SVD and application 奇异值分解及其应用

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文件名称：SVD and application 奇异值分解及其应用
文件大小：744KB
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更新时间：2021-07-25 03:47:45
SVD 奇异值分解 The singular value decomposition (SVD) is not only a classical theory in matrix computation and analysis, but also is a powerful tool in ma- chine learning and modern data analysis. In this tutorial we first study the basic notion of SVD and then show the central role of SVD in ma- trices. Using majorization theory, we consider variational principles of singular values and eigenvalues. Built on SVD and a theory of sym- metric gauge functions, we discuss unitarily invariant norms, which are then used to formulate general results for matrix low rank approxima- tion. We study the subdifferentials of unitarily invariant norms. These results would be potentially useful in many machine learning problems such as matrix completion and matrix data classification. Finally, we discuss matrix low rank approximation and its recent developments such as randomized SVD, approximate matrix multiplication, CUR decomposition, and Nyström approximation. Randomized algorithms are important approaches to large scale SVD as well as fast matrix computations.