K-median clustering, model-based compressive sensing, and sparse recovery for earth mover distance

We initiate the study of sparse recovery problems under the Earth-Mover Distance (EMD). Specifically, we design a distribution over m x n matrices A such that for any x, given Ax, we can recover a k-sparse approximation to x under the EMD distance. One construction yields m=O(k log (n/k)) and a 1 + ε approximation factor, which matches the best achievable bound for other error measures, such as the l1 norm. Our algorithms are obtained by exploiting novel connections to other problems and areas, such as streaming algorithms for k-median clustering and model-based compressive sensing. We also provide novel algorithms and results for the latter problems.