Asymptotic mutual information in quadratic estimation problems over compact groups
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912777644277760 |
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| author | Yang, Kaylee Y. Wee, Timothy L. H. Fan, Zhou |
| author_facet | Yang, Kaylee Y. Wee, Timothy L. H. Fan, Zhou |
| contents | Motivated by applications to group synchronization and quadratic assignment on random data, we study a general problem of Bayesian inference of an unknown ``signal'' belonging to a high-dimensional compact group, given noisy pairwise observations of a featurization of this signal. We establish a quantitative comparison between the signal-observation mutual information in any such problem with that in a simpler model with linear observations, using interpolation methods. For group synchronization, our result proves a replica formula for the asymptotic mutual information and Bayes-optimal mean-squared-error. Via analyses of this replica formula, we show that the conjectural phase transition threshold for computationally-efficient weak recovery of the signal is determined by a classification of the real-irreducible components of the observed group representation(s), and we fully characterize the information-theoretic limits of estimation in the example of angular/phase synchronization over $SO(2)$/$U(1)$. For quadratic assignment, we study observations given by a kernel matrix of pairwise similarities and a randomly permutated and noisy counterpart, and we show in a bounded signal-to-noise regime that the asymptotic mutual information coincides with that in a Bayesian spiked model with i.i.d. signal prior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10169 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic mutual information in quadratic estimation problems over compact groups Yang, Kaylee Y. Wee, Timothy L. H. Fan, Zhou Statistics Theory Information Theory Motivated by applications to group synchronization and quadratic assignment on random data, we study a general problem of Bayesian inference of an unknown ``signal'' belonging to a high-dimensional compact group, given noisy pairwise observations of a featurization of this signal. We establish a quantitative comparison between the signal-observation mutual information in any such problem with that in a simpler model with linear observations, using interpolation methods. For group synchronization, our result proves a replica formula for the asymptotic mutual information and Bayes-optimal mean-squared-error. Via analyses of this replica formula, we show that the conjectural phase transition threshold for computationally-efficient weak recovery of the signal is determined by a classification of the real-irreducible components of the observed group representation(s), and we fully characterize the information-theoretic limits of estimation in the example of angular/phase synchronization over $SO(2)$/$U(1)$. For quadratic assignment, we study observations given by a kernel matrix of pairwise similarities and a randomly permutated and noisy counterpart, and we show in a bounded signal-to-noise regime that the asymptotic mutual information coincides with that in a Bayesian spiked model with i.i.d. signal prior. |
| title | Asymptotic mutual information in quadratic estimation problems over compact groups |
| topic | Statistics Theory Information Theory |
| url | https://arxiv.org/abs/2404.10169 |