We study convolutional neural networks with monomial activation functions. Specifically, we prove that their parameterization map is regular and is an isomorphism almost everywhere, up to rescaling the filters. By leveraging on tools from algebraic geometry, we explore the geometric properties of the image in function space of this map – typically referred to as neuromanifold. In particular, we compute the dimension and the degree of the neuromanifold, which measure the expressivity of the model, and describe its singularities. Moreover, for a generic large dataset, we derive an explicit formula that quantifies the number of critical points arising in the optimization of a regression loss.

We propose HyperSteiner – an efficient heuristic algorithm for computing Steiner minimal trees in the hyperbolic space. HyperSteiner extends the Euclidean Smith-Lee-Liebman algorithm, which is grounded in a divide-and-conquer approach involving the Delaunay triangulation. The central idea is rephrasing Steiner tree problems with three terminals as a system of equations in the Klein-Beltrami model. Motivated by the fact that hyperbolic geometry is well-suited for representing hierarchies, we explore applications to hierarchy discovery in data. Results show that HyperSteiner infers more realistic hierarchies than the Minimum Spanning Tree and is more scalable to large datasets than Neighbor Joining.

Idempotent Generative Networks (IGNs) are deep generative models that also function as local data manifold projectors, mapping arbitrary inputs back onto the manifold. They are trained to act as identity operators on the data and as idempotent operators off the data manifold. However, IGNs suffer from mode collapse, mode dropping, and training instability due to their objectives, which contain adversarial components and can cause the model to cover the data manifold only partially – an issue shared with generative adversarial networks. We introduce Non-Adversarial Idempotent Generative Networks (NAIGNs) to address these issues. Our loss function combines reconstruction with the non-adversarial generative objective of Implicit Maximum Likelihood Estimation (IMLE). This improves on IGN’s ability to restore corrupted data and generate new samples that closely match the data distribution. We moreover demonstrate that NAIGNs implicitly learn the distance field to the data manifold, as well as an energy-based model.

We consider function spaces defined by self-attention networks without normalization, and theoretically analyze their geometry. Since these networks are polynomial, we rely on tools from algebraic geometry. In particular, we study the identifiability of deep attention by providing a description of the generic fibers of the parametrization for an arbitrary number of layers and, as a consequence, compute the dimension of the function space. Additionally, for a single-layer model, we characterize the singular and boundary points. Finally, we formulate a conjectural extension of our results to normalized self-attention networks, prove it for a single layer, and numerically verify it in the deep case.

What features neural networks learn, and how, remains an open question. In this paper, we introduce Alternating Gradient Flows (AGF), an algorithmic framework that describes the dynamics of feature learning in two-layer networks trained from small initialization. Prior works have shown that gradient flow in this regime exhibits a staircase-like loss curve, alternating between plateaus where neurons slowly align to useful directions and sharp drops where neurons rapidly grow in norm. AGF approximates this behavior as an alternating two-step process: maximizing a utility function over dormant neurons and minimizing a cost function over active ones. AGF begins with all neurons dormant, corresponding to an initialization at the origin. At each iteration, a dormant neuron activates, triggering the acquisition of a feature and a drop in the loss. AGF quantifies the order, timing, and magnitude of these drops, matching experiments across several commonly studied architectures. We show that AGF unifies and extends existing saddle-to-saddle analyses in fully connected linear networks and attention-only linear transformers, where the learned features are singular modes and principal components, respectively. In diagonal linear networks, we prove AGF converges to gradient flow in the limit of vanishing initialization. Applying AGF to quadratic networks trained to perform modular addition, we give the first complete characterization of the training dynamics, revealing that networks learn Fourier features in decreasing order of coefficient magnitude. Altogether, AGF offers a promising step towards understanding feature learning in neural networks.

In this work, we formally prove that, under certain conditions, if a neural network is invariant to a finite group then its weights recover the Fourier transform on that group. This provides a mathematical explanation for the emergence of Fourier features – a ubiquitous phenomenon in both biological and artificial learning systems. The results hold even for non-commutative groups, in which case the Fourier transform encodes all the irreducible unitary group representations. Our findings have consequences for the problem of symmetry discovery. Specifically, we demonstrate that the algebraic structure of an unknown group can be recovered from the weights of a network that is at least approximately invariant within certain bounds. Overall, this work contributes to a foundation for an algebraic learning theory of invariant neural network representations.

We introduce a non-parametric density estimator deemed Radial Voronoi Density Estimator (RVDE). RVDE is grounded in the geometry of Voronoi tessellations and as such benefits from local geometric adaptiveness and broad convergence properties. Due to its radial definition RVDE is continuous and computable in linear time with respect to the dataset size. This amends for the main shortcomings of previously studied VDEs, which are highly discontinuous and computationally expensive. We provide a theoretical study of the modes of RVDE as well as an empirical investigation of its performance on high-dimensional data. Results show that RVDE outperforms other non-parametric density estimators, including recently introduced VDEs.

We introduce a general method for learning representations that are equivariant to symmetries of data. Our central idea is to decompose the latent space into an invariant factor and the symmetry group itself. The components semantically correspond to intrinsic data classes and poses respectively. The learner is trained on a loss encouraging equivariance based on supervision from relative symmetry information. The approach is motivated by theoretical results from group theory and guarantees representations that are lossless, interpretable and disentangled. We provide an empirical investigation via experiments involving datasets with a variety of symmetries. Results show that our representations capture the geometry of data and outperform other equivariant representation learning frameworks.

We introduce an algorithm for active function approximation based on nearest neighbor regression. Our Active Nearest Neighbor Regressor (ANNR) relies on the Voronoi-Delaunay framework from computational geometry to subdivide the space into cells with constant estimated function value and select novel query points in a way that takes the geometry of the function graph into account. We consider the recent state-of-the-art active function approximator called DEFER, which is based on incremental rectangular partitioning of the space, as the main baseline. The ANNR addresses a number of limitations that arise from the space subdivision strategy used in DEFER. We provide a computationally efficient implementation of our method, as well as theoretical halting guarantees. Empirical results show that ANNR outperforms the baseline for both closed-form functions and real-world examples, such as gravitational wave parameter inference and exploration of the latent space of a generative model.

Learning from previously collected datasets of expert data offers the promise of acquiring robotic policies without unsafe and costly online explorations. However, a major challenge is a distributional shift between the states in the training dataset and the ones visited by the learned policy at the test time. While prior works mainly studied the distribution shift caused by the policy during the offline training, the problem of recovering from out-of-distribution states at the deployment time is not very well studied yet. We alleviate the distributional shift at the deployment time by introducing a recovery policy that brings the agent back to the training manifold whenever it steps out of the in-distribution states, e.g., due to an external perturbation. The recovery policy relies on an approximation of the training data density and a learned equivariant mapping that maps visual observations into a latent space in which translations correspond to the robot actions. We demonstrate the effectiveness of the proposed method through several manipulation experiments on a real robotic platform. Our results show that the recovery policy enables the agent to complete tasks while the behavioral cloning alone fails because of the distributional shift problem.

The Voronoi Density Estimator (VDE) is an established density estimation technique that adapts to the local geometry of data. However, its applicability has been so far limited to problems in two and three dimensions. This is because Voronoi cells rapidly increase in complexity as dimensions grow, making the necessary explicit computations infeasible. We define a variant of the VDE deemed Compactified Voronoi Density Estimator (CVDE), suitable for higher dimensions. We propose computationally efficient algorithms for numerical approximation of the CVDE and formally prove convergence of the estimated density to the original one. We implement and empirically validate the CVDE through a comparison with the Kernel Density Estimator (KDE). Our results indicate that the CVDE outperforms the KDE on sound and image data.