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Matrix Completion from One-Bit Dither Samples

Arian Eamaz, Farhang Yeganegi, Mojtaba Soltanalian

TL;DR

The paper tackles recovering a low-rank matrix from dithered one-bit samples generated by time-varying thresholds, reframing one-bit matrix completion as a nuclear-norm minimization with linear inequality constraints. It introduces the One-Bit SVT (OB-SVT) family of algorithms, including deterministic and randomized variants, plus an adaptive-thresholding scheme via the Bregman iterative method, to efficiently enforce the one-bit inequalities while promoting low-rank structure. The authors provide recovery guarantees and convergence analyses, showing improved performance over existing MLE-based methods, especially when leveraging multiple dithers and sketching. Empirical results demonstrate robust recovery under noise and across dithering schemes, with significant reductions in computation time due to randomized sketching and efficient SVT updates, indicating strong practical potential for high-rate, coarse-quantized matrix sensing tasks.

Abstract

We explore the impact of coarse quantization on matrix completion in the extreme scenario of dithered one-bit sensing, where the matrix entries are compared with time-varying threshold levels. In particular, instead of observing a subset of high-resolution entries of a low-rank matrix, we have access to a small number of one-bit samples, generated as a result of these comparisons. In order to recover the low-rank matrix using its coarsely quantized known entries, we begin by transforming the problem of one-bit matrix completion (one-bit MC) with time-varying thresholds into a nuclear norm minimization problem. The one-bit sampled information is represented as linear inequality feasibility constraints. We then develop the popular singular value thresholding (SVT) algorithm to accommodate these inequality constraints, resulting in the creation of the One-Bit SVT (OB-SVT). Our findings demonstrate that incorporating multiple time-varying sampling threshold sequences in one-bit MC can significantly improve the performance of the matrix completion algorithm. In pursuit of achieving this objective, we utilize diverse thresholding schemes, namely uniform, Gaussian, and discrete thresholds. To accelerate the convergence of our proposed algorithm, we introduce three variants of the OB-SVT algorithm. Among these variants is the randomized sketched OB-SVT, which departs from using the entire information at each iteration, opting instead to utilize sketched data. This approach effectively reduces the dimension of the operational space and accelerates the convergence. We perform numerical evaluations comparing our proposed algorithm with the maximum likelihood estimation method previously employed for one-bit MC, and demonstrate that our approach can achieve a better recovery performance.

Matrix Completion from One-Bit Dither Samples

TL;DR

The paper tackles recovering a low-rank matrix from dithered one-bit samples generated by time-varying thresholds, reframing one-bit matrix completion as a nuclear-norm minimization with linear inequality constraints. It introduces the One-Bit SVT (OB-SVT) family of algorithms, including deterministic and randomized variants, plus an adaptive-thresholding scheme via the Bregman iterative method, to efficiently enforce the one-bit inequalities while promoting low-rank structure. The authors provide recovery guarantees and convergence analyses, showing improved performance over existing MLE-based methods, especially when leveraging multiple dithers and sketching. Empirical results demonstrate robust recovery under noise and across dithering schemes, with significant reductions in computation time due to randomized sketching and efficient SVT updates, indicating strong practical potential for high-rate, coarse-quantized matrix sensing tasks.

Abstract

We explore the impact of coarse quantization on matrix completion in the extreme scenario of dithered one-bit sensing, where the matrix entries are compared with time-varying threshold levels. In particular, instead of observing a subset of high-resolution entries of a low-rank matrix, we have access to a small number of one-bit samples, generated as a result of these comparisons. In order to recover the low-rank matrix using its coarsely quantized known entries, we begin by transforming the problem of one-bit matrix completion (one-bit MC) with time-varying thresholds into a nuclear norm minimization problem. The one-bit sampled information is represented as linear inequality feasibility constraints. We then develop the popular singular value thresholding (SVT) algorithm to accommodate these inequality constraints, resulting in the creation of the One-Bit SVT (OB-SVT). Our findings demonstrate that incorporating multiple time-varying sampling threshold sequences in one-bit MC can significantly improve the performance of the matrix completion algorithm. In pursuit of achieving this objective, we utilize diverse thresholding schemes, namely uniform, Gaussian, and discrete thresholds. To accelerate the convergence of our proposed algorithm, we introduce three variants of the OB-SVT algorithm. Among these variants is the randomized sketched OB-SVT, which departs from using the entire information at each iteration, opting instead to utilize sketched data. This approach effectively reduces the dimension of the operational space and accelerates the convergence. We perform numerical evaluations comparing our proposed algorithm with the maximum likelihood estimation method previously employed for one-bit MC, and demonstrate that our approach can achieve a better recovery performance.
Paper Structure (26 sections, 10 theorems, 123 equations, 1 figure, 1 table)

This paper contains 26 sections, 10 theorems, 123 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Prior Arts
  3. Motivation and Contributions of the Paper
  4. Organization and Notation
  5. One-Bit Matrix Completion
  6. Dithered One-Bit Sensing
  7. One-Bit MC as Nuclear Norm Minimization Problem
  8. One-Bit MC With Noisy Entries
  9. Proposed Algorithms
  10. OB-SVT-I
  11. OB-SVT-II
  12. Randomized OB-SVT
  13. Adaptive Thresholding With Bregman Iterative Method
  14. Theoretical Discussion
  15. One-Bit MC Theoretical Guarantees
  16. ...and 11 more sections

Key Result

Theorem 1

Consider the set Suppose $m^{\prime}$ entries of $\mathbf{X}$ with locations sampled uniformly at random are compared with $m$ sequences of uniform thresholds generated as $\tau^{(\ell)}_{i,j}\sim \mathcal{U}_{\left[-\alpha,\alpha\right]}$ for all $(i,j)\in\Omega,\ell\in[m]$, resulting in the observed one-bit data. where the required number of samples must satisfy

Figures (1)

  • Figure 1: (a) Comparison between the reconstruction performance of OB-SVT-I, OB-SVT-II and the MLE davenport20141 in terms of the relative error when a $500\times 500$ matrix $\mathbf{X}$ with the rank $r=10$ is corrupted by Gaussian noise. (b) Comparison between the reconstruction performance of OB-SVT-I, OB-SVT-II and the MLE davenport20141 in terms of the relative error when the measurements are corrupted by Poisson noise. (c) Comparing different dithering schemes, Gaussian, uniform and discrete with quantization levels $M=10$ and $M=100$, for the one-bit MC with Gaussian measurements corrupted by Gaussian noise. (d) Comparison between uniform dithering and Gaussian dithering schemes for the input measurements generated based on uniform distribution. (e) The performance of randomized OB-SVT algorithm for different values of $\beta=s/m^{\prime}$ and $\alpha=m^{\prime}/n_1 n_2$. (f) Improvement of reconstruction accuracy through the OB-SVT-II method as the number of dithering sequences grows large, and comparison between the performance of Gaussian dithering and the proposed adaptive dithering scheme.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Corollary 1
  • proof
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • ...and 3 more