When trying tofitadeepneural network(DNN) toaG-invariant targetfunction with G a group, it only makes sense to constrain the DNN to beG-invariant as well.

The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in machine learning. In this work, we establish fundamental tools for investigating universality properties of kernels in Riemannian symmetric spaces, thereby extending the study of this important topic to kernels in non-Euclidean domains. Moreover, we use the developed tools to prove the universality of several recent examples from the literature on positive definite kernels defined on Riemannian symmetric spaces, thus providing theoretical justification for their use in applications involving manifold-valued data.

Neural Processes (NPs) are a rapidly evolving class of models designed to directly model the posterior predictive distribution of stochastic processes. While early architectures were developed primarily as a scalable alternative to Gaussian Processes (GPs), modern NPs tackle far more complex and data hungry applications spanning geology, epidemiology, climate, and robotics. These applications have placed increasing pressure on the scalability of these models, with many architectures compromising accuracy for scalability. In this paper, we demonstrate that this tradeoff is often unnecessary, particularly when modeling fully or partially translation invariant processes. We propose a versatile new architecture, the Biased Scan Attention Transformer Neural Process (BSA-TNP), which introduces Kernel Regression Blocks (KRBlocks), group-invariant attention biases, and memory-efficient Biased Scan Attention (BSA). BSA-TNP is able to: (1) match or exceed the accuracy of the best models while often training in a fraction of the time, (2) exhibit translation invariance, enabling learning at multiple resolutions simultaneously, (3) transparently model processes that evolve in both space and time, (4) support high dimensional fixed effects, and (5) scale gracefully -- running inference with over 1M test points with 100K context points in under a minute on a single 24GB GPU.

With multi-agent systems increasingly deployed autonomously at scale in complex environments, ensuring safety of the data-driven policies is critical. Control Barrier Functions have emerged as an effective tool for enforcing safety constraints, yet existing learning-based methods often lack in scalability, generalization and sampling efficiency as they overlook inherent geometric structures of the system. To address this gap, we introduce symmetries-infused distributed Control Barrier Functions, enforcing the satisfaction of intrinsic symmetries on learnable graph-based safety certificates. We theoretically motivate the need for equivariant parametrization of CBFs and policies, and propose a simple, yet efficient and adaptable methodology for constructing such equivariant group-modular networks via the compatible group actions. This approach encodes safety constraints in a distributed data-efficient manner, enabling zero-shot generalization to larger and denser swarms. Through extensive simulations on multi-robot navigation tasks, we demonstrate that our method outperforms state-of-the-art baselines in terms of safety, scalability, and task success rates, highlighting the importance of embedding symmetries in safe distributed neural policies.

Reconstructing training data from trained neural networks is an active area of research with significant implications for privacy and explainability. Recent advances have demonstrated the feasibility of this process for several data types. However, reconstructing data from group-invariant neural networks poses distinct challenges that remain largely unexplored. This paper addresses this gap by first formulating the problem and discussing some of its basic properties. We then provide an experimental evaluation demonstrating that conventional reconstruction techniques are inadequate in this scenario. Specifically, we observe that the resulting data reconstructions gravitate toward symmetric inputs on which the group acts trivially, leading to poor-quality results. Finally, we propose two novel methods aiming to improve reconstruction in this setup and present promising preliminary experimental results. Our work sheds light on the complexities of reconstructing data from group invariant neural networks and offers potential avenues for future research in this domain.

Recently, it was proved that group equivariance emerges in ensembles of neural networks as the result of full augmentation in the limit of infinitely wide neural networks (neural tangent kernel limit). In this paper, we extend this result significantly. We provide a proof that this emergence does not depend on the neural tangent kernel limit at all. We also consider stochastic settings, and furthermore general architectures. For the latter, we provide a simple sufficient condition on the relation between the architecture and the action of the group for our results to hold. We validate our findings through simple numeric experiments.

Symmetry is ubiquitous in many real-world phenomena and tasks, such as physics, images, and molecular simulations. Empirical studies have demonstrated that incorporating symmetries into generative models can provide better generalization and sampling efficiency when the underlying data distribution has group symmetry. In this work, we provide the first theoretical analysis and guarantees of score-based generative models (SGMs) for learning distributions that are invariant with respect to some group symmetry and offer the first quantitative comparison between data augmentation and adding equivariant inductive bias. First, building on recent works on the Wasserstein-1 ($\mathbf{d}_1$) guarantees of SGMs and empirical estimations of probability divergences under group symmetry, we provide an improved $\mathbf{d}_1$ generalization bound when the data distribution is group-invariant. Second, we describe the inductive bias of equivariant SGMs using Hamilton-Jacobi-Bellman theory, and rigorously demonstrate that one can learn the score of a symmetrized distribution using equivariant vector fields without data augmentations through the analysis of the optimality and equivalence of score-matching objectives. This also provides practical guidance that one does not have to augment the dataset as long as the vector field or the neural network parametrization is equivariant. Moreover, we quantify the impact of not incorporating equivariant structure into the score parametrization, by showing that non-equivariant vector fields can yield worse generalization bounds. This can be viewed as a type of model-form error that describes the missing structure of non-equivariant vector fields. Numerical simulations corroborate our analysis and highlight that data augmentations cannot replace the role of equivariant vector fields.

Weight space symmetries in neural network architectures, such as permutation symmetries in MLPs, give rise to Bayesian neural network (BNN) posteriors with many equivalent modes. This multimodality poses a challenge for variational inference (VI) techniques, which typically rely on approximating the posterior with a unimodal distribution. In this work, we investigate the impact of weight space permutation symmetries on VI. We demonstrate, both theoretically and empirically, that these symmetries lead to biases in the approximate posterior, which degrade predictive performance and posterior fit if not explicitly accounted for. To mitigate this behavior, we leverage the symmetric structure of the posterior and devise a symmetrization mechanism for constructing permutation invariant variational posteriors. We show that the symmetrized distribution has a strictly better fit to the true posterior, and that it can be trained using the original ELBO objective with a modified KL regularization term. We demonstrate experimentally that our approach mitigates the aforementioned biases and results in improved predictions and a higher ELBO.

Recent work in reinforcement learning has leveraged symmetries in the model to improve sample efficiency in training a policy. A commonly used simplifying assumption is that the dynamics and reward both exhibit the same symmetry. However, in many real-world environments, the dynamical model exhibits symmetry independent of the reward model: the reward may not satisfy the same symmetries as the dynamics. In this paper, we investigate scenarios where only the dynamics are assumed to exhibit symmetry, extending the scope of problems in reinforcement learning and learning in control theory where symmetry techniques can be applied. We use Cartan's moving frame method to introduce a technique for learning dynamics which, by construction, exhibit specified symmetries. We demonstrate through numerical experiments that the proposed method learns a more accurate dynamical model.

We ask whether there exists a function or measure that (1) minimizes a given convex functional or risk and (2) satisfies a symmetry property specified by an amenable group of transformations. Examples of such symmetry properties are invariance, equivariance, or quasi-invariance. Our results draw on old ideas of Stein and Le Cam and on approximate group averages that appear in ergodic theorems for amenable groups. A class of convex sets known as orbitopes in convex analysis emerges as crucial, and we establish properties of such orbitopes in nonparametric settings. We also show how a simple device called a cocycle can be used to reduce different forms of symmetry to a single problem. As applications, we obtain results on invariant kernel mean embeddings and a Monge-Kantorovich theorem on optimality of transport plans under symmetry constraints. We also explain connections to the Hunt-Stein theorem on invariant tests.