[Paper Review] Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation

Yonina C. Eldar, Holger Rauhut|ArXiv.org|Apr 3, 2009

Sparse and Compressive Sensing Techniques38 references4 citations

TL;DR

This paper provides an average-case analysis of multichannel sparse recovery using convex relaxation via the mixed $\ell_{2,1}$ norm, showing that under mild conditions on sparsity and dictionary coherence, the probability of recovery failure decays exponentially with the number of channels. This demonstrates that joint recovery is typically superior to single-channel methods, even for small numbers of signals.

ABSTRACT

In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worst-case analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed l_2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.

Motivation & Objective

• To analyze the average-case performance of convex relaxation methods for multichannel sparse recovery, moving beyond worst-case guarantees.
• To explain why joint sparse recovery outperforms single-channel recovery in practice, despite theoretical equivalence in worst-case settings.
• To derive tight, non-asymptotic bounds on the failure probability of multichannel recovery using the mixed $\ell_{2,1}$ norm.
• To extend and tighten existing average-case performance bounds for thresholding and simultaneous orthogonal matching pursuit (SOMP).

Proposed method

• Uses a probabilistic model on the signal vectors to analyze recovery performance in expectation, rather than worst-case over all signals.
• Applies concentration of measure inequalities to bound the probability that the residual in SOMP selects an incorrect index.
• Employs the mixed $\ell_{2,1}$ norm minimization as the convex relaxation for multichannel sparse recovery.
• Derives bounds on the failure probability using the coherence of the measurement matrix and the spectral properties of submatrices.
• Utilizes the expected norm of a Gaussian vector, $C_2(L) = \mathbb{E}\|Z\|_2$, to quantify the signal-to-noise ratio in the residual selection step.
• Applies a union bound over all possible support sets to ensure recovery success across all iterations of SOMP.

Experimental results

Research questions

• RQ1 Under what conditions does the mixed $\ell_{2,1}$ relaxation for multichannel sparse recovery succeed with high probability in the average case?
• RQ2 Why does joint sparse recovery outperform single-channel recovery in practice, despite theoretical worst-case equivalence?
• RQ3 How does the failure probability of SOMP decay with the number of channels under a random signal model?
• RQ4 Can tighter average-case bounds be derived for thresholding and SOMP using the same probabilistic framework?
• RQ5 What role does the coherence of the dictionary play in determining the success rate of multichannel recovery?

Key findings

• The probability of recovery failure for the mixed $\ell_{2,1}$ relaxation decays exponentially with the number of channels, even under mild sparsity and coherence conditions.
• For a fixed number of channels, the failure probability is bounded by $ (|S^c| + 1) \exp(-\epsilon^2 A_L^2) $, where $ A_L $ depends on the number of measurements per channel.
• The analysis shows that joint recovery is typically superior to individual channel recovery because worst-case signal instances are unlikely under random models.
• The derived bounds for SOMP and thresholding are tightened compared to prior work, providing more accurate performance predictions.
• The condition $ (1+\epsilon)\frac{\mu_2(S)}{1-\delta(S)} \geq (1-\epsilon)\left(1 - \frac{\mu_2(S)^2}{1-\delta(S)}\right) $ ensures successful support recovery in SOMP with high probability.
• The results are valid and meaningful even for small numbers of channels, making them practically relevant for real-world applications.