Abstract
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, p1, p2>0 satisfying 1/p=1/p1+1/p2, the Schatten p-(quasi-)norm of any matrix is equivalent to minimizing the product of the Schatten p1-(quasi-)norm and Schatten p2-(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: p1=p2 and p1≠p2. In particular, when p>1/2, there is an equivalence between the Schatten p-quasi-norm of any matrix and the Schatten 2p-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.
Highlights
The affine rank minimization problem arises directly in various areas of science and engineering, including statistics, machine learning, information theory, data mining, medical imaging, and computer vision
When p > 1/2 and by setting the same value for p1 and p2, there is an equivalence between the Schatten p-(quasi-)norm of any matrix and the Schatten 2p-norms of its two factor matrices, where a representative example is the equivalent formulation of the nuclear norm, i.e., k X k∗ = min
The bi-nuclear and Frobenius/nuclear quasi-norms defined in our previous work [22] and the tri-nuclear quasi-norm defined in our previous work [29] are three important special cases of our unified scalable formulations for Schatten quasi-norms
Summary
The affine rank minimization problem arises directly in various areas of science and engineering, including statistics, machine learning, information theory, data mining, medical imaging, and computer vision. Some representative applications include matrix completion [1], robust principal component analysis (RPCA) [2], low-rank representation [3], multivariate regression [4], multi-task learning [5], and system identification [6] To efficiently solve such problems, we mainly relax the rank function to its tractable convex envelope, that is, the nuclear norm k · k∗ (sum of the singular values, known as the trace norm or Schatten 1-norm), which leads to a convex optimization problem [1,7,8,9]. Proposed a family of iteratively re-weighted nuclear norm (IRNN) algorithms to solve various non-convex surrogate (including the Schatten quasi-norm) minimization problems. The bi-nuclear and Frobenius/nuclear quasi-norms defined in our previous work [22] and the tri-nuclear quasi-norm defined in our previous work [29] are three important special cases of our unified scalable formulations for Schatten quasi-norms
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