Supplementary Material of Noisy Sparse Subspace Clustering

Our main deterministic result Theorem 1 is proved by duality. We first establish a set of conditions on the optimal dual variable of D 0 corresponding to all primal solutions satisfying self-expression property. Then we construct such a dual variable ν, hence certify that the optimal solution of P 0 satisfies the LASSO Subspace Detection Property. Define general convex optimization: min c,e c 1 + λ 2 e 2 s.t. x = Ac + e. (A.1) We may state an extension of the Lemma 7.1 in Soltanolkotabi & Candes's SSC Proof. Lemma A.1. Consider a vector y ∈ R d and a matrix A ∈ R d×N. If there exists triplet (c, e, ν) obeying y = Ac + e and c has support S ⊆ T , furthermore the dual certificate vector ν satisfies A T s ν = sgn(c S), ν = λe, A T T ∩S c ν ∞ ≤ 1, A T T c ν ∞ < 1, then all optimal solution (c * , e *) to (A.1) obey c * T c = 0.