Quantization for Low-Rank Matrix Recovery

TL;DR

This paper develops quantization and reconstruction methods for low-rank matrix sampling, achieving polynomial decay in error, exponential accuracy, and robustness to noise, extending classical signal acquisition results.

Contribution

It generalizes quantization techniques from signal and compressed sensing to low-rank matrix recovery, providing error decay rates and near-optimal bit-rate schemes.

Findings

Reconstruction error decays polynomially with oversampling

Achieves root-exponential accuracy through optimized quantization

Random encoding yields near-optimal exponential bit-rate

Abstract

We study Sigma-Delta quantization methods coupled with appropriate reconstruction algorithms for digitizing randomly sampled low-rank matrices. We show that the reconstruction error associated with our methods decays polynomially with the oversampling factor, and we leverage our results to obtain root-exponential accuracy by optimizing over the choice of quantization scheme. Additionally, we show that a random encoding scheme, applied to the quantized measurements, yields a near-optimal exponential bit-rate. As an added benefit, our schemes are robust both to noise and to deviations from the low-rank assumption. In short, we provide a full generalization of analogous results, obtained in the classical setup of bandlimited function acquisition, and more recently, in the finite frame and compressed sensing setups to the case of low-rank matrices sampled with sub-Gaussian linear operators. Finally, we believe our techniques for generalizing results from the compressed sensing setup to the analogous low-rank matrix setup is applicable to other quantization schemes.