Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices (1206.3953v1)

Abstract: Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make less measurements than what was considered necessary to record a signal, enabling faster or more precise measurement protocols in a wide range of applications. Using an interdisciplinary approach, we have recently proposed in [arXiv:1109.4424] a strategy that allows compressed sensing to be performed at acquisition rates approaching to the theoretical optimal limits. In this paper, we give a more thorough presentation of our approach, and introduce many new results. We present the probabilistic approach to reconstruction and discuss its optimality and robustness. We detail the derivation of the message passing algorithm for reconstruction and expectation max- imization learning of signal-model parameters. We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distribution of signals, and discuss the best possible reconstruction performances regardless of the algorithm. We also present new efficient seeding matrices, test them on synthetic data and analyze their performance asymptotically.

• The paper demonstrates that probabilistic inference via belief propagation achieves optimal signal recovery with fewer measurements under noiseless conditions.
• It introduces generalized approximate message passing (G-AMP) and employs expectation-maximization for robust parameter learning in noisy scenarios.
• Phase diagrams and seeding matrices delineate reconstruction thresholds, guiding efficient signal acquisition and advancing algorithmic improvements.

Probabilistic Reconstruction in Compressed Sensing: An Analysis

The paper "Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices" by Krzakala et al. explores a probabilistic approach to signal reconstruction in the field of compressed sensing (CS). It presents a detailed exploration of algorithms based on belief propagation (BP) and outlines the conditions under which these can achieve optimality.

Core Contributions

The paper presents several pivotal contributions to the field of compressed sensing:

1. Probabilistic Inference as Optimality: Under noiseless conditions, the paper affirms that probabilistic inference can achieve optimality in signal reconstruction even when the signal model used does not perfectly align with the true signal distribution. It is shown that belief propagation methods can achieve exact reconstruction with fewer measurements than traditional methods, reflecting significant robustness.
2. Message Passing Algorithms: A simplification of BP, termed generalized approximate message passing (G-AMP), is employed for signal reconstruction. This paper enhances the analysis by providing asymptotic expressions and conditions under which these algorithms converge. The usage of expectation-maximization for parameter learning further optimizes reconstruction in noisy settings.
3. Phase Transitions in CS: Through the derivation of phase diagrams, the authors delineate the regions of parameter space where BP provides exact signal reconstruction versus regions where it fails. The phases illustrate where BP methods meet optimal (Bayesian) signal reconstruction and highlight key transitions such as spinodal points, offering insights into algorithmic limitations and potential improvements.
4. Seeding Matrices: A novel introduction and analysis of seeding matrices allow for achieving the theoretical limits of measurement efficiency in CS. These matrices introduce inhomogeneities in the sensing process that strategically support more efficient reconstruction by overcoming barriers in the BP method.

Implications and Future Work

This work significantly impacts practical and theoretical aspects of compressed sensing:

• Practical Implications: In applications requiring efficient and accurate signal recovery, the probabilistic approaches and seeding matrices discussed can lead to substantial reductions in data acquisition needs. This is especially beneficial in environments where measurement resources are limited or costly.
• Theoretical Insights: The paper enhances the understanding of phase transitions within compressed sensing, offering a relatively complete picture of situations where probabilistic methods meet their optimal inference limits, providing a benchmark for future algorithmic developments.
• Potential Developments: Future research might explore the design and optimization of seeding matrices further, considering specific constraints or objectives in distinct application domains. Moreover, extending this approach in environments with correlated noise or dynamic signal reconstruction could be promising research directions.

In summary, the paper presents a comprehensive framework for applying probabilistic inference in compressed sensing, supported by robust numerical analysis and theoretical insights. It marks a notable advancement in understanding and utilizing compressed sensing methods, aligning well with the current trajectory in signal processing towards more efficient and effective reconstruction techniques.