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We assume data sampled from a mixture of d-dimensional linear subspaces withspherically symmetric distributions within each subspace and an additionaloutlier component with spherically symmetric distribution within the ambientspace (for simplicity we may assume that all distributions are uniform on theircorresponding unit spheres). We also assume mixture weights for the differentcomponents. We say that one of the underlying subspaces of the model is mostsignificant if its mixture weight is higher than the sum of the mixture weightsof all other subspaces. We study the recovery of the most significant subspaceby minimizing the lp-averaged distances of data points from d-dimensionalsubspaces, where p>0. Unlike other lp minimization problems, this minimizationis non-convex for all p>0 and thus requires different methods for its analysis.We show that if 0<p<=1, then for any fraction of outliers the most significantsubspace can be recovered by lp minimization with overwhelming probability(which depends on the generating distribution and its parameters). We show thatwhen adding small noise around the underlying subspaces the most significantsubspace can be nearly recovered by lp minimization for any 0<p<=1 with anerror proportional to the noise level. On the other hand, if p>1 and there ismore than one underlying subspace, then with overwhelming probability the mostsignificant subspace cannot be recovered or nearly recovered. This last resultdoes not require spherically symmetric outliers.

lp-Recovery of the Most Significant Subspace among Multiple Subspaces with Outliers