Matrix denoising is central to signal processing and machine learning. Its analysis when the matrix to infer has a factorised structure with a rank growing proportionally to its dimension remains a challenge, except when it is rotationally invariant. In this case the information theoretic limits and a Bayes-optimal denoising algorithm, called rotational invariant estimator [1,2], are known. Beyond this setting few results can be found. The reason is that the model is not a usual spin system because of the growing rank dimension, nor a matrix model due to the lack of rotation symmetry, but rather a hybrid between the two. In this paper we make progress towards the understanding of Bayesian matrix denoising when the hidden signal is a factored matrix $XX^\intercal$ that is not rotationally invariant. Monte Carlo simulations suggest the existence of a denoising-factorisation transition separating a phase where denoising using the rotational invariant estimator remains Bayes-optimal due to universality properties of the same nature as in random matrix theory, from one where universality breaks down and better denoising is possible by exploiting the signal's prior and factorised structure, though algorithmically hard. We also argue that it is only beyond the transition that factorisation, i.e., estimating $X$ itself, becomes possible up to sign and permutation ambiguities. On the theoretical side, we combine mean-field techniques in an interpretable multiscale fashion in order to access the minimum mean-square error and mutual information. Interestingly, our alternative method yields equations which can be reproduced using the replica approach of [3]. Using numerical insights, we then delimit the portion of the phase diagram where this mean-field theory is reliable, and correct it using universality when it is not. Our ansatz matches well the numerics when accounting for finite size effects.

本研究解决了矩阵去噪中一个重要问题，即在隐藏信号不是旋转不变的情况下进行去噪。提出了一种基于贝叶斯的方法来分析具有因子结构的矩阵去噪，发现存在去噪-因子化转变，并提出在此转变后，通过利用信号的先验信息和因子化结构可以实现更好的去噪效果。这一发现对理解矩阵去噪的理论和算法具有重要影响。

超对称矩阵去噪的相图研究超越旋转不变性