✨ TL;DR
This paper develops a general mathematical theory explaining how symmetries in target distributions and variational families guarantee that variational inference can accurately recover certain statistics, even when the approximation is imperfect. The framework unifies existing results and enables derivation of new recovery guarantees across diverse settings including directional statistics.
Variational inference approximates intractable probability distributions by optimizing over simpler families, but these families often cannot represent the target exactly. This model misspecification raises critical questions about which properties of the target distribution can be reliably captured. While recent work has shown that symmetries can enable recovery of certain statistics despite misspecification, these results are problem-specific and lack a unified theoretical foundation. There is no general understanding of the fundamental mechanism by which symmetry forces statistic recovery, limiting our ability to predict when VI will succeed or fail at capturing specific properties of interest.
The authors develop a general mathematical framework for understanding symmetry-induced statistic recovery in variational inference. Their approach has three main components: First, they characterize when variational minimizers inherit symmetries from the target distribution and establish conditions under which these inherited symmetries pin down identifiable statistics. Second, they demonstrate that their theory subsumes existing results by showing that known recovery guarantees in location-scale families emerge as special cases of their general framework. Third, they apply their theory to new settings, specifically distributions on the sphere, to derive novel guarantees for recovering directional statistics in von Mises-Fisher families. The framework is designed to be modular, providing a systematic blueprint for deriving recovery guarantees across different symmetry settings.
What the paper shows.
The paper establishes a general theory that unifies and extends existing statistic recovery guarantees in variational inference. The authors successfully show that known results for location-scale families emerge naturally from their framework, validating the theory's explanatory power. They derive novel recovery guarantees for directional statistics in von Mises-Fisher families on spheres, demonstrating the framework's ability to generate new results. The theory provides both necessary and sufficient conditions for when symmetries guarantee statistic recovery, offering a complete characterization of the phenomenon. The modular nature of the framework enables systematic derivation of guarantees across different symmetry settings.
The paper focuses on theoretical characterization and does not provide extensive empirical validation of the recovery guarantees in practical applications. The framework requires identifying and verifying symmetries in both target and variational families, which may be non-trivial in complex real-world problems. While the theory is general, deriving specific guarantees for new settings still requires mathematical analysis tailored to each symmetry structure. The paper does not extensively discuss computational aspects of exploiting these symmetries in practical VI algorithms or how to detect when symmetry conditions are violated in practice.
✨ Generated by Claude · Apr 21, 2026 · Read the PDF for authoritative content.