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A 1/R Law for Kurtosis Contrast in Balanced Mixtures

Yuda Bi, Wenjun Xiao, Linhao Bai, Vince D Calhoun

TL;DR

A sharp redundancy law is proved: for a standardized projection with effective width $R_{\mathrm{eff}}$ (participation ratio), the population excess kurtosis obeys O(\kappa_{\max}/R_{\mathrm{eff}})$, yielding the order-tight $O(c_b\kappa_{\max}/R)$ under balance (typically $c_b=O(\log R)$ under balance.

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.

A 1/R Law for Kurtosis Contrast in Balanced Mixtures

TL;DR

A sharp redundancy law is proved: for a standardized projection with effective width (participation ratio), the population excess kurtosis obeys O(\kappa_{\max}/R_{\mathrm{eff}})O(c_b\kappa_{\max}/R)c_b=O(\log R)$ under balance.

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 (participation ratio), the population excess kurtosis obeys , yielding the order-tight under balance (typically ). As an impossibility screen, under standard finite-moment conditions for sample kurtosis estimation, surpassing the estimation scale requires . We also show that \emph{purification} -- selecting sign-consistent sources -- restores -independent contrast , with a simple data-driven heuristic. Synthetic experiments validate the predicted decay, the crossover, and contrast recovery.
Paper Structure (7 sections, 5 theorems, 10 equations, 2 figures)

This paper contains 7 sections, 5 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and Main Results
  3. Setup and Notation
  4. Redundancy Law
  5. Purification via Sign-Consistent Subset Selection
  6. Experiments
  7. Conclusion

Key Result

Proposition 1

Fix a block index set $\mathcal{S}\subseteq\{1,\dots,k\}$ with $|\mathcal{S}|=R$. If, for a nonzero direction $u$, the inner products $c_j:=|a_j^\top u|$ satisfy for some $\rho\ge 1$, then $\max_{j\in\mathcal{S}}|w_j|^2\le \rho/R$. In particular, if $A_{\mathcal{S}}^\top A_{\mathcal{S}}=I_R$ and $u$ is uniform on the unit sphere in $\mathrm{span}(A_{\mathcal{S}})$, then $\max_j|w_j|^2\le C(\log R

Figures (2)

  • Figure 1: Empirical validation (mean $\pm$ SEM). (a) FastICA error $\mathrm{err}(W,A)$ increases with $1/\Delta_\kappa$ ($R^2{=}0.78$). (b) Balanced Student-$t$ mixtures (df$=8$, $\kappa_0{=}1.5$) show $|\hat{\kappa}(y)|\propto 1/R$ for $R=2,\ldots,50$ ($R^2{=}0.986$); unbalanced power-law weights decay more slowly. Inset at $R{=}50$: $\mathrm{std}(\hat{\kappa})\sim \sigma_0/\sqrt{T}$ ($\sigma_0\approx5.3$); dotted line: $|\kappa_0|/R$. (c) At $R{=}50$ (df$\in[6,30]$), purification ($m{=}5$) increases contrast from $\approx0.03$ to $\approx0.43$ (oracle) / $\approx0.44$ (sample-based), $\sim14\times$.
  • Figure 2: Paired kurtosis-gap comparison in COBRE group ICA ($n=155$) for model orders $k=53$ and $k=100$, with FNC insets. Top: two-level FNC summaries (edge-level $S$ and fingerprint-level $F$). Bottom: example connectivity fingerprint ($|z|$) and one representative subject's static FNC matrix for component $k^{*}=51$ in the $k=53$ solution (with $|z|$ clipped at the 98th percentile for visualization). FNC insets are qualitative context only and are not used in the redundancy-law inference.

Theorems & Definitions (10)

  • Proposition 1: Block balance
  • proof
  • Theorem 1: Sharp Redundancy Bound
  • proof
  • Corollary 1: Effective-width law
  • proof
  • Corollary 2: Necessary model-order screening condition
  • proof
  • Theorem 2: Purification Lower Bound
  • proof