We assume data sampled from a mixture of d-dimensional linear subspaces with spherically symmetric outliers. We study the recovery of the global l 0 subspace (i.e., with largest number of points) by minimizing the lp-averaged distances of data points from d-dimensional subspaces of R D , where 0 < p ∈ R. Unlike other lp minimization problems, this minimization is non-convex for all p > 0 and thus requires different methods for its analysis. We show that if 0 < p ≤ 1, then the global l 0 subspace can be recovered by lp minimization with overwhelming probability (which depends on the generating distribution and its parameters). Moreover, when adding homoscedastic noise around the underlying subspaces, then with overwhelming probability the generalized l 0 subspace (with largest number of points "around it") can be nearly recovered by lp minimization with an error proportional to the noise level. On the other hand, if p > 1 and there is more than one underlying subspace, then with overwhelming probability the global l 0 subspace cannot be recovered and the generalized one cannot even be nearly recovered.
G. Lerman and T. Zhang REF studied the problem by minimizing the p -averaged distance of data points from d-dimensional subspaces in high ambient dimensional space.
lp-Recovery of the Most Significant Subspace among Multiple Subspaces with Outliers
Constructive Approximation, December 2014, Volume 40, Issue 3, pp 329-385
Mathematics Computer Science