Reservoir Subspace Injection for Online ICA under Top-n Whitening

Linear Independent Component Analysis (ICA) recovers jointly independent source signals from their linear mixtures. To achieve this, classical ICA algorithms attempt to maximize non-Gaussianity, measured by negentropy, which is linked to independence by information theory. Because exact negentropy optimization is intractable, they rely on proxy contrast functions, such as fourth-order cumulants and parametric log-likelihoods. We propose instead to use the squared $L_2$-Wasserstein distance to a standard Gaussian as the ICA contrast. We show that the Wasserstein distance between a standard normal distribution and linear projections of the data is maximized when the projection recovers an independent component, and that under a regularity condition on the sources this maximum is separated from every genuine mixture by an explicit margin. We uncover the advantageous properties of the resulting estimator: for sources with a smooth density it is $\sqrt{N}$-consistent and asymptotically normal with a closed-form variance, whereas for a source with an atom the contrast has a kink at the true direction, which makes the estimator exact with probability tending to one. The proposed OT-ICA algorithm finds this projection by gradient-based optimization. Empirical evaluation on simulated data shows that OT-ICA outperforms proxy-based methods and that the $W_2^2$ contrast provides more robust signal than proxy contrasts in measuring non-Gaussianity for different distributional mixtures of the latent variables. Application to linear causal disentanglement and EEG artifact removal, along with further applications detailed in the appendix, confirms OT-ICA can be used for applied ICA tasks without distributional assumptions.

Independent Component Analysis (ICA) is a fundamental unsupervised learning technique foruncovering latent structure in data by separating mixed signals into their independent sources. While substantial progress has been made in establishing asymptotic identifiability guarantees for nonlinear ICA, the finite-sample statistical properties of learning algorithms remain poorly understood. This gap poses significant challenges for practitioners who must determine appropriate sample sizes for reliable source recovery. This paper presents a comprehensive finite-sample analysis of nonlinear ICA with neural network encoders, providing the first complete characterization with matching upper and lower bounds. Our theoretical development introduces three key technical contributions. First, we establish a direct relationship between excess risk and identification error that bypasses parameter-space arguments, thereby avoiding the rate degradation that would otherwise yield suboptimal scaling. Second, we prove matching information-theoretic lower bounds that confirm the optimality of our sample complexity results. Third, we extend our analysis to practical SGD optimization, showing that the same sample efficiency can be achieved with finite-iteration gradient descent under standard landscape assumptions. We validate our theoretical predictions through carefully designed simulation experiments. This gap points toward valuable future research on finite-sample behavior of neural network training and highlights the importance of our validated scaling laws for dimension and diversity.

Reservoir computing (RC) trains only a linear readout over a fixed recurrent layer, making it fast and data-efficient for online prediction. However, a static reservoir degrades under system drift, readout-only adaptation is then insufficient, and unconstrained reservoir adaptation can destroy the echo-state and incremental stability properties that make RC reliable. This paper proposes LoRA-RC, which adapts the recurrent matrix through a low-rank correction driven by streaming prediction errors. The base reservoir and adaptation bases are fixed offline; a small core matrix is adapted online, projected onto a spectral-norm ball, and low-pass filtered at each step. The projection guarantees that every applied recurrent matrix remains within a certified contraction set, and an incremental input-to-state stability bound is established for the reservoir along each online adaptation path, with path-independent rate and gain. On a Lorenz system with an abrupt parameter drift, LoRA-RC cuts post-drift prediction error by 56% versus a fixed RC and 51% versus readout-only adaptation; ablations over 20 seeds show that removing the projection inflates this error by more than a factor of 40.