Quantization for low-rank matrix recovery

Abstract

We study Sigma-Delta (ΣΔ) 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 rootexponential 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 band-limited 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.