Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms.

Abstract

We study the problem of residual error estimation for matrix and vector norms using a linear sketch. Such estimates can be used, for example, to quickly assess how useful a more expensive low-rank approximation computation will be. The matrix case concerns the Frobenius norm and the task is to approximate the $k$ -residual ${\left(A-A_k\right)}_F$ of the input matrix $A$ within a $(1+\epsilon )$ -factor, where $A_k$ is the optimal rank- $k$ approximation. We provide a tight bound of $\Theta \left(k^2/{\epsilon }^4\right)$ on the size of bilinear sketches, which have the form of a matrix product $SAT$ . This improves the previous $O\left(k^2/{\epsilon }^6\right)$ upper bound in (Andoni et al. SODA 2013) and gives the first non-trivial lower bound, to the best of our knowledge. In our algorithm, our sketching matrices $S$ and $T$ can both be sparse matrices, allowing for a very fast update time. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. For the vector case, we consider the ${\ell }_p$ -norm for $p>2$ , where the task is to approximate the $k$ -residual ${\left(x-x_k\right)}_p$ up to a constant factor, where $x_k$ is the optimal $k$ -sparse approximation to $x$ . Such vector norms are frequently studied in the data stream literature and are useful for finding frequent items or so-called heavy hitters. We establish an upper bound of $O\left(k^{2/p}{n}^{1-2/p}\right)$ poly $\left(\left(\text{log}n\right)\right)$ for constant $\epsilon$ on the dimension of a linear sketch for this problem. Our algorithm can be extended to the ${\ell }_p$ sparse recovery problem with the same sketching dimension, which seems to be the first such bound for $p>2$ . We also show an $\Omega \left(k^{2/p}{n}^{1-2/p}\right)$ lower bound for the sparse recovery problem, which is tight up to a poly $\left(\text{log}n\right)$ factor.