Equivariant Representation Learning for Symmetry-Aware Inference with Guarantees

2025-05-26

AI Summary

Modeling physical/geometric symmetries as priors is critical for regression, conditional probability estimation, and uncertainty quantification. Method: We propose a representation learning framework that jointly enforces equivariance and disentanglement. We establish the first non-asymptotic statistical learning guarantee for symmetry-aware inference; leverage spectral decomposition of the conditional expectation operator to construct equivariant and disentangled representations across independent subgroups; and integrate operator theory, group representation theory, and geometric deep learning to design spectral approximation and subgroup-disentangling architectures. Contributions/Results: Experiments on synthetic data and real-world robotic tasks demonstrate that our method matches or surpasses state-of-the-art equivariant baselines in regression accuracy, while simultaneously producing well-calibrated, parametric uncertainty estimates.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Representation Learning for Vision
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 subgroups. 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.

Problem

• Exploiting symmetries to improve generalization and sample efficiency
• Providing statistical learning guarantees for equivariant representation learning
• Addressing regression, probability estimation, and uncertainty quantification simultaneously

Innovation

• Equivariant representation learning for symmetry-aware inference
• Spectral decomposition of conditional expectation operator
• Disentangled representations along independent symmetry subgroups