Robust Subspace Clustering via Thresholding Ridge Regression

In this material, we provide the theoretical analyses to show that the trivial coefficients always correspond to the codes over errors. Lemmas 1–3 show that our errors-removing strategy will perform well when the lp-norm is enforced over the representation, where p = {1, 2,∞}. Let x 6= 0 be a data point in the union of subspaces SD that is spanned by D = [Dx D−x], where Dx and D−x consist of the intra-cluster and inter-cluster data points, respectively. Note that, noise and outlier could be regarded as a kind of inter-cluster data point of x. Without loss of generality, let SDx and SD−x be the subspace spanned by Dx and D−x, respectively. Hence, there are only two possibilities for the location of x, i.e., in the intersection between SDx and SD−x (denoted by x ∈ {S|S = SDx ∩ SD−x}), or in SDx except the intersection (denoted by x ∈ {S|S = SDx\SD−x}). Let c∗x and c ∗ −x be the optimal solutions of