Representation Learning for Equivariant Inference with Guarantees
For practitioners in physics, geometry, and robotics, this framework offers a principled way to incorporate symmetries with statistical guarantees, improving generalization and sample efficiency.
The paper introduces an equivariant representation learning framework for regression, conditional probability estimation, and uncertainty quantification, providing first-of-its-kind non-asymptotic statistical learning guarantees. Empirical results on synthetic and real-world robotics tasks show it matches or outperforms existing equivariant baselines in regression while offering well-calibrated uncertainty estimates.
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 empirical advances by incorporating symmetry and geometry priors, 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 providing well-calibrated uncertainty estimates.