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 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 are limited to special cases of noise ensembles. In this paper, using tools from statistical physics (replica method) and random matrix theory (generalized spherical integrals) we establish the first 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.

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。当噪声为高斯维格纳矩阵时，信息论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极具挑战性。为在保持数学可处理性的同时捕捉噪声结构，一系列研究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。本文借助统计物理学（复本方法）与随机矩阵理论（广义球面积分）工具，首次刻画了从一般迹系综中抽取噪声矩阵时的信息论极限。值得注意的是，我们的分析揭示了旋转不变模型与替代高斯模型之间的渐近等价性。最后，我们展示了如何通过受自适应Thouless-Anderson-Palmer（TAP）方程理论启发的有效算法来达到预测的统计极限。