Self-sufficient Independent Component Analysis for Demixing Flows

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.

We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on $\mathbb{R}^d$, including their analyticity and the way in which they determine and characterise the distributions. We then focus on several identifiability results of ICA models with successively strengthened assumptions on the sources: from merely non-constant, to non-Gaussian, to Gaussian-free independent sources. Under the strictest assumptions, we show that the independent sources are identifiable up to translation, permutation, scales and signs, and this even in the presence of additive Gaussian noise. Furthermore, we present the online equivariant gradient descent ICA algorithm for recovering the independent sources from data, in the standard complete noiseless non-Gaussian ICA setting.

Many machine learning systems try to explain complex data - like images or financial time series - in terms of hidden, independent factors that generated them. Recovering the true underlying factors, rather than some scrambled version of them, is the central challenge of nonlinear Independent Component Analysis (nICA). We prove identifiability (exact recovery) up to trivial ambiguities for real analytic generating functions when source probability density functions have a finite number of discontinuities in the first derivative. The Laplace distribution is the most prominent example satisfying this assumption. Our proof relies on the contrast between kinks in the source distribution and the smoothness of real analytic functions. Real analytic functions comprise a broad class of generating mechanisms, and can be approximated with Normalizing Flows or Variational Autoencoders with standard activation functions (e.g., tanh, softplus, GELU), so our result applies with minimal changes to existing training pipelines. We perform experiments on real and synthetic data with both Normalizing Flows and Variational Auto-Encoders demonstrating their identifiability properties. In experiments on CelebA data we recover several interpretable latent factors controlling unique attributes across the dataset.