New method recovers latent components from unlabeled mixtures using marginal independence

The paper addresses the classic problem of recovering latent components and mixing proportions from unlabeled mixtures—without ever seeing clean component samples or labels. The key innovation is leveraging marginal independence: each latent component is assumed to be independent on at least one pair of coordinates. The authors first prove a structural result: under linear independence of univariate marginals, any independent affine combination of product components must collapse to a single component. They then extend this to observable mixtures, showing that when the mixing matrix is full-rank and no cancelation occurs, marginally independent affine combinations recover the latent components. This yields identifiability of all components and the mixing matrix.

To turn theory into practice, the team proposes the Product-Marginal Maximum Mean Discrepancy (PM-MMD) estimator. It searches over affine combinations of the observable mixtures to find those that satisfy marginal independence, using kernel embeddings. They prove uniform convergence and stability even under approximate marginal independence. Experiments on controlled and flow-cytometry data demonstrate that PM-MMD recovers components more reliably than clustering, factorization, or pairwise mixture-proportion baselines. The framework also provides a held-out diagnostic—marginal independence is directly testable, unlike irreducibility assumptions. This work opens new avenues for unsupervised component analysis in biology, genomics, and any domain where unlabeled mixtures arise.