Reservoir Subspace Injection for Online ICA under Top-n Whitening

Reservoir expansion can improve online independent component analysis (ICA) under nonlinear mixing, yet top-$n$ whitening may discard injected features. We formalize this bottleneck as \emph{reservoir subspace injection} (RSI): injected features help only if they enter the retained eigenspace without displacing passthrough directions. RSI diagnostics (IER, SSO, $ρ_x$) identify a failure mode in our top-$n$ setting: stronger injection increases IER but crowds out passthrough energy ($ρ_x: 1.00\!\rightarrow\!0.77$), degrading SI-SDR by up to $2.2$\,dB. A guarded RSI controller preserves passthrough retention and recovers mean performance to within $0.1$\,dB of baseline $1/N$ scaling. With passthrough preserved, RE-OICA improves over vanilla online ICA by $+1.7$\,dB under nonlinear mixing and achieves positive SI-SDR$_{\mathrm{sc}}$ on the tested super-Gaussian benchmark ($+0.6$\,dB).

A 1/R Law for Kurtosis Contrast in Balanced Mixtures

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 \emph{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.