Fast low rank approximations of matrices and tensors

In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given $m\times n$ matrix $A$ by a matrix $B$ of rank at most $k$ which is much smaller than $m$ and $n$. The best rank $k$ approximation can be determined via the singular value decomposition which, however, has prohibitively high computational complexity and storage requirements for very large $m$ and $n$. We present an optimal least squares algorithm for computing a rank $k$ approximation to an $m\times n$ matrix $A$ by reading only a limited number of rows and columns of $A$. The algorithm has complexity $\mathcal O(k^2\max(m,n))$ and allows to iteratively improve given rank $k$ approximations by reading additional rows and columns of $A$. We also show how this approach can be extended to tensors and present numerical results.