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Low-rank Matrix Sensing With Dithered One-Bit Quantization

Farhang Yeganegi, Arian Eamaz, Mojtaba Soltanalian

TL;DR

This work addresses recovering a low-rank matrix from highly quantized, dithered one-bit measurements. It develops SVP-RKA, a hybrid of the Randomized Kaczmarz algorithm and Singular Value Projection, to solve the resulting overdetermined one-bit polyhedron under a rank constraint. The authors prove a uniform reconstruction guarantee with Gaussian sensing and dithering, derive a linear convergence rate for SVP-RKA to a neighborhood of the true matrix, and corroborate these results with extensive numerical experiments showing superior performance to HSVT. Practically, the approach enables accurate low-rank recovery at extreme quantization while using fewer samples, with potential benefits for high-rate, energy-efficient sensing systems.

Abstract

We explore the impact of coarse quantization on low-rank matrix sensing in the extreme scenario of dithered one-bit sampling, where the high-resolution measurements are compared with random time-varying threshold levels. To recover the low-rank matrix of interest from the highly-quantized collected data, we offer an enhanced randomized Kaczmarz algorithm that efficiently solves the emerging highly-overdetermined feasibility problem. Additionally, we provide theoretical guarantees in terms of the convergence and sample size requirements. Our numerical results demonstrate the effectiveness of the proposed methodology.

Low-rank Matrix Sensing With Dithered One-Bit Quantization

TL;DR

This work addresses recovering a low-rank matrix from highly quantized, dithered one-bit measurements. It develops SVP-RKA, a hybrid of the Randomized Kaczmarz algorithm and Singular Value Projection, to solve the resulting overdetermined one-bit polyhedron under a rank constraint. The authors prove a uniform reconstruction guarantee with Gaussian sensing and dithering, derive a linear convergence rate for SVP-RKA to a neighborhood of the true matrix, and corroborate these results with extensive numerical experiments showing superior performance to HSVT. Practically, the approach enables accurate low-rank recovery at extreme quantization while using fewer samples, with potential benefits for high-rate, energy-efficient sensing systems.

Abstract

We explore the impact of coarse quantization on low-rank matrix sensing in the extreme scenario of dithered one-bit sampling, where the high-resolution measurements are compared with random time-varying threshold levels. To recover the low-rank matrix of interest from the highly-quantized collected data, we offer an enhanced randomized Kaczmarz algorithm that efficiently solves the emerging highly-overdetermined feasibility problem. Additionally, we provide theoretical guarantees in terms of the convergence and sample size requirements. Our numerical results demonstrate the effectiveness of the proposed methodology.
Paper Structure (10 sections, 2 theorems, 30 equations, 1 figure)

This paper contains 10 sections, 2 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. One-Bit Quantization With Multiple Varying Thresholds
  4. One-Bit Low-Rank Matrix Sensing
  5. SVP-RKA
  6. Theoretical Guarantees
  7. Uniform Reconstruction Guarantee
  8. Convergence Guarantee for SVP-RKA
  9. Numerical Results
  10. Summary

Key Result

Proposition 1

Let each element of $\mathbf{A}_j\in\mathbb{R}^{n_1\times n_2}$ and each $\tau_j$ for $j\in[n]$ be independently drawn from the standard normal distribution. If $n\geq C\delta^{-4}(n_1+n_2)r$, then with probability at least $1-2e^{-c\delta^4n}$, all $\mathbf{X},\mathbf{X}^{\prime}\in\bar{\mathcal{K} satisfy $\operatorname{vec}(\mathbf{X}^{\prime})\in\mathcal{B}_{\frac{\delta}{4}}(\operatorname{vec

Figures (1)

  • Figure 1: Comparison between the recovery performance of SVP-RKA and HSVT algorithm over different values of oversampling factor $\lambda$.

Theorems & Definitions (2)

  • Proposition 1: Random Hyperplane Tessellations of $\bar{\mathcal{K}}_{n_1,n_2,r}$
  • Lemma 1