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 i.i.d. Gaussian, the more realistic case of structured noise still proves to be challenging. To capture the structure while maintaining mathematical tractability, a line of work has focused on rotationally invariant noise. However, existing studies either provide sub-optimal algorithms or they are limited to a special class of noise ensembles. In this paper, we establish the first characterization of the information-theoretic limits for a noise matrix drawn from a general trace ensemble. These limits are then achieved by an efficient algorithm inspired by the theory of adaptive Thouless-Anderson-Palmer (TAP) equations. Our approach leverages tools from statistical physics (replica method) and random matrix theory (generalized spherical integrals), and it unveils the equivalence between the rotationally invariant model and a surrogate Gaussian model.

翻译：我们考虑一个结构化尖峰模型的贝叶斯推断典型问题：低秩信号被加性噪声破坏。当噪声为独立同分布高斯噪声时，其信息论极限与算法极限已得到充分理解，但更具现实意义的结构化噪声情形仍被证明具有挑战性。为了在保持数学可处理性的同时捕捉噪声结构，一系列研究工作聚焦于旋转不变噪声。然而，现有研究要么提供次优算法，要么局限于特殊的噪声系综类别。本文首次建立了从一般迹系综中抽取噪声矩阵时的信息论极限特征刻画，并通过受自适应Thouless-Anderson-Palmer（TAP）方程理论启发的有效算法达到了这些极限。我们的方法利用了统计物理学（复本方法）和随机矩阵理论（广义球面积分）的工具，并揭示了旋转不变模型与代理高斯模型之间的等价性。