Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation (0904.0494v1)

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.

• The paper demonstrates that average-case analysis reveals exponential decay in failure probability using the mixed ℓ2,1 convex relaxation approach.
• It employs probabilistic models, including Gaussian and spherical distributions, to realistically evaluate multichannel signal recovery.
• The study shows that convex relaxation methods require milder conditions and outperform greedy algorithms in joint sparse recovery.

Analyzing Average Case Performance of Multichannel Sparse Recovery via Convex Relaxation

The paper "Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation" by Yonina C. Eldar and Holger Rauhut addresses the recovery of jointly sparse signals in multichannel systems from insufficient measurements. This is a crucial issue in various signal processing applications such as image denoising, radar, and analog-to-digital conversion. Traditionally, recovering the sparsest signal consistent with given data is known to be NP-hard, motivating the exploration of efficient approximation algorithms like thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation methods.

The research focuses on the mixed ℓ2,1 approach for multichannel recovery, offering insights into how this method improves over treating each channel independently. The authors build upon the probabilistic model imposed on signals, and their primary contribution is the average-case analysis of these algorithms, deviating from previous worst-case assessments.

Key Contributions

1. Average-Case Analysis: The paper departs from worst-case analysis, examining situations where the inputs are treated as random variables. This allows for a more realistic assessment of algorithm performance, akin to understanding how methods behave over a typical set of inputs rather than the hardest possible instance.
2. Probabilistic Models: The authors utilize probabilistic models such as real and complex Gaussian and spherical distributions to simulate the input signals' distribution. This probabilistic assessment enhances the understanding of algorithm behavior over a spectrum of scenarios.
3. Exponential Decay of Failure Probability: A significant result is their establishment that the failure probability of multichannel sparse recovery using mixed ℓ2,1 optimization decays exponentially with the number of channels when certain mild conditions on the measurement matrix and sparsity level are fulfilled.
4. Comparative Analysis: The paper benchmarks the multichannel ℓ2,1 methods against greedy algorithms like thresholding and SOMP. It provides a nuanced perspective on their average performance, highlighting the superior theoretical guarantees of convex relaxation strategies in joint sparse recovery contexts.
5. Weakening of Conditions for Success: The paper introduces improved conditions for the success of recovery compared to traditional worst-case scenarios, which generally imposed more stringent assumptions.
6. Bounding Norm Conditions: By relaxing the previous bounds, the authors demonstrate that ℓ2-norm conditions, as opposed to ℓ1-norm, can ensure recovery success, expanding applicability to single and multiple channel settings.

Implications and Future Directions

The implications of this research are profound for fields relying on efficient data reconstruction from limited measurements. Convex relaxation methods, due to their ability to handle multichannel signals effectively, present a preferential choice over traditional single-channel methods and even some greedy algorithms, particularly when considering average-case scenarios.

From a practical standpoint, these findings encourage transitioning to multichannel approaches where channel count can be leveraged to enhance recovery fidelity. Theoretically, this work lays the groundwork for further exploration of average-case analyses in sparse recovery, inviting advancements that could refine probabilistic models and conditions under which these models excel.

Future research might explore extending these methods to more complex signal structures or settings with noisy measurements to better mimic real-world conditions. Moreover, potential improvements in algorithm design, guided by the insights obtained from average-case behavior, could yield even more efficient and robust sparse recovery techniques applicable across various technological landscapes.