SVD for Very Large Matrices: An Approach with Polar Decomposition and Polynomial Approximation

Abstract

We propose a fast approximation method for singular value decomposition (SVD) based on polar decomposition (PD), which utilizes Chebyshev polynomial approximation (CPA). SVD is widely used in various applications in data mining, machine learning, and signal/image processing. However, very large computation times are usually required when large matrices need to be decomposed. SVD based on PD (SVD_PD) is computationally efficient compared to the traditional SVD approach, as the eigenvalue decomposition in its algorithm can be easily parallelized. However, the PD part of the algorithm still needs to be accelerated due to the requirement of QR decomposition. This computational burden poses a problem both for the traditional SVD and SVD_PD. In this paper, we propose an approximation method for PD using CPA. It can bypass QR decomposition, thus leading to the acceleration of SVD_PD. First, we show that PD can be performed by filtering the eigenvalues of DTD, where D is the target matrix to be decomposed. Then, the actual algorithm and approximate error bound are shown. The proposed approximate SVD can be implemented only by multiplying the matrices; that is, QR decomposition is no longer required. As a result, the approximation of PD with the proposed method significantly reduces the computation cost and can be easily parallelized. Experimental results show that the proposed CPA-based SVD_PD can decompose a matrix faster than the conventional approach, and the application of the proposed method does not affect performance.