diagrams are the core data structure of DisCoPy, they allow to reason about computation at a high level of abstraction. We show how they can encode both grammatical structures and quantum circuits, but also logical formulae, neural networks or arbitrary Python code. Monoidal functors allow to translate these abstract diagrams into concrete computation, interfacing with optimised task-specific libraries. The second chapter uses DisCopy to implement QNLP models as parameterised functors from grammar to quantum circuits. It gives a first proof-of-concept for the more general concept of functorial learning: generalising machine learning from functions to functors by learning from diagram-like data. In order to learn optimal functor parameters via gradient descent, we introduce the notion of diagrammatic differentiation: a graphical calculus for computing the gradients of parameterised diagrams.

From the Bayesian perspective, the category of conditional probabilities (a variant of the Kleisli category of the Giry monad, whose objects are measurable spaces and arrows are Markov kernels) gives a nice framework for conceptualization and analysis of many aspects of machine learning. Using categorical methods, we construct models for parametric and nonparametric Bayesian reasoning on function spaces, thus providing a basis for the supervised learning problem. In particular, stochastic processes are arrows to these function spaces which serve as prior probabilities. The resulting inference maps can often be analytically constructed in this symmetric monoidal weakly closed category. We also show how to view general stochastic processes using functor categories and demonstrate the Kalman filter as an archetype for the hidden Markov model.

Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as geometric graphs, i.e., graphs with nodes positioned in the Euclidean space given an arbitrarily chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as Galilean invariance. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

We present our position on the elusive quest for a general-purpose framework for specifying and studying deep learning architectures. Our opinion is that the key attempts made so far lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations. Focusing on building a such a bridge, we propose to apply category theory -- precisely, the universal algebra of monads valued in a 2-category of parametric maps -- as a single theory elegantly subsuming both of these flavours of neural network design. To defend our position, we show how this theory recovers constraints induced by geometric deep learning, as well as implementations of many architectures drawn from the diverse landscape of neural networks, such as RNNs. We also illustrate how the theory naturally encodes many standard constructs in computer science and automata theory.

Neural graphics primitives are faster and achieve higher quality when their neural networks are augmented by spatial data structures that hold trainable features arranged in a grid. However, existing feature grids either come with a large memory footprint (dense or factorized grids, trees, and hash tables) or slow performance (index learning and vector quantization). In this paper, we show that a hash table with learned probes has neither disadvantage, resulting in a favorable combination of size and speed. Inference is faster than unprobed hash tables at equal quality while training is only 1.2-2.6x slower, significantly outperforming prior index learning approaches. We arrive at this formulation by casting all feature grids into a common framework: they each correspond to a lookup function that indexes into a table of feature vectors. In this framework, the lookup functions of existing data structures can be combined by simple arithmetic combinations of their indices, resulting in Pareto optimal compression and speed.

We introduce in this paper the mechanism of graph random features (GRFs). GRFs can be used to construct unbiased randomized estimators of several important kernels defined on graphs' nodes, in particular the regularized Laplacian kernel. As regular RFs for non-graph kernels, they provide means to scale up kernel methods defined on graphs to larger networks. Importantly, they give substantial computational gains also for smaller graphs, while applied in downstream applications. Consequently, GRFs address the notoriously difficult problem of cubic (in the number of the nodes of the graph) time complexity of graph kernels algorithms. We provide a detailed theoretical analysis of GRFs and an extensive empirical evaluation: from speed tests, through Frobenius relative error analysis to kmeans graph-clustering with graph kernels. We show that the computation of GRFs admits an embarrassingly simple distributed algorithm that can be applied if the graph under consideration needs to be split across several machines. We also introduce a (still unbiased) quasi Monte Carlo variant of GRFs, q-GRFs, relying on the so-called reinforced random walks, that might be used to optimize the variance of GRFs. As a byproduct, we obtain a novel approach to solve certain classes of linear equations with positive and symmetric matrices.

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Frechet mean. In this work, we equip a set of graphs with the pseudometric defined by the norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Frechet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric.

Numerical simulations in climate, chemistry, or astrophysics are computationally too expensive for uncertainty quantification or parameter-exploration at high-resolution. Reduced-order or surrogate models are multiple orders of magnitude faster, but traditional surrogates are inflexible or inaccurate and pure machine learning (ML)-based surrogates too data-hungry. We propose a hybrid, flexible surrogate model that exploits known physics for simulating large-scale dynamics and limits learning to the hard-to-model term, which is called parametrization or closure and captures the effect of fine- onto large-scale dynamics. Leveraging neural operators, we are the first to learn grid-independent, non-local, and flexible parametrizations. Our multiscale neural operator is motivated by a rich literature in multiscale modeling, has quasilinear runtime complexity, is more accurate or flexible than state-of-the-art parametrizations and demonstrated on the chaotic equation multiscale Lorenz96.

Machine learning (ML) and deep learning (DL) techniques have gained significant attention as reduced order models (ROMs) to computationally expensive structural analysis methods, such as finite element analysis (FEA). Graph neural network (GNN) is a particular type of neural network which processes data that can be represented as graphs. This allows for efficient representation of complex geometries that can change during conceptual design of a structure or a product. In this study, we propose a novel graph embedding technique for efficient representation of 3D stiffened panels by considering separate plate domains as vertices. This approach is considered using Graph Sampling and Aggregation (GraphSAGE) to predict stress distributions in stiffened panels with varying geometries. A comparison between a finite-element-vertex graph representation is conducted to demonstrate the effectiveness of the proposed approach. A comprehensive parametric study is performed to examine the effect of structural geometry on the prediction performance. Our results demonstrate the immense potential of graph neural networks with the proposed graph embedding method as robust reduced-order models for 3D structures.

Compared to the wide array of advanced Monte Carlo methods supported by modern probabilistic programming languages (PPLs), PPL support for variational inference (VI) is less developed: users are typically limited to a predefined selection of variational objectives and gradient estimators, which are implemented monolithically (and without formal correctness arguments) in PPL backends. In this paper, we propose a more modular approach to supporting variational inference in PPLs, based on compositional program transformation. In our approach, variational objectives are expressed as programs, that may employ first-class constructs for computing densities of and expected values under user-defined models and variational families. We then transform these programs systematically into unbiased gradient estimators for optimizing the objectives they define. Our design enables modular reasoning about many interacting concerns, including automatic differentiation, density accumulation, tracing, and the application of unbiased gradient estimation strategies. Additionally, relative to existing support for VI in PPLs, our design increases expressiveness along three axes: (1) it supports an open-ended set of user-defined variational objectives, rather than a fixed menu of options; (2) it supports a combinatorial space of gradient estimation strategies, many not automated by today's PPLs; and (3) it supports a broader class of models and variational families, because it supports constructs for approximate marginalization and normalization (previously introduced only for Monte Carlo inference). We implement our approach in an extension to the Gen probabilistic programming system (genjax.vi, implemented in JAX), and evaluate on several deep generative modeling tasks, showing minimal performance overhead vs. hand-coded implementations and performance competitive with well-established open-source PPLs.

Recently similarity graphs became the leading paradigm for efficient nearest neighbor search, outperforming traditional tree-based and LSH-based methods. Similarity graphs perform the search via greedy routing: a query traverses the graph and in each vertex moves to the adjacent vertex that is the closest to this query. In practice, similarity graphs are often susceptible to local minima, when queries do not reach its nearest neighbors, getting stuck in suboptimal vertices. In this paper we propose to learn the routing function that overcomes local minima via incorporating information about the graph global structure. In particular, we augment the vertices of a given graph with additional representations that are learned to provide the optimal routing from the start vertex to the query nearest neighbor. By thorough experiments, we demonstrate that the proposed learnable routing successfully diminishes the local minima problem and significantly improves the overall search performance.

Despite recent advancements, most computational methods for molecule optimization are constrained to single- or double-property optimization tasks and suffer from poor scalability and generalizability to novel optimization tasks. Meanwhile, Large Language Models (LLMs) demonstrate remarkable out-of-domain generalizability to novel tasks. To demonstrate LLMs' potential for molecule optimization, we introduce MoMUInstruct, the first high-quality instruction-tuning dataset specifically focused on complex multi-property molecule optimization tasks. Leveraging MoMUInstruct, we develop GeLLM^3Os, a series of instruction-tuned LLMs for molecule optimization. Extensive evaluations across 5 in-domain and 5 out-of-domain tasks demonstrate that GeLLM^3Os consistently outperform state-of-the-art baselines. GeLLM^3Os also exhibit outstanding zero-shot generalization to unseen tasks, significantly outperforming powerful closed-source LLMs. Such strong generalizability demonstrates the tremendous potential of GeLLM^3Os as foundational models for molecule optimization, thereby tackling novel optimization tasks without resource-intensive retraining. MoMUInstruct, models, and code are accessible through https://github.com/ninglab/GeLLMO.

We define and investigate the problem of c-approximate window search: approximate nearest neighbor search where each point in the dataset has a numeric label, and the goal is to find nearest neighbors to queries within arbitrary label ranges. Many semantic search problems, such as image and document search with timestamp filters, or product search with cost filters, are natural examples of this problem. We propose and theoretically analyze a modular tree-based framework for transforming an index that solves the traditional c-approximate nearest neighbor problem into a data structure that solves window search. On standard nearest neighbor benchmark datasets equipped with random label values, adversarially constructed embeddings, and image search embeddings with real timestamps, we obtain up to a 75times speedup over existing solutions at the same level of recall.

We give a geometry of interaction model for a typed lambda-calculus endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The model is based on the category of measurable spaces and partial measurable functions, and is proved adequate with respect to both a distribution-based and a sampling based operational semantics.

Representations learned via self-supervised learning (SSL) can be susceptible to dimensional collapse, where the learned representation subspace is of extremely low dimensionality and thus fails to represent the full data distribution and modalities. Dimensional collapse also known as the "underfilling" phenomenon is one of the major causes of degraded performance on downstream tasks. Previous work has investigated the dimensional collapse problem of SSL at a global level. In this paper, we demonstrate that representations can span over high dimensional space globally, but collapse locally. To address this, we propose a method called local dimensionality regularization (LDReg). Our formulation is based on the derivation of the Fisher-Rao metric to compare and optimize local distance distributions at an asymptotically small radius for each data point. By increasing the local intrinsic dimensionality, we demonstrate through a range of experiments that LDReg improves the representation quality of SSL. The results also show that LDReg can regularize dimensionality at both local and global levels.

Second-order optimization has been developed to accelerate the training of deep neural networks and it is being applied to increasingly larger-scale models. In this study, towards training on further larger scales, we identify a specific parameterization for second-order optimization that promotes feature learning in a stable manner even if the network width increases significantly. Inspired by a maximal update parameterization, we consider a one-step update of the gradient and reveal the appropriate scales of hyperparameters including random initialization, learning rates, and damping terms. Our approach covers two major second-order optimization algorithms, K-FAC and Shampoo, and we demonstrate that our parameterization achieves higher generalization performance in feature learning. In particular, it enables us to transfer the hyperparameters across models with different widths.

We prove rich algebraic structures of the solution space for 2-layer neural networks with quadratic activation and L_2 loss, trained on reasoning tasks in Abelian group (e.g., modular addition). Such a rich structure enables analytical construction of global optimal solutions from partial solutions that only satisfy part of the loss, despite its high nonlinearity. We coin the framework as CoGO (Composing Global Optimizers). Specifically, we show that the weight space over different numbers of hidden nodes of the 2-layer network is equipped with a semi-ring algebraic structure, and the loss function to be optimized consists of monomial potentials, which are ring homomorphism, allowing partial solutions to be composed into global ones by ring addition and multiplication. Our experiments show that around 95% of the solutions obtained by gradient descent match exactly our theoretical constructions. Although the global optimizers constructed only required a small number of hidden nodes, our analysis on gradient dynamics shows that over-parameterization asymptotically decouples training dynamics and is beneficial. We further show that training dynamics favors simpler solutions under weight decay, and thus high-order global optimizers such as perfect memorization are unfavorable.

The inference of topological principles is a key problem in structured reconstruction. We observe that wrongly predicted topological relationships are often incurred by the lack of holistic geometry clues in low-level features. Inspired by the fact that massive signals can be compactly described with frequency analysis, we experimentally explore the efficiency and tendency of learning structure geometry in the frequency domain. Accordingly, we propose a frequency-domain feature learning strategy (F-Learn) to fuse scattered geometric fragments holistically for topology-intact structure reasoning. Benefiting from the parsimonious design, the F-Learn strategy can be easily deployed into a deep reconstructor with a lightweight model modification. Experiments demonstrate that the F-Learn strategy can effectively introduce structure awareness into geometric primitive detection and topology inference, bringing significant performance improvement to final structured reconstruction. Code and pre-trained models are available at https://github.com/Geo-Tell/F-Learn.

Complex networks can be used for modeling street meshes and urban agglomerates. With such a model, many aspects of a city can be investigated to promote a better quality of life to its citizens. Along these lines, this paper proposes a set of distance-based pattern-discovery algorithmic instruments to improve urban structures modeled as complex networks, detecting nodes that lack access from/to points of interest in a given city. Furthermore, we introduce a greedy algorithm that is able to recommend improvements to the structure of a city by suggesting where points of interest are to be placed. We contribute to a thorough process to deal with complex networks, including mathematical modeling and algorithmic innovation. The set of our contributions introduces a systematic manner to treat a recurrent problem of broad interest in cities.

Engineering efficient implementations of compact and succinct structures is a time-consuming and challenging task, since there is no standard library of easy-to- use, highly optimized, and composable components. One consequence is that measuring the practical impact of new theoretical proposals is a difficult task, since older base- line implementations may not rely on the same basic components, and reimplementing from scratch can be very time-consuming. In this paper we present a framework for experimentation with succinct data structures, providing a large set of configurable components, together with tests, benchmarks, and tools to analyze resource requirements. We demonstrate the functionality of the framework by recomposing succinct solutions for document retrieval.

Building credible simulators from data is difficult because structure design, parameter calibration, and out-of-distribution (OOD) robustness are tightly coupled. We introduce SOCIA (Simulation Orchestration for Computational Intelligence with Agents), a framework that treats simulator construction as joint structure-parameter co-optimization: it elicits mechanism-rich blueprints, exposes explicit tunable parameters, and instantiates a calibration schema, producing an executable simulator with built-in calibration hooks. SOCIA couples Bayesian Optimization for sample-efficient point calibration with Simulation-Based Inference for uncertainty-aware fitting; diagnostics trigger targeted structural edits in an outer refinement loop to co-optimize design and parameters under tight budgets. Across three diverse tasks, SOCIA consistently outperforms strong baselines, excelling on both in-distribution (ID) fitting and OOD shift. Ablations that weaken structure, calibration design, or tuning yield near-monotone degradations, underscoring the necessity of unified structure-parameter optimization. We will release the code soon.

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

When considering a model architecture, there are several ways to reduce its memory footprint. Historically, popular approaches included selecting smaller architectures and creating sparse networks through pruning. More recently, randomized parameter-sharing (RPS) methods have gained traction for model compression at start of training. In this paper, we comprehensively assess the trade-off between memory and accuracy across RPS, pruning techniques, and building smaller models. Our findings demonstrate that RPS, which is both data and model-agnostic, consistently outperforms/matches smaller models and all moderately informed pruning strategies, such as MAG, SNIP, SYNFLOW, and GRASP, across the entire compression range. This advantage becomes particularly pronounced in higher compression scenarios. Notably, even when compared to highly informed pruning techniques like Lottery Ticket Rewinding (LTR), RPS exhibits superior performance in high compression settings. This points out inherent capacity advantage that RPS enjoys over sparse models. Theoretically, we establish RPS as a superior technique in terms of memory-efficient representation when compared to pruning for linear models. This paper argues in favor of paradigm shift towards RPS based models. During our rigorous evaluation of RPS, we identified issues in the state-of-the-art RPS technique ROAST, specifically regarding stability (ROAST's sensitivity to initialization hyperparameters, often leading to divergence) and Pareto-continuity (ROAST's inability to recover the accuracy of the original model at zero compression). We provably address both of these issues. We refer to the modified RPS, which incorporates our improvements, as STABLE-RPS.

Hyperparameter optimization (HPO) is concerned with the automated search for the most appropriate hyperparameter configuration (HPC) of a parameterized machine learning algorithm. A state-of-the-art HPO method is Hyperband, which, however, has its own parameters that influence its performance. One of these parameters, the maximal budget, is especially problematic: If chosen too small, the budget needs to be increased in hindsight and, as Hyperband is not incremental by design, the entire algorithm must be re-run. This is not only costly but also comes with a loss of valuable knowledge already accumulated. In this paper, we propose incremental variants of Hyperband that eliminate these drawbacks, and show that these variants satisfy theoretical guarantees qualitatively similar to those for the original Hyperband with the "right" budget. Moreover, we demonstrate their practical utility in experiments with benchmark data sets.

Large language models deliver strong reasoning and tool-use skills, yet their computational demands make them impractical for edge or cost-sensitive deployments. We present Xmodel-2.5, a 1.3-billion-parameter small language model designed as a drop-in agent core. Training with maximal-update parameterization (μP) allows hyper-parameters tuned on a 20M-parameter proxy to transfer directly to the full model, even under the parameter-tied tie-word-embedding architecture. A 1.4T-token Warmup--Stable--Decay curriculum is used, and we further show that switching from AdamW to Muon during the decay phase improves the 13-task reasoning average by 4.58\,\% while keeping every other hyper-parameter fixed, verifying that early AdamW stability can be paired with late Muon sharpening for better downstream performance. FP8-mixed-precision training balances accuracy and throughput. All checkpoints, recipes, and evaluation code are released under the Apache-2.0 license.https://huggingface.co/XiaoduoAILab/Xmodel-2.5 and https://huggingface.co/XiaoduoAILab/Xmodel-2.5-history (training checkpoints). Training code and evaluation harness: https://github.com/XiaoduoAILab/Xmodel-2.5.

Asymmetrical distance structures (quasimetrics) are ubiquitous in our lives and are gaining more attention in machine learning applications. Imposing such quasimetric structures in model representations has been shown to improve many tasks, including reinforcement learning (RL) and causal relation learning. In this work, we present four desirable properties in such quasimetric models, and show how prior works fail at them. We propose Interval Quasimetric Embedding (IQE), which is designed to satisfy all four criteria. On three quasimetric learning experiments, IQEs show strong approximation and generalization abilities, leading to better performance and improved efficiency over prior methods. Project Page: https://www.tongzhouwang.info/interval_quasimetric_embedding Quasimetric Learning Code Package: https://www.github.com/quasimetric-learning/torch-quasimetric

It is often useful to compactly summarize important properties of model parameters and training data so that they can be used later without storing and/or iterating over the entire dataset. As a specific case, we consider estimating the Function Space Distance (FSD) over a training set, i.e. the average discrepancy between the outputs of two neural networks. We propose a Linearized Activation Function TRick (LAFTR) and derive an efficient approximation to FSD for ReLU neural networks. The key idea is to approximate the architecture as a linear network with stochastic gating. Despite requiring only one parameter per unit of the network, our approach outcompetes other parametric approximations with larger memory requirements. Applied to continual learning, our parametric approximation is competitive with state-of-the-art nonparametric approximations, which require storing many training examples. Furthermore, we show its efficacy in estimating influence functions accurately and detecting mislabeled examples without expensive iterations over the entire dataset.

Hypergraphs are a powerful abstraction for representing higher-order interactions between entities of interest. To exploit these relationships in making downstream predictions, a variety of hypergraph neural network architectures have recently been proposed, in large part building upon precursors from the more traditional graph neural network (GNN) literature. Somewhat differently, in this paper we begin by presenting an expressive family of parameterized, hypergraph-regularized energy functions. We then demonstrate how minimizers of these energies effectively serve as node embeddings that, when paired with a parameterized classifier, can be trained end-to-end via a supervised bilevel optimization process. Later, we draw parallels between the implicit architecture of the predictive models emerging from the proposed bilevel hypergraph optimization, and existing GNN architectures in common use. Empirically, we demonstrate state-of-the-art results on various hypergraph node classification benchmarks. Code is available at https://github.com/yxzwang/PhenomNN.

We propose an algorithm for inexpensive gradient-based hyperparameter optimization that combines the implicit function theorem (IFT) with efficient inverse Hessian approximations. We present results about the relationship between the IFT and differentiating through optimization, motivating our algorithm. We use the proposed approach to train modern network architectures with millions of weights and millions of hyper-parameters. For example, we learn a data-augmentation network - where every weight is a hyperparameter tuned for validation performance - outputting augmented training examples. Jointly tuning weights and hyperparameters with our approach is only a few times more costly in memory and compute than standard training.