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arXiv · 2602.22334

A 1/R Law for Kurtosis Contrast in Balanced Mixtures

Yuda Bi·Wenjun Xiao·Linhao Bai·Vince D Calhoun
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Abstract

Kurtosis-based Independent Component Analysis (ICA) weakens in wide, balanced mixtures. We prove a sharp redundancy law: for a standardized projection with effective width $R_{\mathrm{eff}}$ (participation ratio), the population excess kurtosis obeys $|\kappa(y)|=O(\kappa_{\max}/R_{\mathrm{eff}})$, yielding the order-tight $O(c_b\kappa_{\max}/R)$ under balance (typically $c_b=O(\log R)$). As an impossibility screen, under standard finite-moment conditions for sample kurtosis estimation, surpassing the $O(1/\sqrt{T})$ estimation scale requires $R\lesssim \kappa_{\max}\sqrt{T}$. We also show that \emph{purification} -- selecting $m\!\ll\!R$ sign-consistent sources -- restores $R$-independent contrast $\Omega(1/m)$, with a simple data-driven heuristic. Synthetic experiments validate the predicted decay, the $\sqrt{T}$ crossover, and contrast recovery.

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Yuda Bi, Wenjun Xiao, Linhao Bai, Vince D Calhoun. 2026-02-25. A 1/R Law for Kurtosis Contrast in Balanced Mixtures. https://arxiv.org/abs/2602.22334

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