Split-and-Combine Singular Value Decomposition for Large-Scale Matrix

Abstract

The singular value decomposition (SVD) is a fundamental matrix decomposition in linear algebra. It is widely applied in many modern techniques, for example, high- dimensional data visualization, dimension reduction, data mining, latent semantic analysis, and so forth. Although the SVD plays an essential role in these fields, its apparent weakness is the order three computational cost. This order three computational cost makes many modern applications infeasible, especially when the scale of the data is huge and growing. Therefore, it is imperative to develop a fast SVD method in modern era. If the rank of matrix is much smaller than the matrix size, there are already some fast SVD approaches. In this paper, we focus on this case but with the additional condition that the data is considerably huge to be stored as a matrix form. We will demonstrate that this fast SVD result is sufficiently accurate, and most importantly it can be derived immediately. Using this fast method, many infeasible modern techniques based on the SVD will become viable.

A novel recursive method for computing the null-space of random matrices of arbitrary size

“…Eq. (14) automatically leads us to this result. By setting the stepsize less that the trace of the matrix , Eq.…”

“…The step-size parameter is selected for the misadjustment values 10%, 20%, and 30% according to the Eq. (14). [ ] is initialized to zero plus a small constant to avoid division by zero.…”

“…By setting the stepsize less that the trace of the matrix , Eq. (14) guarantees both the convergence and the stability of the method.…”

“…SVD decomposes a rectangular matrix into diagonal matrix ∑ of singular values and two orthonromal matrices U and V that contain the null-space basis corresponding to the zero singular values. SVD is computationally expensive [14]. Krylov methods are iterative methods to solve large scale systems and take advantage of the sparsity of the matrix to generate Krylov sub-spaces and find a null-space solution therein.…”

Fast approximate truncated SVD

“…Several stochastic methods were proposed during last decade (see the papers and references within). The best one known to the authors (referred, henceforth, as Randomized algorithm ) requires operations .…”