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arxiv: 2604.18310 · v1 · submitted 2026-04-20 · 📊 stat.ML · cs.LG

Symmetry Guarantees Statistic Recovery in Variational Inference

Daniel Marks , Dario Paccagnan , Mark van der Wilk This is my paper

Pith reviewed 2026-05-10 03:46 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords variational inferencesymmetrystatistic recoverymisspecificationidentifiable statisticsvon Mises-Fisherlocation-scale families
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The pith

Symmetries of the target distribution ensure variational minimizers recover identifiable statistics under misspecification.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a general theory showing that symmetries in the target density are often preserved in the optimal variational approximation. This preservation pins down certain statistics, such as location or direction, that can be recovered accurately even if the full distribution cannot be represented exactly. The theory explains why this happens in location-scale families and extends it to von Mises-Fisher distributions on the sphere. A reader should care because it provides a systematic way to know what information VI can reliably extract without solving the full inference problem.

Core claim

We characterise when variational minimisers inherit the symmetries of the target and establish conditions under which these pin down identifiable statistics. We unify existing results by showing that previously known statistic recovery guarantees in location-scale families arise as special cases of our theory. We apply our framework to distributions on the sphere to obtain novel guarantees for directional statistics in von Mises-Fisher families.

What carries the argument

Symmetry inheritance by the variational minimizer, which constrains the approximation to preserve certain group actions from the target and thereby identifies recoverable statistics.

If this is right

  • Recovery guarantees for statistics in location-scale families follow directly as special cases of the general theory.
  • Novel guarantees can be obtained for directional statistics when approximating spherical distributions with von Mises-Fisher families.
  • The framework supplies a modular method for deriving new statistic recovery results in other settings that possess symmetry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Choosing a variational family whose symmetry group matches the target's could guarantee statistic recovery without requiring exact density matching.
  • The same inheritance mechanism might be checked in other approximate inference procedures to identify recoverable quantities.
  • A direct test would involve optimizing the ELBO for a new symmetric family and verifying whether the minimizer respects the group action.

Load-bearing premise

Variational minimizers exist and inherit the symmetries of the target under the conditions specified in the theory.

What would settle it

A concrete example of a symmetric target distribution where the ELBO minimizer in a symmetric variational family fails to inherit the symmetry, causing the predicted recoverable statistics to deviate from the target's true values.

Figures

Figures reproduced from arXiv: 2604.18310 by Daniel Marks, Dario Paccagnan, Mark van der Wilk.

Figure 1
Figure 1. Figure 1: Unique variational minimisers q from a Gaussian family exactly recover symmetry￾determined statistics of highly non-Gaussian targets p. Left: under even symmetry, the mean is recovered exactly as per Corollary 4.3. Right: under elliptical symmetry, the mean and correlation are recovered exactly, whereas the covariance is recovered only up to scale, as per Corollary 4.5. By (4), Condition (3) allows us to d… view at source ↗
Figure 2
Figure 2. Figure 2: Log-density contours of the target and variational posterior are shown using a Lambert [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a general theory of symmetry-induced statistic recovery in variational inference. It first characterizes conditions (including group-invariance of the variational family and lower semi-continuity of the objective) under which variational minimizers exist and inherit the symmetries of the target distribution, then shows that these inherited symmetries force recovery of the corresponding identifiable statistics. The theory is used to unify prior location-scale results as direct special cases of the same lemmas and to derive new recovery guarantees for directional statistics in von Mises-Fisher families on the sphere.

Significance. If the central results hold, the paper supplies a modular, assumption-light blueprint for deriving statistic-recovery guarantees in variational inference across symmetry settings. It unifies previously scattered location-scale results without additional ad-hoc assumptions, supplies explicit existence and inheritance conditions, and yields novel directional-statistic guarantees; these strengths make the framework a useful reference for both theoretical analysis and variational-family design.

minor comments (3)
  1. [§3.2] §3.2: the statement that the recovered statistic is 'identifiable' would benefit from an explicit definition or reference to the precise identifiability notion used (e.g., up to the group action).
  2. [Figure 2] Figure 2: the caption does not indicate whether the plotted curves are exact or Monte-Carlo estimates; adding this detail would improve reproducibility.
  3. [§2 and §5] Notation: the symbol for the group action on measures is introduced in §2 but reused with a slightly different meaning in the von Mises-Fisher application; a short clarifying sentence would prevent confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of the manuscript, as well as the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation starts from explicit assumptions (group-invariance of the variational family and lower semi-continuity of the objective) and mathematically characterizes when minimizers inherit target symmetries, then proves these symmetries determine identifiable statistics. Known location-scale recovery results are recovered directly as special cases of the same lemmas, and the von Mises-Fisher guarantees on the sphere are derived analogously without introducing fitted parameters, self-referential definitions, or load-bearing self-citations. All steps are forward mathematical implications from stated conditions rather than reductions to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the theory rests on domain assumptions about symmetry inheritance in variational optimization.

axioms (2)
  • domain assumption Variational minimizers inherit the symmetries of the target distribution under certain conditions.
    This is the first main step stated in the abstract and is load-bearing for all subsequent claims.
  • domain assumption Inherited symmetries pin down identifiable statistics.
    Second main step in the abstract; required for the recovery guarantees.

pith-pipeline@v0.9.0 · 5511 in / 1260 out tokens · 34659 ms · 2026-05-10T03:46:14.262813+00:00 · methodology

discussion (0)

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