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Block coordinate descent is an optimization paradigm that iteratively updatesone block of variables at a time, making it quite amenable to big dataapplications due to its scalability and performance. Its convergence behaviorhas been extensively studied in the (block-wise) convex case, but it is muchless explored in the non-convex case. In this paper we analyze the convergenceof block coordinate methods on non-convex sets and derive convergence rates onsmooth manifolds under natural or weaker assumptions than prior work. Ouranalysis applies to many non-convex problems, including ones that seeklow-dimensional structures (e.g., maximal coding rate reduction, neuralcollapse, reverse engineering adversarial attacks, generalized PCA, alternatingprojection); ones that seek combinatorial structures (homomorphic sensing,regression without correspondences, real phase retrieval, robust pointmatching); ones that seek geometric structures from visual data (e.g.,essential matrix estimation, absolute pose estimation); and ones that seekinliers sparsely hidden in a large number of outliers (e.g., outlier-robustestimation via iteratively-reweighted least-squares). While our convergencetheory applies to all these problems, yielding novel corollaries, it alsoapplies to other, perhaps more familiar, problems (e.g., optimal transport,matrix factorization, Burer-Monteiro factorization), recovering previouslyknown results.

Block Coordinate Descent on Smooth Manifolds: Convergence Theory and Twenty-One Examples