A Unified Scalable Equivalent Formulation for Schatten Quasi-Norms

The Schatten quasi-norm is an approximation of the rank, which is tighter than the nuclear norm. However, most Schatten quasi-norm minimization (SQNM) algorithms suffer from high computational cost to compute the singular value decomposition (SVD) of large matrices at each iteration. In this paper, we prove that for any p , p 1 , p 2 > 0 satisfying 1 / p = 1 / p 1 + 1 / p 2 , the Schatten p -(quasi-)norm of any matrix is equivalent to minimizing the product of the Schatten p 1 -(quasi-)norm and Schatten p 2 -(quasi-)norm of its two much smaller factor matrices. Then, we present and prove the equivalence between the product and its weighted sum formulations for two cases: p 1 = p 2 and p 1 ≠ p 2 . In particular, when p > 1 / 2 , there is an equivalence between the Schatten p -quasi-norm of any matrix and the Schatten 2 p -norms of its two factor matrices. 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 of the Schatten ( ⌊ 1 / p ⌋ + 1 ) p -norms of ⌊ 1 / p ⌋ + 1 factor matrices, where ⌊ 1 / p ⌋ denotes the largest integer not exceeding 1 / p .