Compressed Randomized UTV Decompositions for Low-Rank Matrix Approximations

Abstract

Low-rank matrix approximations play a fundamental role in numerical linear algebra and signal processing applications. This paper introduces a novel rank-revealing matrix decomposition algorithm termed Compressed Randomized UTV (CoR-UTV) decomposition along with a CoR-UTV variant aided by the power method technique. CoR-UTV is primarily developed to compute an approximation to a low-rank input matrix by making use of random sampling schemes. Given a large and dense matrix of size m × n with numerical rank k, where k ≪ min{m, n}, CoR-UTV requires a few passes over the data, and runs in O(mnk) floating-point operations. Furthermore, CoR-UTV can exploit modern computational platforms and, consequently, can be optimized for maximum efficiency. CoR-UTV is simple and accurate, and outperforms reported alternative methods in terms of efficiency and accuracy. Simulations with synthetic data as well as real data in image reconstruction and robust principal component analysis applications support our claims.

Projection-based QLP Algorithm for Efficiently Computing Low-Rank Approximation of Matrices

“…Lastly, a truncated SVD follows to construct an approximate SVD of the given matrix. The work in [5] presented a rank-revealing algorithm using the randomized scheme termed compressed randomized UTV (CoR-UTV) decomposition. This algorithm is described in Algorithm 2.…”

“…The accuracy of SRQR can match that of CPQR. (For a comparison of CPQR, the SVD and randomized methods in terms of approximation accuracy, see, e.g., [20], [5].) The authors in [46] presented the Flip-Flop SRQR factorization algorithm.…”

“…M ATRICES with low-rank structure are omnipresent in signal processing, data analysis, scientific computing and machine learning applications including system identification [1], subspace clustering [2], matrix completion [3], background subtraction [4], [5], least-squares regression [6], hyperspectral imaging [7], [8], anomaly detection [9], [10], subspace estimation over networks [11], genomics [12], [13], tensor decompositions [14], and sparse matrix problems [15].…”

“…Randomized methods [23], [24], [25], [6], [26], [5], [27], [28], [29], [22] provide an approximation to the foregoing deterministic decompositions. The general strategy of this type of methods is as follows: first, the input matrix is transformed to a lower-dimensional space by means of random sampling.…”

Compressed Randomized Utv Decompositions for Low-rank Matrix Approximations in Data Science

“…In this section, we present a randomized rank-revealing decomposition algorithm termed compressed randomized UTV (CoR-UTV) decomposition [44], which computes a low-rank approximation of a given matrix. We focus on the matrix A with m ≥ n, where CoR-UTV, in the form of (1), produces an upper triangular middle matrix T. The modifications required for a CoR-UTV for the case m < n that produces a lower triangular middle matrix T is straightforward.…”