Separation Is a Riesz Energy: How Many Environments Does ICA Need?

원문 초록

How many environments and samples does it take to separate $d$ Gaussian sources whose variances change across environments? Each source has a signature, its vector of variances. The Riesz $2$-energy $E_j$ of source $j$ sums the inverse Fisher informations for rotating it towards other sources. With known whitening, we prove a worst-case squared-angle risk lower bound of order $E_j/(n+E_j)$ for orthogonal estimators at every $n$ and $d$. A quadratic pencil given the signatures attains $E_j/n$ up to a constant above a separation-dependent sample threshold, even with estimated whitening. For well-spread signatures at fixed $K$, the energy grows as $d^2$, $d\log d$ and $d$ for $K=2$, $3$ and $K\ge4$, respectively. A design with $K\asymp\log d$ and pairwise Fisher information bounded below by a constant makes $d\log d$ samples sufficient. The eigenvectors of a single fixed linear combination have an average first-order error coefficient of at least order $d^2$. Simulations up to $d=1000$ support the first-order predictions, and on task fMRI the energy helps rank split-half instability.