A SPARSE DECOMPOSITION OF LOW RANK SYMMETRIC POSITIVE SEMIDEFINITE MATRICES

作者：Hou Thomas Y^*; Li Qin; Zhang Pengchuan

来源：Multiscale Modeling and Simulation, 2017, 15(1): 410-444.

DOI：10.1137/16M107760X

• intrinsic sparse mode decomposition
• principal component analysis
• sparse PCA
• joint diagonalization
• pivoted Cholesky decomposition
• matrix factorization

Suppose that A is an element of R-N x N is symmetric positive semidefinite with rank K <= N. Our goal is to decompose A into K rank-one matrices Sigma(K)(k-1) gkgT(k) where the modes {gk}(K)(k=1) are required to be as sparse as possible. In contrast to eigendecomposition, these sparse modes are not required to be orthogonal. Such a problem arises in random field parametrization where A is the covariance function and is intractable to solve in general. In this paper, we partition the indices from 1 to N into several patches and propose to quantify the sparseness of a vector by the number of patches on which it is nonzero, which is called patchwise sparseness. Our aim is to find the decomposition which minimizes the total patchwise sparseness of the decomposed modes. We propose a domain decomposition type method, called intrinsic sparse mode decomposition (ISMD), which follows the "local-modes-construction broken vertical bar patching-up" procedure. The key step in the ISMD is to construct local pieces of the intrinsic sparse modes by a joint diagonalization problem. Thereafter, a pivoted Cholesky decomposition is utilized to glue these local pieces together. Optimal sparse decomposition, consistency with different domain decomposition, and robustness to small perturbation are proved under the so-called regular-sparse assumption (see Definition 1.2). We provide simulation results to show the efficiency and robustness of the ISMD. We also compare the ISMD to other existing methods, e.g., eigendecomposition, pivoted Cholesky decomposition, and convex relaxation of sparse principal component analysis [R. LAI, J. LU, AND S. OSHER, Comm. Math. Sci., to appear; V. Q. Vu, J. CHO, J. LEI, AND K. ROHE, Fantope projection and selection: A near-optimal convex relaxation of sparse PCA, in Proceedings in Advances in Neural Information Processing Systems 26, 2013, pp. 2670-2678].

• 出版日期2017