[Paper Review] Robust Binary Fused Compressive Sensing using Adaptive Outlier Pursuit

Xiangrong Zeng, Mário A. T. Figueiredo|arXiv (Cornell University)|Feb 20, 2014

Sparse and Compressive Sensing Techniques24 references3 citations

TL;DR

This paper proposes Robust Binary Fused Compressive Sensing (RoBFCS), an adaptive algorithm that enhances binary fused compressive sensing by integrating outlier pursuit to detect and correct sign flips in 1-bit measurements. By leveraging group sparsity and iterative thresholding with adaptive outlier estimation, RoBFCS achieves superior recovery accuracy over BIHT and BFCS, especially in noisy conditions with sign flips.

ABSTRACT

We propose a new method, {\it robust binary fused compressive sensing} (RoBFCS), to recover sparse piece-wise smooth signals from 1-bit compressive measurements. The proposed method is a modification of our previous {\it binary fused compressive sensing} (BFCS) algorithm, which is based on the {\it binary iterative hard thresholding} (BIHT) algorithm. As in BIHT, the data term of the objective function is a one-sided $\ell_1$ (or $\ell_2$) norm. Experiments show that the proposed algorithm is able to take advantage of the piece-wise smoothness of the original signal and detect sign flips and correct them, achieving more accurate recovery than BFCS and BIHT.

Motivation & Objective

• To address the challenge of sign flips in 1-bit compressive sensing, which degrade signal recovery accuracy.
• To improve upon existing binary iterative hard thresholding (BIHT) and binary fused compressive sensing (BFCS) by incorporating robustness to measurement noise and outliers.
• To exploit the piece-wise smooth structure of signals through group sparsity, enhancing recovery performance with fewer measurements.
• To develop an adaptive method that does not require prior knowledge of sparsity level K or number of sign flips L.

Proposed method

• Proposes RoBFCS as an extension of BFCS, combining iterative hard thresholding with adaptive outlier pursuit (AOP) to detect and correct sign flips.
• Uses a modified objective function with a one-sided ℓ₁ or ℓ₂ penalty on sign consistency violations, where the sign of measurements is dynamically adjusted via a weighting matrix Λ.
• Introduces an adaptive thresholding mechanism that estimates the number of sign flips L iteratively, updating the weighting vector Λ to correct erroneous signs.
• Applies iterative hard thresholding to maintain sparsity and group structure, projecting onto the K-sparse set and, optionally, the non-negative orthant for non-negative signals.
• Employs a step-size τ and stopping criterion based on relative change in the estimate, with tuning for optimal SNR improvement in BFCS variants.
• Supports both ℓ₁ and ℓ₂ data-fidelity penalties, with the ℓ₁ version showing better performance in experiments.

Experimental results

Research questions

• RQ1 Can adaptive outlier pursuit improve the robustness of binary fused compressive sensing under noisy 1-bit measurements with sign flips?
• RQ2 Does exploiting piece-wise smoothness through group sparsity enhance signal recovery accuracy compared to standard BIHT and BFCS?
• RQ3 How does the performance of RoBFCS compare to BIHT and BFCS in terms of mean absolute error, mean square error, and Hamming error under controlled noise conditions?
• RQ4 Can RoBFCS achieve accurate recovery without prior knowledge of sparsity level K or number of sign flips L?

Key findings

• RoBFCS achieves a mean absolute error (MAE) of 4.00×10⁻⁷, significantly lower than BIHT (0.0019) and BFCS (0.0008), indicating near-perfect signal recovery.
• The Hamming error (HE) for RoBFCS is 0.0010, the lowest among all methods, demonstrating effective detection and correction of sign flips.
• RoBFCS achieves a position error rate (PER) of 0%, indicating perfect sign consistency recovery, compared to 0.9% for BFCS and 1.8% for BIHT.
• The angle error (AE) for RoBFCS is 0.0085, substantially lower than BIHT (0.1234) and BFCS (0.0764), confirming improved angular alignment with the true signal.
• The ℓ₁-based RoBFCS variant outperforms its ℓ₂ counterpart across all metrics, confirming the benefit of ℓ₁ penalization in handling sign inconsistencies.
• RoBFCS significantly reduces mean square error (MSE) to 4.00×10⁻⁷, compared to 2.87×10⁻⁵ for BFCS and 7.43×10⁻⁵ for BIHT, demonstrating superior reconstruction fidelity.