K-Median Clustering, Model-Based Compressive Sensing, and Sparse Recovery for Earth Mover Distance

TL;DR

This paper introduces new methods for sparse recovery under the Earth Mover Distance, achieving near-optimal bounds by leveraging connections to clustering and compressive sensing.

Contribution

It presents a novel approach to sparse recovery under EMD with a distribution over matrices that achieves optimal bounds, connecting to clustering and compressive sensing.

Findings

Achieves m = O(k log(n/k)) with 1 + epsilon approximation for EMD

Develops algorithms linking EMD recovery to clustering and compressive sensing

Provides new algorithms for model-based compressive sensing

Abstract

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 + epsilon approximation factor, which matches the best achievable bound for other error measures, such as the L_1 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.