A Unified Scalable Equivalent Formulation for Schatten Quasi-Norms
| Autor: | Fanhua Shang, Yuanyuan Liu, Fanjie Shang, Hongying Liu, Lin Kong, Licheng Jiao |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Factor matrix
equivalent formulations General Mathematics lcsh:Mathematics Matrix norm 020206 networking & telecommunications 02 engineering and technology factor matrix lcsh:QA1-939 Combinatorics Matrix (mathematics) Large matrices Norm (mathematics) Singular value decomposition Scalability 0202 electrical engineering electronic engineering information engineering Computer Science (miscellaneous) rank function 020201 artificial intelligence & image processing Schatten quasi-norm Minification nuclear norm Engineering (miscellaneous) Mathematics |
| Zdroj: | Mathematics, Vol 8, Iss 1325, p 1325 (2020) Mathematics Volume 8 Issue 8 |
| ISSN: | 2227-7390 |
| Popis: | 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&ne 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 (&lfloor 1/p&rfloor +1)p-norms of &lfloor +1 factor matrices, where &lfloor denotes the largest integer not exceeding 1/p. |
| Databáze: | OpenAIRE |
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