Separation Is a Riesz Energy: How Many Environments Does ICA Need?
Yuda Bi, Yubo Guo, Jiaxiao Cai, Xiaonai Li, Zhuolin Li, Vince D. Calhoun
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: $d^2/n$, $d\log d/n$ and $d/n$ for $K=2$, $3$ and $K\ge4$ well-spread environments. What counts is the dimension the signatures span, not their number, and $\log d$ well-designed environments make $d\log d$ 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 $K\ge4$ 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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