On the phase diagram of extensive-rank symmetric matrix denoising beyond rotational invariance

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.

Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise

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.

Fundamental limits of overparametrized shallow neural networks for supervised learning

We carry out an information-theoretical analysis of a two-layer neural network trained from input-output pairs generated by a teacher network with matching architecture, in overparametrized regimes. Our results come in the form of bounds relating i) the mutual information between training data and network weights, or ii) the Bayes-optimal generalization error, to the same quantities but for a simpler (generalized) linear model for which explicit expressions are rigorously known. Our bounds, which are expressed in terms of the number of training samples, input dimension and number of hidden units, thus yield fundamental performance limits for any neural network (and actually any learning procedure) trained from limited data generated according to our two-layer teacher neural network model. The proof relies on rigorous tools from spin glasses and is guided by ``Gaussian equivalence principles'' lying at the core of numerous recent analyses of neural networks. With respect to the existing literature, which is either non-rigorous or restricted to the case of the learning of the readout weights only, our results are information-theoretic (i.e. are not specific to any learning algorithm) and, importantly, cover a setting where all the network parameters are trained.

Mismatched estimation of non-symmetric rank-one matrices corrupted by structured noise

We study the performance of a Bayesian statistician who estimates a rank-one signal corrupted by non-symmetric rotationally invariant noise with a generic distribution of singular values. As the signal-to-noise ratio and the noise structure are unknown, a Gaussian setup is incorrectly assumed. We derive the exact analytic expression for the error of the mismatched Bayes estimator and also provide the analysis of an approximate message passing (AMP) algorithm. The first result exploits the asymptotic behavior of spherical integrals for rectangular matrices and of low-rank matrix perturbations; the second one relies on the design and analysis of an auxiliary AMP. The numerical experiments show that there is a performance gap between the AMP and Bayes estimators, which is due to the incorrect estimation of the signal norm.

Bayes-optimal limits in structured PCA, and how to reach them

We study the paradigmatic spiked matrix model of principal components analysis, where the rank-one signal is corrupted by additive noise. While the noise is typically taken from a Wigner matrix with independent entries, here the potential acting on the eigenvalues has a quadratic plus a quartic component. The quartic term induces strong correlations between the matrix elements, which makes the setting relevant for applications but analytically challenging. Our work provides the first characterization of the Bayes-optimal limits for inference in this model with structured noise. If the signal prior is rotational-invariant, then we show that a spectral estimator is optimal. In contrast, for more general priors, the existing approximate message passing algorithm (AMP) falls short of achieving the information-theoretic limits, and we provide a justification for this sub-optimality. Finally, by generalizing the theory of Thouless-Anderson-Palmer equations, we cure the issue by proposing a novel AMP which matches the theoretical limits. Our information-theoretic analysis is based on the replica method, a powerful heuristic from statistical mechanics; instead, the novel AMP comes with a rigorous state evolution analysis tracking its performance in the high-dimensional limit. Even if we focus on a specific noise distribution, our methodology can be generalized to a wide class of trace ensembles, at the cost of more involved expressions.

The price of ignorance: how much does it cost to forget noise structure in low-rank matrix estimation?

We consider the problem of estimating a rank-1 signal corrupted by structured rotationally invariant noise, and address the following question: how well do inference algorithms perform when the noise statistics is unknown and hence Gaussian noise is assumed? While the matched Bayes-optimal setting with unstructured noise is well understood, the analysis of this mismatched problem is only at its premises. In this paper, we make a step towards understanding the effect of the strong source of mismatch which is the noise statistics. Our main technical contribution is the rigorous analysis of a Bayes estimator and of an approximate message passing (AMP) algorithm, both of which incorrectly assume a Gaussian setup. The first result exploits the theory of spherical integrals and of low-rank matrix perturbations; the idea behind the second one is to design and analyze an artificial AMP which, by taking advantage of the flexibility in the denoisers, is able to "correct" the mismatch. Armed with these sharp asymptotic characterizations, we unveil a rich and often unexpected phenomenology. For example, despite AMP is in principle designed to efficiently compute the Bayes estimator, the former is outperformed by the latter in terms of mean-square error. We show that this performance gap is due to an incorrect estimation of the signal norm. In fact, when the SNR is large enough, the overlaps of the AMP and the Bayes estimator coincide, and they even match those of optimal estimators taking into account the structure of the noise.

Statistical limits of dictionary learning: random matrix theory and the spectral replica method

We consider increasingly complex models of matrix denoising and dictionary learning in the Bayes-optimal setting, in the challenging regime where the matrices to infer have a rank growing linearly with the system size. This is in contrast with most existing literature concerned with the low-rank (i.e., constant-rank) regime. We first consider a class of rotationally invariant matrix denoising problems whose mutual information and minimum mean-square error are computable using standard techniques from random matrix theory. Next, we analyze the more challenging models of dictionary learning. To do so we introduce a novel combination of the replica method from statistical mechanics together with random matrix theory, coined spectral replica method. It allows us to conjecture variational formulas for the mutual information between hidden representations and the noisy data as well as for the overlaps quantifying the optimal reconstruction error. The proposed methods reduce the number of degrees of freedom from $\Theta(N^2)$ (matrix entries) to $\Theta(N)$ (eigenvalues or singular values), and yield Coulomb gas representations of the mutual information which are reminiscent of matrix models in physics. The main ingredients are the use of HarishChandra-Itzykson-Zuber spherical integrals combined with a new replica symmetric decoupling ansatz at the level of the probability distributions of eigenvalues (or singular values) of certain overlap matrices.

Performance of Bayesian linear regression in a model with mismatch

For a model of high-dimensional linear regression with random design, we analyze the performance of an estimator given by the mean of a log-concave Bayesian posterior distribution with gaussian prior. The model is mismatched in the following sense: like the model assumed by the statistician, the labels-generating process is linear in the input data, but both the classifier ground-truth prior and gaussian noise variance are unknown to her. This inference model can be rephrased as a version of the Gardner model in spin glasses and, using the cavity method, we provide fixed point equations for various overlap order parameters, yielding in particular an expression for the mean-square reconstruction error on the classifier (under an assumption of uniqueness of solutions). As a direct corollary we obtain an expression for the free energy. Similar models have already been studied by Shcherbina and Tirozzi and by Talagrand, but our arguments are more straightforward and some assumptions are relaxed. An interesting consequence of our analysis is that in the random design setting of ridge regression, the performance of the posterior mean is independent of the noise variance (or "temperature") assumed by the statistician, and matches the one of the usual (zero temperature) ridge estimator.

High-dimensional inference: a statistical mechanics perspective

Statistical inference is the science of drawing conclusions about some system from data. In modern signal processing and machine learning, inference is done in very high dimension: very many unknown characteristics about the system have to be deduced from a lot of high-dimensional noisy data. This "high-dimensional regime" is reminiscent of statistical mechanics, which aims at describing the macroscopic behavior of a complex system based on the knowledge of its microscopic interactions. It is by now clear that there are many connections between inference and statistical physics. This article aims at emphasizing some of the deep links connecting these apparently separated disciplines through the description of paradigmatic models of high-dimensional inference in the language of statistical mechanics. This article has been published in the issue on artificial intelligence of Ithaca, an Italian popularization-of-science journal. The selected topics and references are highly biased and not intended to be exhaustive in any ways. Its purpose is to serve as introduction to statistical mechanics of inference through a very specific angle that corresponds to my own tastes and limited knowledge.

Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures

We consider a generic class of log-concave, possibly random, (Gibbs) measures. Using a new type of perturbation we prove concentration of an infinite family of order parameters called multioverlaps. These completely parametrise the quenched Gibbs measure of the system, so that their self-averaging behavior implies a simple representation of asymptotic Gibbs measures, as well as decoupling of the variables at hand in a strong sense. Our concentration results may prove themselves useful in several contexts. In particular in machine learning and high-dimensional inference, log-concave measures appear in convex empirical risk minimisation, maximum a-posteriori inference or M-estimation. We believe that our results may be applicable in establishing some type of "replica symmetric formulas" for the free energy, inference or generalisation error in such settings.