# A 1/R Law for Kurtosis Contrast in Balanced Mixtures

arXiv: https://arxiv.org/abs/2602.22334 · HTML: https://mlsanity.com/paper/2602.22334/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 25 references that are in the map.
- First posted on arXiv 25 Feb 2026.

## Builds on (most cited first, 6 of 6)

- [Nonlinear Independent Component Analysis for Principled Disentanglement in Unsupervised Deep Learning](https://mlsanity.com/paper/2303.16535/) (2023)
- [Large Dimensional Independent Component Analysis: Statistical Optimality and Computational Tractability](https://mlsanity.com/paper/2303.18156/) (2023)
- [Independent Vector Analysis: Identification Conditions and Performance Bounds](https://mlsanity.com/paper/1303.7474/) (2013)
- [Feature learning from non-Gaussian inputs: the case of Independent Component Analysis in high dimensions](https://mlsanity.com/paper/2503.23896/) (2025)
- [Multidataset Independent Subspace Analysis with Application to Multimodal Fusion](https://mlsanity.com/paper/1911.04048/) (2019)
- [Stochastic Approximation for Online Tensorial Independent Component Analysis](https://mlsanity.com/paper/2012.14415/) (2020)

## Abstract (arXiv)

Kurtosis-based Independent Component Analysis (ICA) weakens in wide, balanced mixtures. We prove a sharp redundancy law: for a standardized projection with effective width Reff (participation ratio), the population excess kurtosis obeys |kappa(y)|=O(kappamax/Reff), yielding the order-tight O(cbkappamax/R) under balance (typically cb=O(log R)). As an impossibility screen, under standard finite-moment conditions for sample kurtosis estimation, surpassing the O(1/√T) estimation scale requires Rlesssim kappamax√T. We also show that 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 √T crossover, and contrast recovery.