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.

本文主要研究了贝叶斯推断中结构化尖峰模型的一个典型问题：低秩信号被加性噪声所污染。通过使用统计物理和随机矩阵理论的工具，我们建立了从一个通用迹集合中绘制的噪声矩阵的信息理论限制的第一个表征，并通过启发自自适应 Thoulless-Anderson-Palmer（TAP）方程的理论的高效算法实现了这些限制。同时，我们揭示了旋转不变模型与一个替代的高斯模型之间的等效性。