Federated Independent Component Analysis via Spectral Alignment and Robust Aggregation

This paper studies robust estimation for Independent Component Analysis (ICA) in the federated learning setting, where data are locally distributed across clients and may exhibit substantial heterogeneity. The goal is to recover a common global mixing matrix by aggregating local estimators computed by individual clients. The main difficulty is that local ICA estimators are identifiable only up to sign flips and column permutations and may have highly heterogeneous estimation quality. We propose a novel three-step aggregation method that first aligns the permutations across all local estimators via a particular spectral clustering approach, then aligns the signs within each estimated cluster via another tailored spectral approach, and finally applies the geometric median for robust aggregation. The proposed estimator is shown to remain accurate even when a substantial fraction of local estimators are of low quality or inconsistent, as long as each cluster contains a majority of accurate estimators. This contrasts with its analogue based on simple averaging, whose performance is determined by the worst local estimator. In the homogeneous setting, the robust estimator is also shown to be minimax optimal, up to a logarithmic factor, both when the number of clients remains fixed and when it diverges. The theoretical findings are corroborated by simulation studies and a real-data analysis.