Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.

Towards more efficient SPSD matrix approximation and CUR matrix decomposition

Abstract: Symmetric positive semi-definite (SPSD) matrix approximation methods have been extensively used to speed up large-scale eigenvalue computation and kernel learning methods. The standard sketch based method, which we call the prototype model, produces relatively accurate approximations, but is inefficient on large square matrices. The Nystrom method is highly efficient, but can only achieve low accuracy. In this paper we propose a novel model that we call the fast SPSD matrix approximation model. The fast model is nearly as efficient as the Nystrom method and as accurate as the prototype model. We show that the fast model can potentially solve eigenvalue problems and kernel learning problems in linear time with respect to the matrix size n to achieve 1 + e relative-error, whereas both the prototype model and the Nystrom method cost at least quadratic time to attain comparable error bound. Empirical comparisons among the prototype model, the Nystrom method, and our fast model demonstrate the superiority of the fast model. We also contribute new understandings of the Nystrom method. The Nystrom method is a special instance of our fast model and is approximation to the prototype model. Our technique can be straightforwardly applied to make the CUR matrix decomposition more efficiently computed without much affecting the accuracy.

Singular Value Decomposition

Abstract: In this chapter we discuss reduction of matrices to the canonical form by use of orthogonal transformations in the spaces of images and preimages. Such canonical form is called the singular value decomposition. In what follows we will use the well-known polar decomposition, which is recalled in Section 1 in course of discussion of singular value decomposition of square matrices.