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  • Open Access

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

Jean Barbier1,*, Francesco Camilli1,†, Justin Ko2,‡, and Koki Okajima3,§

  • *Contact author: jbarbier@ictp.it
  • †Contact author: fcamilli@ictp.it
  • ‡Contact author: justin.ko@uwaterloo.ca
  • §Contact author: koki.okajima.25@gmail.com
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Phys. Rev. X 15, 021085 – Published 6 June, 2025

DOI: https://doi.org/10.1103/PhysRevX.15.021085

Abstract

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 XX⊺ 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 X 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 Ansatz matches well the numerics in the whole phase diagram when considering finite-size effects.

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Popular Summary

Denoising large data matrices is a fundamental problem across diverse fields such as signal processing, finance, and neuroscience. In many cases, the underlying matrix has a factorized structure, such as when it represents an empirical covariance matrix. Recovering the matrix itself can be useful as not only a primary goal but also a way to better estimate the original covariance. However, due to various invariances, the matrix cannot be uniquely identified without incorporating some prior knowledge. This is where Bayesian inference is essential. In this work, we quantify the performance of Bayesian-optimal methods for denoising such factorized matrices corrupted by noise.

Despite significant prior research, a complete statistical analysis of this denoising problem when the matrix to recover has extensive rank has remained elusive due to the complex structure of the model, which behaves more like a “matrix glass” than a vector of spins. Drawing on numerical insights, we develop a theory that reveals two distinct phases in the denoising process: In one phase, optimal performance does not require leveraging the matrix’s factorized structure, while in the other, using this structure becomes crucial to achieving the best results.

The resulting phase diagram brings together various recent findings into a unified framework that clarifies when and how the internal structure of a matrix influences denoising performance. Beyond this specific application, the methods we introduce have broader relevance, offering tools to analyze learning transitions in other complex systems such as neural networks.

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Phys. Rev. X 15, 021085– Published 6 June, 2025
  • Received 12 November 2024
  • Accepted 6 May 2025
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