On the Convergence of Approximate Message Passing with Arbitrary Matrices (1402.3210v3)

Abstract: Approximate message passing (AMP) methods and their variants have attracted considerable recent attention for the problem of estimating a random vector $\mathbf{x}$ observed through a linear transform $\mathbf{A}$. In the case of large i.i.d. zero-mean Gaussian $\mathbf{A}$, the methods exhibit fast convergence with precise analytic characterizations on the algorithm behavior. However, the convergence of AMP under general transforms $\mathbf{A}$ is not fully understood. In this paper, we provide sufficient conditions for the convergence of a damped version of the generalized AMP (GAMP) algorithm in the case of quadratic cost functions (i.e., Gaussian likelihood and prior). It is shown that, with sufficient damping, the algorithm is guaranteed to converge, although the amount of damping grows with peak-to-average ratio of the squared singular values of the transforms $\mathbf{A}$. This result explains the good performance of AMP on i.i.d. Gaussian transforms $\mathbf{A}$, but also their difficulties with ill-conditioned or non-zero-mean transforms $\mathbf{A}$. A related sufficient condition is then derived for the local stability of the damped GAMP method under general cost functions, assuming certain strict convexity conditions.

• The paper derives sufficient damping conditions that ensure convergence of the AMP algorithm under quadratic cost functions and variable matrix properties.
• It conducts a local stability analysis demonstrating how convexity conditions and perturbation responses secure reliable convergence.
• Numerical results reveal that matrices with favorable singular value distributions require minimal damping, broadening AMP's applicability in compressed sensing.

On the Convergence of Approximate Message Passing with Arbitrary Matrices: A Summary

The paper "On the Convergence of Approximate Message Passing with Arbitrary Matrices" addresses an important aspect of approximate message passing (AMP) algorithms, specifically focusing on their convergence behavior under arbitrary matrix transformations. AMP algorithms have been extensively employed in the estimation of random vectors observed through a linear transformation, particularly in contexts such as compressed sensing and high-dimensional statistical inference.

This research explores the behavior of AMP methods when dealing with matrix transformations beyond the well-studied large random i.i.d. zero-mean Gaussian matrices. The authors identify that while AMP methods show effective convergence with i.i.d. Gaussian matrices due to their specific statistical properties, their convergence can be problematic with matrices that possess ill-conditioning or non-zero mean components.

Core Contributions

1. Convergence Conditions for Gaussian-Like Matrices: The paper rigorously derives sufficient conditions under which a damped version of the generalized AMP (GAMP) algorithm exhibits guaranteed convergence, particularly when quadratic cost functions (Gaussian likelihood and priors) are involved. This involves the derivation of a relationship between damping and the peak-to-average ratio of the matrix's squared singular values.
2. Local Stability Analysis: The authors extend the analysis to propose conditions for the local stability of the damped GAMP algorithm when faced with general cost functions, provided certain convexity conditions are met. Here, "local stability" refers to small perturbations in initial conditions leading to convergence to an optimal solution, a notion crucial for the robustness of AMP methods.
3. Implications for Various Matrix Types: Through analytical and numerical exploration, the authors illustrate how different matrix classes such as low-rank, subsampled unitary, and matrices used in linear filtering contexts influence the damping requirements and convergence behavior of GAMP. The paper elucidates that matrices with a favorable peak-to-average ratio in their singular value distribution (such as walk-summable matrices) tend to require less or no damping to maintain convergence.

Implications and Future Work

The findings significantly impact both theoretical explorations and practical implementations of AMP methods. The theoretical contribution provides a pathway to apply AMP methods with improved reliability on real-world data, where matrices often do not conform to idealized random distributions.

The results highlight the necessity for careful tuning of the damping factor, which acts as a convergence facilitator, modulating the algorithm to adjust to matrix conditions. This underscores potential research directions focusing on adaptive damping strategies that automatically adjust based on runtime estimations of matrix behavior, facilitating more scalable and robust AMP solutions.

Further research might explore extending these convergence guarantees to more complex, structured sparse domains or circuits with non-linearities, prevalent in emerging applications like neural network inference and complex systems modeling.

In conclusion, this paper enhances our understanding of AMP algorithm dynamics, broadening its applicability in scenarios beyond conventional Gaussian settings, and enabling its usage in more diverse problem settings where structured matrices are commonly encountered. This contribution stands as a pivotal piece in the theoretical foundation of modern compressed sensing and statistical inference methods.