  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 $K \ge 2$
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 $\Theta(\log\log N)$ 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.
