Efficient volume sampling for row/column subset selection

We give efficient algorithms for volume sampling, i.e., for picking $k$-subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala. Our first algorithm for volume sampling $k$-subsets of rows from an $m$-by-$n$ matrix runs in $O(kmn^{\omega} \log n)$ arithmetic operations and a second variant of it for $(1+\epsilon)$-approximate volume sampling runs in $O(mn \log m \cdot k^{2}/\epsilon^{2} + m \log^{\omega} m \cdot k^{2\omega+1}/\epsilon^{2\omega} \cdot \log(k \epsilon^{-1} \log m))$ arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small $k$. Our efficient volume sampling algorithms imply several interesting results for low-rank matrix approximation.