[Paper Review] Robust one-bit compressed sensing with partial circulant matrices

Sjoerd Dirksen, Shahar Mendelson|arXiv (Cornell University)|Dec 17, 2018

Sparse and Compressive Sensing Techniques4 citations

TL;DR

This paper presents a robust one-bit compressed sensing framework using partial circulant measurement matrices that are both structured and efficient to implement. It achieves optimal sample complexity under realistic noise conditions—handling both analog pre-quantization noise and adversarial bit flips—by employing dithered quantization and convex optimization, ensuring accurate recovery of sparse signals even with high noise levels.

ABSTRACT

We present optimal sample complexity estimates for one-bit compressed sensing problems in a realistic scenario: the procedure uses a structured matrix (a randomly sub-sampled circulant matrix) and is robust to analog pre-quantization noise as well as to adversarial bit corruptions in the quantization process. Our results imply that quantization is not a statistically expensive procedure in the presence of nontrivial analog noise: recovery requires the same sample size one would have needed had the measurement matrix been Gaussian and the noisy analog measurements been given as data.

Motivation & Objective

• To address the gap in one-bit compressed sensing by enabling robust recovery under realistic noise conditions, including significant analog pre-quantization noise and adversarial bit corruptions.
• To develop a practical solution using structured measurement matrices—specifically partial circulant matrices—instead of i.i.d. Gaussian matrices, which are infeasible in real-world applications.
• To achieve optimal sample complexity in the presence of non-trivial noise levels, matching the theoretical performance of Gaussian matrices while maintaining computational and implementation feasibility.
• To demonstrate that quantization is not statistically costly when analog noise is present, by showing that recovery requires the same number of samples as in the unquantized Gaussian case.

Proposed method

• Uses a partial circulant matrix as the measurement matrix, which is structured and efficiently realizable in physical systems, enabling practical deployment.
• Employs dithered quantization via random thresholds drawn uniformly from $[- ho, ho]$, which stabilizes the quantization process and enables robustness to noise.
• Applies convex optimization via the generalized Lasso or $ ho$-regularized program to recover the sparse signal from one-bit measurements corrupted by noise and bit flips.
• Establishes recovery guarantees using a modified growth property for structured matrices, showing that the image of sparse sets under the matrix satisfies subgaussian-type concentration.
• Derives sample complexity bounds using subgaussian and heavy-tailed noise models, with explicit dependence on noise level, signal sparsity, and desired accuracy.
• Uses a duality-based analysis framework to bound the error in terms of the Gaussian width of the set of interest, adapted to structured matrices and quantized data.

Experimental results

Research questions

• RQ1 Can one-bit compressed sensing achieve optimal sample complexity when using structured measurement matrices like partial circulant matrices, rather than i.i.d. Gaussian matrices?
• RQ2 How can robustness to both analog pre-quantization noise and adversarial bit corruptions be achieved in one-bit compressed sensing?
• RQ3 What is the minimal number of one-bit measurements required to recover a sparse signal with error at most $\rho$, under realistic noise conditions?
• RQ4 Does dithering significantly improve the robustness of one-bit compressed sensing for non-Gaussian structured matrices, and can this be formalized with tight sample complexity bounds?
• RQ5 Can the recovery performance of one-bit compressed sensing with structured matrices match that of the ideal Gaussian case in terms of sample complexity?

Key findings

• The proposed method achieves optimal sample complexity up to logarithmic factors, with $m \geq c_3 \gamma_1^2 \gamma_2^2 \frac{\lambda^2 s \log(en/s)}{\rho^4}$ measurements required for $\rho$-accurate recovery.
• The recovery error is bounded by $\|x^\# - x\|_2 \leq \rho$ with high probability $1 - (s/\rho^2 n)^2$, under mild conditions on the noise and threshold parameters.
• The method is robust to both subgaussian and heavier-tailed noise, provided the dithering level $\lambda$ is chosen appropriately to compensate for noise tail behavior.
• The results are minimax optimal up to logarithmic factors, meaning no other method can achieve better sample complexity in this setting.
• The use of partial circulant matrices enables practical implementation while preserving theoretical guarantees, making the framework suitable for real-world signal acquisition.
• Alternative recovery programs such as the generalized Lasso and $\ell_2$-projection-based solvers also achieve the same recovery guarantees, offering flexibility in implementation.