Unified Scalable Equivalent Formulations for Schatten Quasi-Norms

The Schatten quasi-norm can be used to bridge the gap between the nuclear norm and rank function. However, most existing algorithms are too slow or even impractical for large-scale problems, due to the singular value decomposition (SVD) or eigenvalue decomposition (EVD) of the whole matrix in each iteration. In this paper, we rigorously prove that for any 0< p≤ 1, p1, p2 > 0 satisfying 1/p= 1/p1+1/p2, the Schatten-p quasi-norm (or norm) of any matrix is equivalent to the minimization of the product of the Schatten-p1 norm (or quasi-norm) and Schatten-p2 norm (or quasi-norm) of its two factor matrices. Then we present and prove the equivalence relationship between the product formula of the Schatten quasi-norm and its sum formula for the two cases of p1 and p2: p1=p2 and p1 6=p2. Especially, when p>1/2, there is an equivalence between the Schatten-p quasi-norm (or norm) of any matrix and the Schatten-2p norms of its two factor matrices, where the widely used equivalent formulation of the nuclear norm, i.e., ‖X‖∗ = minX=UV T (‖U‖2F +‖V ‖ 2 F )/2, can be viewed as a special case. That is, various Schatten-p quasi-norm minimization problems with p>1/2 can be transformed into the one only involving the smooth, convex norms of two factor matrices, which can lead to simpler and more efficient algorithms than conventional methods. We further extend the theoretical results of two factor matrices to the cases of three and more factor matrices, from which we can see that for any 0<p<1, the Schatten-p quasi-norm of any matrix is the minimization of the mean to the power of (⌊1/p⌋+1) of the Schatten-(⌊1/p⌋+1)p norms of all factor matrices, where ⌊1/p⌋ denotes the largest integer not exceeding 1/p. In other words, for any 0<p<1, the Schatten-p quasi-norm minimization can be transformed into an optimization problem only involving the smooth, convex norms of multiple factor matrices. In addition, we also present some representative examples for two and three factor matrices. Naturally, the bi-nuclear and Frobenius/nuclear quasi-norms defined in our previous paper [1] and the tri-nuclear quasi-norm defined in our previous paper [2] are three important special cases.