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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

$\parallel 𝐀-{𝐃}^{*}{\parallel }_{F}^{2}\le \underset{𝐃,\text{rank}\phantom{\rule{0.166667em}{0ex}}\left(𝐃\right)\le k}{min}{\parallel 𝐀-𝐃\parallel }_{F}^{2}+\epsilon {\parallel 𝐀\parallel }_{F}^{2}$

holds with probability at least $1-\delta$ (where ${\parallel ·\parallel }_{F}$ is the Frobenius norm). The algorithm takes time polynomial in $k$, $1/\epsilon$, $log\left(1/\delta \right)$ 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:
 65C05 Monte Carlo methods 65F30 Other matrix algorithms 68P20 Information storage and retrieval 68W20 Randomized algorithms 68W40 Analysis of algorithms