[Paper Review] Correcting Errors in Linear Measurements and Compressed Sensing of Multiple Sources

Alexander Petukhov, Inna Kozlov|arXiv (Cornell University)|Apr 15, 2013

Sparse and Compressive Sensing Techniques9 references3 citations

TL;DR

This paper proposes the ℓ¹-greedy-generous algorithm (LGGA), a novel decoding method for compressed sensing that efficiently corrects sparse errors in linear measurements without additional encoding. By combining greedy and generous strategies, LGGA adapts to non-uniform information distribution across sources, significantly outperforming existing methods in reconstructing sparse Gaussian signals even under noisy conditions, demonstrating strong intrinsic error-correcting capability through residual redundancy.

ABSTRACT

We present an algorithm for finding sparse solutions of the system of linear equations $Φ\mathbf{x}=\mathbf{y}$ with rectangular matrices $Φ$ of size $n imes N$, where $n

Motivation & Objective

• To address the challenge of reconstructing sparse signals from linear measurements corrupted by sparse errors, especially when sources have non-uniform sparsity.
• To develop a decoding algorithm that operates directly on linear measurements without requiring redundant channel encoding.
• To improve reconstruction accuracy and robustness in compressed sensing when information content varies across data blocks, such as in image or multichannel source encoding.
• To enable efficient joint source-channel encoding by leveraging the natural error-correcting properties of compressed sensing.

Proposed method

• The LGGA algorithm combines greedy pursuit with a generous strategy to iteratively identify and correct sparse error components in the measurement vector.
• It uses ℓ¹-minimization as a foundation but enhances it with adaptive weighting to prioritize regions of higher information content.
• The algorithm dynamically estimates the distribution of information across blocks during iterations, adjusting weights to improve recovery of non-uniformly sparse sources.
• It operates directly on the measurement vector y, exploiting residual redundancy to detect and correct errors without side information from the encoder.
• The method is applied to multichannel source encoding by treating each source as a separate sparse component in a joint measurement model.
• Theoretical and numerical analysis shows that LGGA maintains high recovery performance even under noise, with precision limited only by fundamental information-theoretic bounds.

Experimental results

Research questions

• RQ1 Can a compressed sensing decoder correct sparse errors in linear measurements without requiring additional redundant encoding?
• RQ2 How does the performance of LGGA compare to standard ℓ¹-minimization and greedy algorithms when sources have non-uniform sparsity?
• RQ3 To what extent can the natural redundancy in linear measurements be exploited for intrinsic error correction in compressed sensing?
• RQ4 Can the algorithm adapt to unknown or varying information distribution across data blocks during decoding?

Key findings

• LGGA significantly outperforms traditional ℓ¹-minimization and greedy algorithms in reconstructing sparse Gaussian signals, especially when the sparsity distribution is non-uniform across blocks.
• The algorithm achieves high recovery accuracy even under measurement noise, with reconstruction error close to the theoretical information-theoretic limit of 2^{NH(k/N)/n}√nσ.
• Numerical experiments confirm that the 'best case scenario' estimate based on information theory closely predicts actual reconstruction precision, validating the theoretical bounds.
• The algorithm demonstrates robustness to noise, with success rates decreasing progressively and predictably as noise increases, indicating stable performance degradation.
• LGGA enables efficient joint source-channel encoding by implicitly encoding both amplitude and index information in linear measurements, eliminating the need for separate index encoding.