From Quasi-Isometric Embeddings to Finite-Volume Property: A Theoretical Framework for Quantized Matrix Completion

TL;DR

This paper develops a theoretical framework for quantized matrix completion, analyzing the effects of scalar quantization and dithering on nuclear norm minimization recovery, with guarantees and robustness insights.

Contribution

It provides the first comprehensive theoretical guarantees for low-rank matrix recovery under coarse quantization and dithering schemes without regularization.

Findings

Theoretical guarantees for quantized matrix completion with dithering.

Analysis of sign flips and prequantization noise effects.

Recovery performance bounds under various quantization scenarios.

Abstract

We delve into the impact of memoryless scalar quantization on matrix completion. Our primary motivation for this research is to evaluate the recovery performance of nuclear norm minimization in handling quantized matrix problems without the use of any regularization terms such as those stemming from maximum likelihood estimation. We broaden our theoretical discussion to encompass the coarse quantization scenario with a dithering scheme, where the only available information for low-rank matrix recovery is a few-bit low-resolution data. We furnish theoretical guarantees for both scenarios: when access to dithers is available during the reconstruction process, and when we have access solely to the statistical properties of the dithers. Additionally, we conduct a comprehensive analysis of the effects of sign flips and prequantization noise on the recovery performance, particularly when the impact of sign flips is quantified using the well-known Hamming distance in the upper bound of recovery error.