Sharp mean-field analysis of permutation mixtures and permutation-invariant decisions
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.
Cite
@article{arxiv.2509.12584,
title = {Sharp mean-field analysis of permutation mixtures and permutation-invariant decisions},
author = {Yiguo Liang and Yanjun Han},
journal= {arXiv preprint arXiv:2509.12584},
year = {2025}
}