Asymptotic Analysis of Complex LASSO via Complex Approximate Message Passing (CAMP)

Abstract: Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complex. We study the popular reconstruction method of $\ell_1$-regularized least squares or LASSO. While several studies have shown that the LASSO algorithm offers desirable solutions under certain conditions, the precise asymptotic performance of this algorithm in the complex setting is not yet known. In this paper, we extend the approximate message passing (AMP) algorithm to the complex signals and measurements and obtain the complex approximate message passing algorithm (CAMP). We then generalize the state evolution framework recently introduced for the analysis of AMP, to the complex setting. Using the state evolution, we derive accurate formulas for the phase transition and noise sensitivity of both LASSO and CAMP.

• The paper introduces the CAMP algorithm, extending AMP to complex-valued LASSO with rigorous state evolution analysis.
• It precisely characterizes phase transitions and noise sensitivity, aiding in determining optimal undersampling conditions.
• Simulation results demonstrate the algorithm's robustness and universal performance across different matrix and coefficient distributions.

Asymptotic Analysis of Complex LASSO via Complex Approximate Message Passing (CAMP)

This paper presents a comprehensive study on the recovery of sparse signals from undersampled complex-valued linear measurements, a core problem within the compressed sensing domain. It extends the conventional analysis by considering both the signal and measurements in their complex forms, addressing a gap where previous works predominantly focused on real-valued domains. The authors develop a new algorithm, termed Complex Approximate Message Passing (CAMP), to solve the complex-valued LASSO problem and provide insights into its theoretical performance.

Main Contributions and Theoretical Advancements

The chief contribution of this work is the introduction and asymptotic analysis of the CAMP algorithm, offering an extension of the real-valued approximate message passing to the complex domain. The authors leverage the state evolution framework, demonstrating its efficacy in predicting the performance of the CAMP algorithm asymptotically and providing accurate formulas for the phase transition and noise sensitivity.

1. Extension to Complex Domain: The adaptation of the AMP algorithm to complex numbers, resulting in CAMP, is a notable theoretical advancement. This adaptation is not straightforward due to the inherent mathematical complexities when transitioning from real to complex domains.
2. State Evolution Framework: The application of state evolution in this context allows for precise characterizations of the phase transitions and noise sensitivities, key performance metrics when evaluating LASSO-type algorithms. This extension confirms that the algorithm retains desirable properties, such as concavity and fixed-point convergence.
3. Performance Predictions: The paper introduces implicit equations that describe the phase transition curve in terms of undersampling rate. This curve is crucial for evaluating the algorithm's capabilities in signal recovery, especially in determining the sparsity level that can be effectively reconstructed from a given number of measurements.
4. Matrix and Coefficient Universality: Through extensive simulations, the authors demonstrate "universal" empirical performance across different types of matrices and coefficient distributions. This further solidifies the CAMP algorithm's practical relevance and robustness, asserting that its performance does not hinge on specific matrix constructions, expanding its applicability.
5. Noise Sensitivity Analysis: The calculations of the noise sensitivity provide insights into how the algorithm performs in the presence of noise, a realistic scenario in many practical applications. The results suggest a robust performance against noisy measurements, maintaining a favorable noise sensitivity across tested ranges.

Implications and Future Directions

The findings presented in this paper have multiple implications for both theoretical exploration and practical implementation within compressed sensing and related fields:

• Theoretical Implications: The paper enhances our understanding of complex-valued LASSO problems, paving the way for future advancements in solving other complex optimization problems using message passing techniques.
• Practical Implications: The CAMP algorithm's capability to handle complex-valued signals directly ties into applications like MRI and radar systems, offering a tool that aligns with the data characteristics in these fields.
• Potential for Further Research: Future research could explore optimizing the threshold parameters used in the CAMP iterations to refine performance further. There is also room for exploring the integration of CAMP within larger signal processing frameworks, especially in adaptive systems.
• Broader AI Implications: The robustness of the CAMP algorithm against different signal and noise variability suggests potential applications in real-time AI systems managing large-scale data, necessitating efficient compressed sensing solutions that operate reliably under less-than-ideal conditions.

In summary, this paper makes significant strides in expanding LASSO-type solutions to complex fields and provides a solid foundation for both evaluating and improving complex signal processing methodologies in practice. The blend of algorithmic development, theoretical purview, and empirical validation offers a holistic approach to understanding and leveraging complex approximations in signal recovery tasks.