[Paper Review] Recovery of signals under the condition on RIC and ROC via prior support information

Wengu Chen, Yaling Li|arXiv (Cornell University)|Feb 27, 2016

Sparse and Compressive Sensing Techniques17 references3 citations

TL;DR

This paper establishes improved sufficient conditions for stable and robust recovery of sparse signals using weighted $l_1$ minimization with prior support information. By leveraging restricted isometry constants (RIC) and restricted orthogonality constants (ROC), it derives a less restrictive recovery condition and tighter error bounds than standard $l_1$ minimization, especially when prior support estimates are at least 50% accurate.

ABSTRACT

In this paper, the sufficient condition in terms of the RIC and ROC for the stable and robust recovery of signals in both noiseless and noisy settings was established via weighted $l_{1}$ minimization when there is partial prior information on support of signals. An improved performance guarantee has been derived. We can obtain a less restricted sufficient condition for signal reconstruction and a tighter recovery error bound under some conditions via weighted $l_{1}$ minimization. When prior support estimate is at least $50\%$ accurate, the sufficient condition is weaker than the analogous condition by standard $l_{1}$ minimization method, meanwhile the reconstruction error upper bound is provably to be smaller under additional conditions. Furthermore, the sufficient condition is also proved sharp.

Motivation & Objective

• To improve the performance guarantees of sparse signal recovery in compressed sensing by incorporating prior support information.
• To derive a less restrictive sufficient condition for stable and robust recovery using weighted $l_1$ minimization compared to standard $l_1$ minimization.
• To establish tighter upper bounds on reconstruction error under prior support information.
• To prove that the derived sufficient condition is sharp, meaning it cannot be further relaxed without losing recovery guarantees.
• To analyze the impact of prior support accuracy (especially ≥50%) on improving recovery performance in both noiseless and noisy settings.

Proposed method

• Proposes a weighted $l_1$ minimization framework that incorporates prior support information through adaptive weights in the optimization objective.
• Uses restricted isometry constants (RIC) $\delta_a$ and restricted orthogonality constants (ROC) $\theta_{a,b}$ as key matrix properties to derive recovery conditions.
• Introduces a modified constant $C_{a,b,k}^{\alpha,\omega}$ that depends on the prior support accuracy $\alpha$ and weight parameter $\omega$, refining the standard $C_{a,b,k}$.
• Derives a sufficient condition $\delta_a + C_{a,b,k}^{\alpha,\omega} \theta_{a,b} < 1$ for stable and robust recovery, which is weaker than the standard $l_1$ condition when prior support is sufficiently accurate.
• Establishes recovery error bounds $D_0$, $D_0'$, and $D_1$ that are provably smaller than their standard $l_1$ counterparts $C_0$, $C_0'$, and $C_1$ under the same conditions.
• Employs a contradiction-based proof strategy to show the sharpness of the derived condition by constructing counterexamples where recovery fails when the condition is violated.

Experimental results

Research questions

• RQ1 How does prior support information improve the sufficient condition for stable and robust recovery in compressed sensing?
• RQ2 What is the quantitative improvement in the recovery condition and error bound when using weighted $l_1$ minimization with prior support information?
• RQ3 Under what conditions is the weighted $l_1$ minimization condition strictly weaker than the standard $l_1$ minimization condition?
• RQ4 How does the accuracy of the prior support estimate (e.g., ≥50%) affect the tightness of the recovery guarantee?
• RQ5 Is the derived sufficient condition for recovery via weighted $l_1$ minimization sharp, meaning it cannot be further relaxed?

Key findings

• The sufficient condition $\delta_a + C_{a,b,k}^{\alpha,\omega} \theta_{a,b} < 1$ is strictly weaker than the standard $\delta_a + C_{a,b,k} \theta_{a,b} < 1$ condition when prior support accuracy $\alpha > 1/2$ and $\omega < 1$.
• When prior support estimate is at least 50% accurate, the recovery error upper bound $D_0$ is provably smaller than the standard bound $C_0$.
• The derived condition is proven to be sharp, as equality $\delta_a + C_{a,b,k}^{\alpha,\omega} \theta_{a,b} = 1$ leads to failure of exact and stable recovery.
• The error bounds $D_0$, $D_0'$, and $D_1$ are strictly tighter than $C_0$, $C_0'$, and $C_1$ respectively, under the same matrix and signal conditions.
• For $\alpha > 1/2$ and $\omega < 1$, the modified constant $C_{a,b,k}^{\alpha,\omega}$ is always smaller than $C_{a,b,k}$, leading to improved theoretical guarantees.
• The improvement in recovery performance is most significant when the prior support is highly accurate, and the condition remains valid even under noisy measurements with $\ell_2$ and Dantzig selector noise models.