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arXiv · 2509.12584

Sharp mean-field analysis of permutation mixtures and permutation-invariant decisions

Yiguo Liang·Yanjun Han
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Abstract

We develop sharp bounds on the statistical distance between high-dimensional permutation mixtures and their i.i.d. counterparts. Our approach establishes a new geometric link between the spectrum of a complex channel overlap matrix and the information geometry of the channel, yielding tight dimension-independent bounds that close gaps left by previous work. Within this geometric framework, we also derive dimension-dependent bounds that uncover phase transitions in dimensionality for Gaussian and Poisson families. Applied to compound decision problems, this refined control of permutation mixtures enables sharper mean-field analyses of permutation-invariant decision rules, yielding strong non-asymptotic equivalence results between two notions of compound regret in Gaussian and Poisson models.

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Yiguo Liang, Yanjun Han. 2026-08-31. Sharp mean-field analysis of permutation mixtures and permutation-invariant decisions. https://arxiv.org/abs/2509.12584

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