One Wasserstein criterion, two methods reinvented

A single contrast-free principle — the Wasserstein distance to the Gaussian — that reinvents two methods at once: source separation and causal inference.

Independent Component Analysis (ICA) is a mature field. Since FastICA, source separation has relied on maximizing non-Gaussianity through carefully designed contrast functions such as kurtosis, negentropy or log-cosh. Their effectiveness, however, depends on how well the chosen contrast matches the source distributions.

In a recent preprint with Félix Laplante and Pierre Humbert, we replace these contrasts by a single quantity: the squared 2-Wasserstein distance to the standard Gaussian, \(W_2^2(Y,\mathcal N(0,1))\).

Under the standard ICA assumption that at most one source is Gaussian, we prove that mixing independent standardized sources strictly decreases the total Wasserstein non-Gaussianity. Consequently, the unmixing matrix is the unique population maximizer (up to signed permutation).

More importantly, the same optimization criterion,

\[ \max_{W\in\mathcal O(d)} \sum_{j=1}^d W_2^2(Y_j,\mathcal N(0,1)), \]

unifies two classical problems.

Interpreting the maximizer as an unmixing matrix recovers the independent sources. Interpreting the ordering of least-squares residuals recovers the causal order. Source separation and causal discovery therefore emerge from the same contrast-free principle.

Figure 1: One Wasserstein criterion, two problems: from whitened data, maximizing non-Gaussianity either recovers the ICA sources (source separation) or orders the least-squares residuals into a causal DAG (causal discovery).

The paper also introduces practical algorithms together with finite-sample consistency guarantees. The implementations are available as two scikit-learn-compatible Python packages: otica (GitHub) for Optimal Transport ICA, and otlingam (GitHub) for Optimal Transport LiNGAM.

Preprint: https://arxiv.org/abs/2607.12832