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Fast Monte-Carlo algorithms for finding low-rank approximations. (English) Zbl 1125.65005

Summary: We consider the problem of approximating a given m×n matrix 𝐀 by another matrix of specified rank k, which is smaller than m and n. The singular value decomposition (SVD) can be used to find the ‘best’ such approximation. However, it takes time polynomial in m, n which is prohibitive for some modern applications. In this article, we develop an algorithm that is qualitatively faster, provided we may sample the entries of the matrix in accordance with a natural probability distribution. In many applications, such sampling can be done efficiently. Our main result is a randomized algorithm to find the description of a matrix 𝐃 * of rank at most k so that

𝐀-𝐃 * F 2 min 𝐃,rank(𝐃)k 𝐀-𝐃 F 2 +ε𝐀 F 2

holds with probability at least 1-δ (where · F is the Frobenius norm). The algorithm takes time polynomial in k, 1/ε, log(1/δ) only and is independent of m and n. In particular, this implies that in constant time, it can be determined if a given matrix of arbitrary size has a good low-rank approximation.


MSC:
65C05Monte Carlo methods
65F30Other matrix algorithms
68P20Information storage and retrieval
68W20Randomized algorithms
68W40Analysis of algorithms