Symmetry Guarantees Statistic Recovery in Variational Inference

2026-04-20

AI Summary

Variational inference often fails to accurately recover key statistics of the target distribution due to model misspecification. This work proposes a symmetry-driven theoretical framework that establishes precise conditions under which the variational optimum inherits the symmetries of the target distribution, thereby enabling exact recovery of identifiable statistics. The framework unifies and generalizes existing results for location-scale families and extends to directional models such as the von Mises–Fisher distribution on the sphere, accommodating a broad class of symmetric settings. By integrating group-theoretic symmetry analysis, variational optimization, and statistical identifiability theory, the study rigorously proves exact statistic recovery across diverse symmetry configurations, demonstrating the method’s universality and theoretical soundness in both classical and novel variational families.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Computer Vision: Visual Reasoning & Symbolic 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: User privacy protection in personalized systems
• Security and Privacy: Large-scale security measurements

Abstract

Variational inference (VI) is a central tool in modern machine learning, used to approximate an intractable target density by optimising over a tractable family of distributions. As the variational family cannot typically represent the target exactly, guarantees on the quality of the resulting approximation are crucial for understanding which of its properties VI can faithfully capture. Recent work has identified instances in which symmetries of the target and the variational family enable the recovery of certain statistics, even under model misspecification. However, these guarantees are inherently problem-specific and offer little insight into the fundamental mechanism by which symmetry forces statistic recovery. In this paper, we overcome this limitation by developing a general theory of symmetry-induced statistic recovery in variational inference. First, we characterise when variational minimisers inherit the symmetries of the target and establish conditions under which these pin down identifiable statistics. Second, we unify existing results by showing that previously known statistic recovery guarantees in location-scale families arise as special cases of our theory. Third, we apply our framework to distributions on the sphere to obtain novel guarantees for directional statistics in von Mises-Fisher families. Together, these results provide a modular blueprint for deriving new recovery guarantees for VI in a broad range of symmetry settings.

Problem

• symmetry
• variational inference
• statistic recovery
• model misspecification
• identifiable statistics

Innovation

• symmetry
• variational inference
• statistic recovery
• identifiability
• directional statistics