SVD update methods for large matrices and applications

Abstract

We consider the problem of updating the SVD when augmenting a "tall thin" matrix, i.e., a rectangular matrix A ∈ R m×n with m n. Supposing that an SVD of A is already known, and given a matrix B ∈ R m×n , we derive an efficient method to compute and efficiently store the SVD of the augmented matrix [AB] ∈ R m×(n+n ) . This is an important tool for two types of applications: in the context of principal component analysis, the dominant left singular vectors provided by this decomposition form an orthonormal basis for the best linear subspace of a given dimension, while from the right singular vectors one can extract an orthonormal basis of the kernel of the matrix. We also describe two concrete applications of these concepts which motivated the development of our method and to which it is very well adapted.

Background Subtraction using Adaptive Singular Value Decomposition

“…Our background subtraction method is based on an iterative calculation of an SVD for matrices augmented by columns, cf. [23]. In this section, we revise the essential statements and the advantages of using this method for calculating the SVD.…”

“…For details, see [23]. In the original version of the iterative SVD, the matrix U k is (formally) of dimension d × d.…”

“…The algorithm in [23] was initially designed with the goal to determine the kernels of a sequence of columnwise augmented matrices using the V matrix of the SVDs. In background subtraction, on the other hand, we are interested in finding a low-rank approximation of the column space of A and therefore concentrate on the U matrix of the SVD which will allow us to avoid the computation and storage of V .…”

“…3.1. The details of this algorithm based on Householder representation can be found in [23]. By the thresholding procedure from Sect.…”

Signal de-noising in gear pitting fault identification by an improved singular value decomposition method

“…It retains the effective signal value and rid the noise signal value, then reconstructs the effective signal in a suitable order. Essentially, SVD is a method of matrix orthogonalization in the mathematical view, which decomposes the given matrix into two matrixes U m×m and V n×n , as shown: [29][30][31][32].…”