Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise
We consider a prototypical problem of Bayesian inference for a structured spiked model: a low-rank signal is corrupted by additive noise. While both information-theoretic and algorithmic limits are well understood when the noise is a Gaussian Wigner matrix, the more realistic case of structured nois...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
American Physical Society
2025-01-01
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| Series: | Physical Review Research |
| Online Access: | http://doi.org/10.1103/PhysRevResearch.7.013081 |
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| Summary: | We consider a prototypical problem of Bayesian inference for a structured spiked model: a low-rank signal is corrupted by additive noise. While both information-theoretic and algorithmic limits are well understood when the noise is a Gaussian Wigner matrix, the more realistic case of structured noise still remains challenging. To capture the structure while maintaining mathematical tractability, a line of work has focused on rotationally invariant noise. However, existing studies either provide suboptimal algorithms or are limited to a special class of noise ensembles. In this paper, using tools from statistical physics (replica method) and random matrix theory (generalized spherical integrals) we establish the characterization of the information-theoretic limits for a noise matrix drawn from a general trace ensemble. Remarkably, our analysis unveils the asymptotic equivalence between the rotationally invariant model and a surrogate Gaussian one. Finally, we show how to saturate the predicted statistical limits using an efficient algorithm inspired by the theory of adaptive Thouless-Anderson-Palmer (TAP) equations. |
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| ISSN: | 2643-1564 |