This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image, or sketch, of the matrix. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.
@cite_8: 
@cite_21: Randomized algorithms for very large matrix problems have received a great deal of attention in recent years. Much of this work was motivated by problems in large-scale data analysis, and this work was performed by individuals from many different research communities. This monograph will provide a detailed overview of recent work on the theory of randomized matrix algorithms as well as the application of those ideas to the solution of practical problems in large-scale data analysis. An emphasis will be placed on a few simple core ideas that underlie not only recent theoretical advances but also the usefulness of these tools in large-scale data applications. Crucial in this context is the connection with the concept of statistical leverage. This concept has long been used in statistical regression diagnostics to identify outliers; and it has recently proved crucial in the development of improved worst-case matrix algorithms that are also amenable to high-quality numerical implementation and that are useful to domain scientists. Randomized methods solve problems such as the linear least-squares problem and the low-rank matrix approximation problem by constructing and operating on a randomized sketch of the input matrix. Depending on the specifics of the situation, when compared with the best previously-existing deterministic algorithms, the resulting randomized algorithms have worst-case running time that is asymptotically faster; their numerical implementations are faster in terms of clock-time; or they can be implemented in parallel computing environments where existing numerical algorithms fail to run at all. Numerous examples illustrating these observations will be described in detail.
@cite_0: 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 holds with probability at least 1 − Δ (where v·vF is the Frobenius norm). The algorithm takes time polynomial in k,1 e, 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.
@cite_19: This survey highlights the recent advances in algorithms for numericallinear algebra that have come from the technique of linear sketching,whereby given a matrix, one first compresses it to a much smaller matrixby multiplying it by a (usually) random matrix with certain properties.Much of the expensive computation can then be performed onthe smaller matrix, thereby accelerating the solution for the originalproblem. In this survey we consider least squares as well as robust regressionproblems, low rank approximation, and graph sparsification.We also discuss a number of variants of these problems. Finally, wediscuss the limitations of sketching methods.
@cite_24: Abstract Given an m × n matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying A T to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient whenever A and A T can be applied rapidly to arbitrary vectors. The discrepancy between A and Z is of the same order as l m times the ( k + 1 ) st greatest singular value σ k + 1 of A, with negligible probability of even moderately large deviations. The actual estimates derived in the paper are fairly complicated, but are simpler when l − k is a fixed small nonnegative integer. For example, according to one of our estimates for l − k = 20 , the probability that the spectral norm ‖ A − Z ‖ is greater than 10 ( k + 20 ) m σ k + 1 is less than 10 − 17 . The paper contains a number of estimates for ‖ A − Z ‖ , including several that are stronger (but more detailed) than the preceding example; some of the estimates are effectively independent of m. Thus, given a matrix A of limited numerical rank, such that both A and A T can be applied rapidly to arbitrary vectors, the scheme provides a simple, efficient means for constructing an accurate approximation to a singular value decomposition of A. Furthermore, the algorithm presented here operates reliably independently of the structure of the matrix A. The results are illustrated via several numerical examples.
Randomized algorithms for matrix approximation date back to research @cite_8 @cite_0 in theoretical computer science (TCS) in the late 1990s. Starting around 2004, this work inspired numerical analysts to develop practical algorithms for matrix approximation and related problems @cite_24 . See the paper [Sec. 2] HMT11:Finding-Structure for a comprehensive historical discussion. The surveys @cite_21 @cite_19 provide more details about the development of these ideas within the TCS literature.