[Paper Review] Compressed Sensing with Prior Information via Maximizing Correlation

Xu Zhang, Wei Cui+1 more|arXiv (Cornell University)|May 22, 2017

Sparse and Compressive Sensing Techniques14 references3 citations

TL;DR

This paper proposes a novel compressed sensing method that improves signal reconstruction by maximizing correlation between the estimated signal and prior information, using a modified Lasso formulation with a correlation-promoting term. Theoretical analysis under sub-Gaussian measurements shows that good prior information significantly enhances recovery performance, validated by simulations across various prior types.

ABSTRACT

Compressed sensing (CS) with prior information concerns the problem of reconstructing a sparse signal with the aid of a similar signal which is known beforehand. We consider a new approach to integrate the prior information into CS via maximizing the correlation between the prior knowledge and the desired signal. We then present a geometric analysis for the proposed method under sub-Gaussian measurements. Our results reveal that if the prior information is good enough, then the proposed approach can improve the performance of the standard CS. Simulations are provided to verify our results.

Motivation & Objective

• To improve compressed sensing reconstruction by integrating prior information beyond standard sparsity constraints.
• To develop a new optimization framework that explicitly maximizes correlation between the signal and prior knowledge.
• To provide theoretical performance guarantees using geometric analysis under sub-Gaussian measurements.
• To validate the method’s superiority over existing approaches under diverse prior information conditions.

Proposed method

• Proposes a new optimization problem: minimizing the ℓ₁-norm of the signal while maximizing its inner product with prior information, formulated as min ||x||₁ − λ⟨x, φ⟩ subject to ||y − Ax||₂ ≤ δ.
• Uses geometric analysis based on tangent and normal cones, Gaussian width, and sub-Gaussian random matrix properties to derive performance bounds.
• Applies matrix deviation inequality to ensure restricted eigenvalue conditions for sub-Gaussian sensing matrices.
• Derives a performance threshold via the effective dimension v, which determines the number of measurements required for successful recovery.
• Analyzes the impact of different prior shifts (on support, complement, or arbitrary) on the recovery threshold v.
• Employs phase transition curves and mean squared error comparisons to evaluate performance across methods.

Experimental results

Research questions

• RQ1 Can maximizing correlation between the signal and prior information improve compressed sensing recovery performance?
• RQ2 How does the quality of prior information affect the number of measurements required for successful reconstruction?
• RQ3 How does the proposed correlation-maximizing method compare to standard Lasso and ℓ₁-ℓ₁/ℓ₁-ℓ₂ minimization under varying prior types?
• RQ4 What is the theoretical performance threshold (in terms of effective dimension v) for the proposed method under sub-Gaussian measurements?
• RQ5 Under what conditions does the proposed method outperform existing prior-based compressed sensing techniques?

Key findings

• The proposed method achieves better recovery performance than standard Lasso when the prior information is sufficiently accurate and highly correlated with the true signal.
• Phase transition curves show that shifts on the support (especially in the same direction as the true signal) improve recovery, while shifts on the complement of the support increase required measurements.
• For arbitrary shifts, the method outperforms standard CS when the prior is aligned with the signal structure, as seen in simulations with v = n − 3s/4 + 9/16.
• When prior information is sparse (e.g., ℓ₁-ℓ₁ favorable), ℓ₁-ℓ₁ minimization may outperform the correlation method; however, the correlation method excels when prior is dense or mismatched in structure.
• Theoretical analysis confirms that if the prior is good enough, the method reduces the effective dimension v, thereby lowering the required number of measurements.
• Simulations demonstrate that the correlation-maximizing method achieves the best performance in cases where prior information is well-aligned with the signal, particularly in scenarios with non-sparse differences between signal and prior.