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Asymptotic Mutual Information and Minimum Mean-Squared Error in Bayesian Inference of Signals over Compact Groups


Core Concepts
The asymptotic mutual information between an unknown signal belonging to a high-dimensional compact group and its noisy pairwise observations can be approximated by the mutual information in a simpler linear observation model. This result enables the characterization of the information-theoretic limits of estimation in group synchronization and quadratic assignment problems.
Abstract
The content presents a general framework for studying Bayesian inference problems where the unknown parameter of interest is an element of a high-dimensional compact group, given noisy pairwise observations of a featurization of this parameter. Key highlights: The authors establish a quantitative comparison between the signal-observation mutual information in the original quadratic observation model and that in a simpler linear observation model. This is done using interpolation methods. For group synchronization problems, the authors provide a rigorous proof of a replica formula for the asymptotic mutual information and Bayes-optimal minimum mean-squared error (MMSE). They analyze the optimization landscape of the replica potential to characterize the information-theoretic limits of inference. For quadratic assignment problems, the authors show that the asymptotic mutual information coincides with that in a low-rank matrix estimation model with i.i.d. signal prior, under a bounded signal-to-noise regime. The authors also establish an overlap concentration result, showing that the posterior overlap between the estimated signal and the true signal concentrates near the set of near-maximizers of the replica potential.
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Deeper Inquiries

What are some potential applications of the general framework beyond the group synchronization and quadratic assignment problems considered in the content

The general framework presented in the context can have various applications beyond group synchronization and quadratic assignment problems. One potential application is in the field of network analysis, where the framework can be used to infer latent structures in complex networks. By modeling the network data as observations from a high-dimensional group and applying Bayesian inference techniques, the framework can help uncover hidden patterns, communities, or relationships within the network. Another application could be in the field of image processing and computer vision. By treating images as elements of a high-dimensional group and using pairwise observations, the framework can aid in tasks such as image registration, object recognition, and image alignment. This approach can improve the accuracy and efficiency of image analysis algorithms by leveraging the information-theoretic limits provided by the framework. Furthermore, the framework can be applied in signal processing and sensor networks for estimating parameters or signals from noisy measurements. By formulating the estimation problem within the framework of Bayesian inference on a compact group, it can help improve the robustness and accuracy of signal estimation algorithms in noisy environments.

How can the insights from the analysis of the replica potential be leveraged to design computationally efficient algorithms for signal estimation in these problems

Insights from the analysis of the replica potential can be instrumental in designing computationally efficient algorithms for signal estimation in synchronization and assignment problems. By identifying the critical points and phase transition thresholds of the replica potential, algorithm designers can focus on developing optimization algorithms that target these critical points to achieve optimal performance. For example, in the context of group synchronization, understanding the conditions under which the replica potential is maximized or minimized can guide the development of optimization algorithms that converge to the global maximum or minimum efficiently. This can lead to the design of novel algorithms, such as Approximate Message Passing (AMP) algorithms, that exploit the structure of the replica potential to achieve accurate and fast signal estimation. Similarly, in quadratic assignment problems, leveraging the insights from the analysis of the replica potential can help in designing algorithms that efficiently estimate the latent permutation or matching between objects. By incorporating the information-theoretic limits provided by the replica potential, algorithm designers can develop robust and scalable methods for solving complex assignment problems.

What are the implications of the overlap concentration result for the design of practical inference methods that can provably achieve the information-theoretic limits

The overlap concentration result has significant implications for the design of practical inference methods that aim to achieve the information-theoretic limits in synchronization and assignment problems. By showing that the posterior overlap with the true signal concentrates near the set of global maximizers of the replica potential, the result suggests that inference methods should focus on estimating the signal components that align with these maximizers. Practical inference methods can leverage this result by incorporating strategies that prioritize the estimation of signal components that align with the global maximizers of the replica potential. This can lead to the development of algorithms that are more robust, accurate, and efficient in estimating the latent signals in synchronization and assignment problems. Additionally, the concentration of the posterior overlap near the global maximizers can guide the design of sampling algorithms, optimization techniques, and regularization methods to improve the performance of inference methods in real-world applications.
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