Nearly-optimal bounds for sparse recovery in generic norms, with applications to $k$-median sketching

We initiate the study of trade-offs between sparsity and the number of measurements in sparse recovery schemes for generic norms. Specifically, for a norm $\|\cdot\|$, sparsity parameter $k$, approximation factor $K>0$, and probability of failure $P>0$, we ask: what is the minimal value of $m$ so that there is a distribution over $m \times n$ matrices $A$ with the property that for any $x$, given $Ax$, we can recover a $k$-sparse approximation to $x$ in the given norm with probability at least $1-P$? We give a partial answer to this problem, by showing that for norms that admit efficient linear sketches, the optimal number of measurements $m$ is closely related to the doubling dimension of the metric induced by the norm $\|\cdot\|$ on the set of all $k$-sparse vectors. By applying our result to specific norms, we cast known measurement bounds in our general framework (for the $\ell_p$ norms, $p \in [1,2]$) as well as provide new, measurement-efficient schemes (for the Earth-Mover Distance norm). The latter result directly implies more succinct linear sketches for the well-studied planar $k$-median clustering problem. Finally, our lower bound for the doubling dimension of the EMD norm enables us to address the open question of [Frahling-Sohler, STOC'05] about the space complexity of clustering problems in the dynamic streaming model.