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Harnessing the Power of Sample Abundance: Theoretical Guarantees and Algorithms for Accelerated One-Bit Sensing

Arian Eamaz, Farhang Yeganegi, Deanna Needell, Mojtaba Soltanalian

TL;DR

This work addresses recovering signals from one-bit measurements by leveraging time-varying dithers to create highly overdetermined linear systems. It introduces ORKA, a suite of randomized Kaczmarz-based algorithms (and variants) that convert costly constraint problems into linear feasibility tasks, enabling accurate recovery under sample abundance and robustness to noise. A central theoretical contribution is the finite volume property (FVP), which provides uniform reconstruction guarantees and sample-complexity bounds for arbitrary and structured signal sets, including low-rank matrices and sparse vectors, across isotropic sensing matrices and even deterministic transforms like the DCT. Complementary algorithmic advances include storage-friendly SKM variants, low-rank factorization (SVP-ORKA), and adaptive thresholding to shrink the feasible region, all demonstrated with numerical experiments showing superior performance over existing one-bit sensing methods. The results have practical implications for high-rate, low-power sensing systems where dithering and randomized iterative methods can robustly recover high-dimensional signals from highly quantized measurements.

Abstract

One-bit quantization with time-varying sampling thresholds (also known as random dithering) has recently found significant utilization potential in statistical signal processing applications due to its relatively low power consumption and low implementation cost. In addition to such advantages, an attractive feature of one-bit analog-to-digital converters (ADCs) is their superior sampling rates as compared to their conventional multi-bit counterparts. This characteristic endows one-bit signal processing frameworks with what one may refer to as sample abundance. We show that sample abundance plays a pivotal role in many signal recovery and optimization problems that are formulated as (possibly non-convex) quadratic programs with linear feasibility constraints. Of particular interest to our work are low-rank matrix recovery and compressed sensing applications that take advantage of one-bit quantization. We demonstrate that the sample abundance paradigm allows for the transformation of such problems to merely linear feasibility problems by forming large-scale overdetermined linear systems -- thus removing the need for handling costly optimization constraints and objectives. To make the proposed computational cost savings achievable, we offer enhanced randomized Kaczmarz algorithms to solve these highly overdetermined feasibility problems and provide theoretical guarantees in terms of their convergence, sample size requirements, and overall performance. Several numerical results are presented to illustrate the effectiveness of the proposed methodologies.

Harnessing the Power of Sample Abundance: Theoretical Guarantees and Algorithms for Accelerated One-Bit Sensing

TL;DR

This work addresses recovering signals from one-bit measurements by leveraging time-varying dithers to create highly overdetermined linear systems. It introduces ORKA, a suite of randomized Kaczmarz-based algorithms (and variants) that convert costly constraint problems into linear feasibility tasks, enabling accurate recovery under sample abundance and robustness to noise. A central theoretical contribution is the finite volume property (FVP), which provides uniform reconstruction guarantees and sample-complexity bounds for arbitrary and structured signal sets, including low-rank matrices and sparse vectors, across isotropic sensing matrices and even deterministic transforms like the DCT. Complementary algorithmic advances include storage-friendly SKM variants, low-rank factorization (SVP-ORKA), and adaptive thresholding to shrink the feasible region, all demonstrated with numerical experiments showing superior performance over existing one-bit sensing methods. The results have practical implications for high-rate, low-power sensing systems where dithering and randomized iterative methods can robustly recover high-dimensional signals from highly quantized measurements.

Abstract

One-bit quantization with time-varying sampling thresholds (also known as random dithering) has recently found significant utilization potential in statistical signal processing applications due to its relatively low power consumption and low implementation cost. In addition to such advantages, an attractive feature of one-bit analog-to-digital converters (ADCs) is their superior sampling rates as compared to their conventional multi-bit counterparts. This characteristic endows one-bit signal processing frameworks with what one may refer to as sample abundance. We show that sample abundance plays a pivotal role in many signal recovery and optimization problems that are formulated as (possibly non-convex) quadratic programs with linear feasibility constraints. Of particular interest to our work are low-rank matrix recovery and compressed sensing applications that take advantage of one-bit quantization. We demonstrate that the sample abundance paradigm allows for the transformation of such problems to merely linear feasibility problems by forming large-scale overdetermined linear systems -- thus removing the need for handling costly optimization constraints and objectives. To make the proposed computational cost savings achievable, we offer enhanced randomized Kaczmarz algorithms to solve these highly overdetermined feasibility problems and provide theoretical guarantees in terms of their convergence, sample size requirements, and overall performance. Several numerical results are presented to illustrate the effectiveness of the proposed methodologies.
Paper Structure (47 sections, 27 theorems, 158 equations, 5 figures, 4 algorithms)

This paper contains 47 sections, 27 theorems, 158 equations, 5 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Prior Art
  3. Motivations and Contributions of the Paper
  4. Organization and Notations
  5. ORKA: Towards Circumventing Costly Constraints
  6. One-Bit Quantization With Multiple Thresholding
  7. Randomized Kaczmarz Algorithm
  8. Sampling Kaczmarz Motzkin Method
  9. Preconditioned SKM
  10. Storage-Friendly PrSKM
  11. Block SKM
  12. Comparing RKA, SKM, PrSKM and Block SKM
  13. ORKA With Noisy Measurements
  14. Robustness of ORKA Against Noise
  15. Upper Quantile-Based ORKA
  16. ...and 32 more sections

Key Result

Theorem 1

The infimum scaled condition number of a matrix $\mathbf{C}\in\mathbb{R}^{m\times n}$ is given by which is achieved if and only if $\mathbf{C}$ is of the form $\mathbf{C}=\alpha \mathbf{U}$, where $\mathbf{U}$ is an orthonormal column matrix and $\alpha\in\mathbb{R}$ is a scalar.

Figures (5)

  • Figure 1: Comparing the recovery performance of the two proposed Kaczmarz algorithms, namely the PrSKM and the Block SKM, with that of SKM and RKA for a linear inequality system.
  • Figure 2: Shrinkage of the one-bit polyhedron (\ref{['eq:80n']}) in blue, ultimately placed within the unit ball of the nuclear norm $\left\|\mathbf{X}\right\|_{\star}\leq 1$ shown with black cylindrical region and its red contours, when the number of constraints (samples) grows large. The arrows point to the half-space associated with each inequality constraint. The evolution of the feasible regime is depicted with increasing samples in three cases: (a) and (d) small sample-size regime, constraints not forming a finite-value polyhedron; (b) and (e) medium sample-size regime, constraints forming a finite-volume polyhedron, parts of which are outside the cylinder; (c) and (f) large sample-size regime, constraints forming a finite-volume polyhedron inside the nuclear norm cylinder, making its constraint redundant. The original signal representing the signal to be recovered is shown by yellow.
  • Figure 3: Comparison between the recovery performance of Block SKM using random thresholds and adaptive thresholds in the sample abundance scenario for (a) one-bit low-rank matrix sensing, and (ii) one-bit CS.
  • Figure 4: (a) Comparison between the recovery performance of SVP-ORKA and HSVT algorithm over different values of oversampling factor $\lambda$. (b) Recovery performance of Algorithm \ref{['algorithm_200']} (ORKA with low-rank matrix factorization) over different values of sampling factor $\beta$.
  • Figure 5: Comparison between the recovery performance of HT-ORKA, ST-ORKA and (a) BIHT with random thresholds, and (b) NBIHT in ditherless scenario.

Theorems & Definitions (31)

  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 1
  • Proposition 1
  • Definition 1
  • Definition 2: Consistent Reconstruction Property
  • Definition 3: Isotropic Property
  • Theorem 3: FVP for an Arbitrary Set
  • Theorem 4: FVP for Low-Rank Matrices
  • ...and 21 more