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.

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.

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 Equivariant Isomorphic Networks (EquIN) – a method for learning representations that are equivariant with respect to general group actions over data. Differently from existing equivariant representation learners, EquIN is suitable for group actions that are not free, i.e., that stabilize data via nontrivial symmetries. EquIN is theoretically grounded in the orbit-stabilizer theorem from group theory. This guarantees that an ideal learner infers isomorphic representations while trained on equivariance alone and thus fully extracts the geometric structure of data. We provide an empirical investigation on image datasets with rotational symmetries and show that taking stabilizers into account improves the quality of the representations.

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.

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.