A 1/R Law for Kurtosis Contrast in Balanced Mixtures
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908853056045056 |
|---|---|
| author | Bi, Yuda Xiao, Wenjun Bai, Linhao Calhoun, Vince D |
| author_facet | Bi, Yuda Xiao, Wenjun Bai, Linhao Calhoun, Vince D |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22334 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A 1/R Law for Kurtosis Contrast in Balanced Mixtures Bi, Yuda Xiao, Wenjun Bai, Linhao Calhoun, Vince D Machine Learning Artificial Intelligence 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. |
| title | A 1/R Law for Kurtosis Contrast in Balanced Mixtures |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2602.22334 |