Title: Representation Learning for Equivariant Inference with Guarantees
Primary area: learning theory
Keywords: Geometric deep learning, equivariance, symmetry, operator theory, representation learning, conditional expectation operator, contrastive learning, density ratio fitting, spectral representation learning

Abstract: In many real-world applications of regression, conditional probability estimation, and uncertainty quantification, exploiting symmetries rooted in physics or geometry can dramatically improve generalization and sample efficiency. While geometric deep learning has made significant empirical advances by incorporating group-theoretic structure, less attention has been given to statistical learning guarantees. In this paper, we introduce an equivariant representation learning framework that simultaneously addresses regression, conditional probability estimation, and uncertainty quantification while providing first-of-its-kind non-asymptotic statistical learning guarantees. Grounded in operator and group representation theory, our framework approximates the spectral decomposition of the conditional expectation operator, building representations that are both equivariant and disentangled along independent symmetry quotient groups. Empirical evaluations on synthetic datasets and real-world robotics applications confirm the potential of our approach, matching or outperforming existing equivariant baselines in regression while additionally providing well-calibrated parametric uncertainty estimates.

TL;DR: We introduce a novel equivariant deep-representation-learning algorithm that can be used to provide estimates of regression, conditional probabilities, and uncertainty quantification with firts-of-its-kind statistical learning guarantees.