A 1/R Law for Kurtosis Contrast in Balanced Mixtures

• URL: http://arxiv.org/abs/2602.22334v1
• Date: Wed, 25 Feb 2026 19:01:01 GMT
• Title: A 1/R Law for Kurtosis Contrast in Balanced Mixtures
• Abstract summary: Kurtosis-based Independent Component Analysis weakens in wide, balanced mixtures.<n>We prove a sharp redundancy law: for a standardized projection with effective width $R_mathrmeff$, the population excess kurtosis obeys $|(y)|=O(_max/R_mathrmeff)$.<n>We also show that emphpurification -- selecting $m!ll!R$ sign-consistent sources -- restores $R$-independent contrast $ (1/m)$, with a simple data
• Score: 15.412858526786978
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $|κ(y)|=O(κ_{\max}/R_{\mathrm{eff}})$, yielding the order-tight $O(c_bκ_{\max}/R)$ 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 $R\lesssim κ_{\max}\sqrt{T}$. We also show that \emph{purification} -- selecting $m\!\ll\!R$ sign-consistent sources -- restores $R$-independent contrast $Ω(1/m)$, with a simple data-driven heuristic. Synthetic experiments validate the predicted decay, the $\sqrt{T}$ crossover, and contrast recovery.

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