Recovering Compressively Sampled Signals Using Partial Support Information (1010.4612v2)

Abstract: In this paper we study recovery conditions of weighted $\ell_1$ minimization for signal reconstruction from compressed sensing measurements when partial support information is available. We show that if at least 50% of the (partial) support information is accurate, then weighted $\ell_1$ minimization is stable and robust under weaker conditions than the analogous conditions for standard $\ell_1$ minimization. Moreover, weighted $\ell_1$ minimization provides better bounds on the reconstruction error in terms of the measurement noise and the compressibility of the signal to be recovered. We illustrate our results with extensive numerical experiments on synthetic data and real audio and video signals.

• The paper demonstrates that weighted ℓ₁ minimization leveraging partial support information outperforms standard ℓ₁ methods when the support estimate is over 50% accurate.
• It establishes specific RIP conditions under which the recovery error is minimized and the reconstruction stability improves in noisy environments.
• Empirical evidence from synthetic and real-world experiments, including audio and video data, validates the enhanced performance, suggesting practical applications in imaging and remote sensing.

Weighted $\ell_1$ Minimization in Compressed Sensing with Partial Support Information

The paper by Friedlander, Mansour, Saab, and Yılmaz focuses on the recovery of signals in the context of compressed sensing, particularly leveraging partial support information through an extension known as weighted $\ell_1$ minimization. This approach is intended to reconstruct signals from fewer measurements than traditional methods by exploiting prior knowledge about the signal's support.

Key Contributions

The authors present an analysis of weighted $\ell_1$ minimization, showing that it outperforms standard $\ell_1$ minimization under less stringent conditions when partial support information is at least 50% accurate. They demonstrate that, compared to standard $\ell_1$ minimization, their method results in improved stability and reduced reconstruction error, which depends on the noise in the measurements and the signal’s compressibility. The core theoretical achievement is the derivation of a condition on the Restricted Isometry Property (RIP) for the measurement matrix that guarantees the success of their approach.

Theoretical Framework

For a given signal representation, recovering the signal from compressed measurements typically involves minimizing its $\ell_1$ norm due to its robustness and computational feasibility. However, in many practical scenarios, partial information about the signal's support may be available, paving the way for weighted $\ell_1$ minimization. This technique assigns different weights to the support estimates based on their expected significance in the recovery process.

The main theorem asserts that if the measurement matrix satisfies a specific RIP condition, weighted $\ell_1$ minimization recovers signals with an error bound that is refined by the accuracy of the support information. The weighting function penalizes less those indices believed to be in the support, thus taking advantage of available prior information.

Numerical Results

The theoretical assertions are corroborated by numerical experiments conducted on both synthetic and real-world data, including audio and video signals. These experiments illustrate a notable enhancement in signal recovery quality, reinforcing their theoretical findings. Notably, the recovery is effectively robust to noise, which is a critical factor in practical applications.

Implications and Future Directions

The implications of this research are significant both theoretically and practically. From a theoretical standpoint, this work contributes to the broader understanding of signal recovery in compressed sensing, particularly in realistic scenarios where partial support information is accessible. Practically, the proposed method can be instrumental in fields like medical imaging, remote sensing, and multimedia processing, where compressed sensing is increasingly adopted.

Future research could explore adaptive schemes for determining optimal weights dynamically as more data is gathered, or leveraging machine learning techniques to predict support information from past patterns. Moreover, extending the evaluation to other types of sparsifying transforms or non-linear systems could further broaden the method's applicability.

In conclusion, this paper advances the paper of signal recovery in compressed sensing by integrating prior support knowledge into the recovery process. The derived theoretical conditions, combined with empirical evidence, suggest that weighted $\ell_1$ minimization holds promise for enhancing the efficiency and accuracy of compressed signal recovery methods.