Separation Is a Riesz Energy: How Many Environments Does ICA Need?
Abstract
Can independent components be recovered when all of them are Gaussian? Yes, if their variances change across environments; but how many environments, and how many samples as the number of components d grows? We answer both for the joint diagonalization of K covariances. Each component has a signature, its variances across the environments. Separating component j costs the Riesz 2-energy Ej of this signature, a sum, over the other components, of inverse pairwise Fisher informations. For Gaussian sources no estimator in O(d) beats cEj/(n+Ej) at any n and d, and above an explicit sample size a spectral estimator attains Ej/n up to a constant, for every mixing and without the signatures (given a lower bound on the pairwise information). Since signatures live in K−1 dimensions, the rate obeys a dimension law: d2/n, dlog d/n and d/n for K=2, $3$ and K≥4 well-spread environments. What counts is the dimension the signatures span, not their number, and log d well-designed environments make dlog d samples suffice. Projecting the covariances onto one linear combination pays d2/n at first order instead. In simulations up to d=1000 the predictions hold and joint diagonalization with K≥4 attains the rate once n/d is in the hundreds; on task fMRI the energy says which components are hard to separate.
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BibTeX
@misc{bi2026separation,
title = {{Separation Is a Riesz Energy: How Many Environments Does ICA Need?}},
author = {Yuda Bi and Yubo Guo and Jiaxiao Cai and Xiaonai Li and Zhuolin Li and Vince D. Calhoun},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06228},
eprint = {2611.06228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06228},
}
RIS
TY - UNPB
TI - Separation Is a Riesz Energy: How Many Environments Does ICA Need?
AU - Yuda Bi
AU - Yubo Guo
AU - Jiaxiao Cai
AU - Xiaonai Li
AU - Zhuolin Li
AU - Vince D. Calhoun
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06228
UR - https://arxiv.org/abs/2611.06228
AB - Can independent components be recovered when all of them are Gaussian? Yes, if their variances change across environments; but how many environments, and how many samples as the number of components d grows? We answer both for the joint diagonalization of K covariances. Each component has a signature, its variances across the environments. Separating component j costs the Riesz 2-energy E_j of this signature, a sum, over the other components, of inverse pairwise Fisher informations. For Gaussian sources no estimator in O(d) beats cE_j/(n+E_j) at any n and d, and above an explicit sample size a spectral estimator attains E_j/n up to a constant, for every mixing and without the signatures (given a lower bound on the pairwise information). Since signatures live in K−1 dimensions, the rate obeys a dimension law: d^2/n, dlog d/n and d/n for K=2, $3$ and K≥4 well-spread environments. What counts is the dimension the signatures span, not their number, and log d well-designed environments make dlog d samples suffice. Projecting the covariances onto one linear combination pays d^2/n at first order instead. In simulations up to d=1000 the predictions hold and joint diagonalization with K≥4 attains the rate once n/d is in the hundreds; on task fMRI the energy says which components are hard to separate.
ER -