Phase Diagram of Extensive-Rank Symmetric Matrix Denoising beyond Rotational Invariance

Matrix denoising is central to signal processing and machine learning. Its statistical analysis when the matrix to infer has a factorized 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 an efficient Bayes-optimal denoising algorithm, called the rotational invariant estimator, 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 (as appearing in high-energy physics) due to the lack of rotation symmetry, but rather a hybrid between the two. In this paper, we make progress toward the understanding of Bayesian matrix denoising when the hidden signal is a factored matrix <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mrow><a:mi mathvariant="bold">X</a:mi></a:mrow><a:msup><a:mrow><a:mi mathvariant="bold">X</a:mi></a:mrow><a:mrow><a:mo>⊺</a:mo></a:mrow></a:msup></a:mrow></a:math> that is not rotationally invariant. Monte Carlo simulations suggest the existence of a denoising-factorization 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, though algorithmically hard. We also argue that it is only beyond the transition that factorization, i.e., estimating <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi mathvariant="bold">X</e:mi></e:mrow></e:math> itself, becomes possible up to irresolvable 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 reproducible by the replica approach of Sakata and Kabashima. Using numerical insights, we delimit the portion of phase diagram where we conjecture the mean-field theory to be exact and correct it using universality when it is not. Our complete matches well the numerics in the whole phase diagram when considering finite-size effects.