A 1/R Law for Kurtosis Contrast in Balanced Mixtures

arXiv:2602.22334v1 Announce Type: cross
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 $O(c_bkappa_{max}/R)$ 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 $Rlesssim kappa_{max}sqrt{T}$. 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.