A 1/R Law for Kurtosis Contrast in Balanced Mixtures

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

cs.LGFeb 25, 2026

A 1/R Law for Kurtosis Contrast in Balanced Mixtures

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

Organizations: Tri-Institutional Center for Translational Research in Neuroimaging and Data Science (TReNDS), Georgia State University, Georgia Institute of Technology, and Emory University, Atlanta, GA 30303 USA · Department of Computer Science, The George Washington University, Washington, DC 20052 USA · School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA

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 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.

Figures & tables

Fig. 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}\approx 5.3

); dotted line:

|\kappa_{0}|/R

. (c) At

R{=}50

(df

\in[6,30]

), purification (

m{=}5

) increases contrast from

\approx 0.03

\approx 0.43

(oracle) /

\approx 0.44

(sample-based),

\sim 14\times

Fig. 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.

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Independent Component Analysis

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Linear Independent Component Analysis (ICA) recovers jointly independent source signals from their linear mixtures. To achieve this, classical ICA algorithms attempt to maximize non-Gaussianity, measured by negentropy, which is linked to independence by information theory. Because exact negentropy optimization is intractable, they rely on proxy contrast functions, such as fourth-order cumulants and parametric log-likelihoods. We propose instead to use the squared

L_2

-Wasserstein distance to a standard Gaussian as the ICA contrast. We show that the Wasserstein distance between a standard normal distribution and linear projections of the data is maximized when the projection recovers an independent component, and that under a regularity condition on the sources this maximum is separated from every genuine mixture by an explicit margin. We uncover the advantageous properties of the resulting estimator: for sources with a smooth density it is

\sqrt{N}

-consistent and asymptotically normal with a closed-form variance, whereas for a source with an atom the contrast has a kink at the true direction, which makes the estimator exact with probability tending to one. The proposed OT-ICA algorithm finds this projection by gradient-based optimization. Empirical evaluation on simulated data shows that OT-ICA outperforms proxy-based methods and that the

W_2^2

contrast provides more robust signal than proxy contrasts in measuring non-Gaussianity for different distributional mixtures of the latent variables. Application to linear causal disentanglement and EEG artifact removal, along with further applications detailed in the appendix, confirms OT-ICA can be used for applied ICA tasks without distributional assumptions.

Abstract

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