Robust recovery of multiple subspaces by geometric lp minimization

Abstract

We assume i.i.d. data sampled from a mixture distribution with K components along fixed d-dimensional linear subspaces and an additional outlier component. For p > 0, we study the simultaneous recovery of the K fixed subspaces by minimizing the l_p-averaged distances of the sampled data points from any K subspaces. Under some conditions, we show that if 0 < p ≤ 1, then all underlying subspaces can be precisely recovered by l_p minimization with overwhelming probability. On the other hand, if K > 1 and p > 1, then the underlying subspaces cannot be recovered or even nearly recovered by l_p minimization. The results of this paper partially explain the successes and failures of the basic approach of l_p energy minimization for modeling data by multiple subspaces.