[edit]
Convergence and Recovery Guarantees of the K-Subspaces Method for Subspace Clustering
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:22884-22918, 2022.
Abstract
The K-subspaces (KSS) method is a generalization of the K-means method for subspace clustering. In this work, we present local convergence analysis and a recovery guarantee for KSS, assuming data are generated by the semi-random union of subspaces model, where N points are randomly sampled from K≥2 overlapping subspaces. We show that if the initial assignment of the KSS method lies within a neighborhood of a true clustering, it converges at a superlinear rate and finds the correct clustering within Θ(loglogN) iterations with high probability. Moreover, we propose a thresholding inner-product based spectral method for initialization and prove that it produces a point in this neighborhood. We also present numerical results of the studied method to support our theoretical developments.