Algorithmic linear dimension reduction in the l_1 norm for sparse vectors

TL;DR

This paper introduces a fast, uniform, and space-efficient algorithm for sparse signal recovery in the l_1 norm, enabling near-optimal dimension reduction and robust reconstruction with polylogarithmic distortion.

Contribution

It presents a novel sublinear-time reconstruction algorithm for sparse signals with near-optimal measurement complexity and a small-space implementation for efficient dimension reduction in the l_1 norm.

Findings

Reconstruction algorithm runs in O(m log^2(m) log^2(d)) time.

Uses O(m log^2(d)) measurements, close to optimal.

Achieves stable and robust recovery with polylogarithmic distortion.

Abstract

This paper develops a new method for recovering m-sparse signals that is simultaneously uniform and quick. We present a reconstruction algorithm whose run time, O(m log^2(m) log^2(d)), is sublinear in the length d of the signal. The reconstruction error is within a logarithmic factor (in m) of the optimal m-term approximation error in l_1. In particular, the algorithm recovers m-sparse signals perfectly and noisy signals are recovered with polylogarithmic distortion. Our algorithm makes O(m log^2 (d)) measurements, which is within a logarithmic factor of optimal. We also present a small-space implementation of the algorithm. These sketching techniques and the corresponding reconstruction algorithms provide an algorithmic dimension reduction in the l_1 norm. In particular, vectors of support m in dimension d can be linearly embedded into O(m log^2 d) dimensions with polylogarithmic distortion. We can reconstruct a vector from its low-dimensional sketch in time O(m log^2(m) log^2(d)). Furthermore, this reconstruction is stable and robust under small perturbations.