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Symmetry Guarantees Statistic Recovery in Variational Inference
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Develops a symmetry-based theory showing that unique f-divergence variational minimisers inherit target symmetries, recovering known location-scale mean/covariance results and giving new axis recovery guarantees for von Mises-Fisher families.
Problem
Which statistics does variational inference recover exactly under a symmetry of the target, even when the variational family is misspecified?
Approach
A general statistic recovery theorem: if the target is invariant under a group, the variational family is stable under that group, and the statistic is compatible with pushforwards, then a unique f-divergence minimiser inherits the invariance and its symmetry-determined statistics lie in a fixed set. Known location-scale results follow as special cases, and new axis recovery results follow for vMF
Key results
- Under even symmetry, a unique location-scale f-divergence minimiser exactly recovers the mean (Corollary recovering Margossian and Saul Theorem 10).
- Under elliptical symmetry, a unique minimiser recovers the mean exactly, the covariance up to a positive scalar, and the correlation exactly.
- For rotationally symmetric targets, a unique von Mises-Fisher minimiser recovers the symmetry axis span(u); a one-parameter reverse KL example shows a threshold eta_c beyond which the minimiser set is non-singleton.
Surprising findings
- Correlation is recovered exactly even though it does not satisfy the framework's Condition (3) for d >= 2, so the sufficient conditions are not necessary.
- Beyond a threshold in a misspecified vMF example, the reverse KL minimisers form a latitude circle and none recover the axis.
Limitations
- Results require uniqueness of the variational minimiser, which is assumed rather than verified in general.
- Conditions are sufficient but not always necessary, as the correlation example shows.
- Theory covers exact symmetries only, not approximately symmetric targets.
Experimental setup
Builds on: Margossian and Saul location-scale recovery results f-divergence reverse KL variational inference
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