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This paper first describes, from a high-level viewpoint, the main challenges that had to be solved in order to develop a theory of spin glasses in the last fifty years. It then explains how important inference problems, notably those occurring in machine learning, can be formulated as problems in statistical physics of disordered systems. However, the main questions that we face in the analysis of deep networks require to develop a new chapter of spin glass theory, which will address the challenge of structured data.

We study generative diffusion models in the regime where both the data dimension and the sample size are large, and the score function is trained optimally. Using statistical physics methods, we identify three distinct dynamical regimes during the generative diffusion process. The generative dynamics, starting from pure noise, first encounters a speciation transition, where the broad structure of the data emerges, akin to symmetry breaking in phase transitions. This is followed by a collapse phase, where the dynamics is attracted to a specific training point through a mechanism similar to condensation in a glass phase. The speciation time can be obtained from a spectral analysis of the data’s correlation matrix, while the collapse time relates to an excess entropy measure, and reveals the existence of a curse of dimensionality for diffusion models. These theoretical findings are supported by analytical solutions for Gaussian mixtures and confirmed by numerical experiments on real datasets. Diffusion methods are widely used for generating data in AI applications. Here, authors show that optimally trained diffusion models exhibit three dynamical regimes: starting from pure noise, they reach a regime where the main data class is sealed, and finally collapse onto one training point.

Understanding the reasons for the success of deep neural networks trained using stochastic gradient-based methods is a key open problem for the nascent theory of deep learning. The types of data where these networks are most successful, such as images or sequences of speech, are characterized by intricate correlations. Yet, most theoretical work on neural networks does not explicitly model training data or assumes that elements of each data sample are drawn independently from some factorized probability distribution. These approaches are, thus, by construction blind to the correlation structure of real-world datasets and their impact on learning in neural networks. Here, we introduce a generative model for structured datasets that we call the hidden manifold model. The idea is to construct high-dimensional inputs that lie on a lower-dimensional manifold, with labels that depend only on their position within this manifold, akin to a single-layer decoder or generator in a generative adversarial network. We demonstrate that learning of the hidden manifold model is amenable to an analytical treatment by proving a “Gaussian equivalence property” (GEP), and we use the GEP to show how the dynamics of two-layer neural networks trained using one-pass stochastic gradient descent is captured by a set of integro-differential equations that track the performance of the network at all times. This approach permits us to analyze in detail how a neural network learns functions of increasing complexity during training, how its performance depends on its size, and how it is impacted by parameters such as the learning rate or the dimension of the hidden manifold.

Following the 2015 migration wave to Europe, numerous French academic institutions organized themselves to welcome refugee students and researchers. As witnessed in the past, initiatives coming from universities largely preceded national dispositions, which took place in a second phase and worked towards reinforcing them. These initiatives provide some examples demonstrating the commitment of academic communities as a whole to crucial societal issues.

As a flood of data pours from scientific and medical experiments, researchers crave more efficient computational methods to organize and analyze it. Mezard presents information on how a fast way of finding representative examples in complex data sets may be applicable to a wide range of difficult problems.

Contact tracing is an essential tool to mitigate the impact of a pandemic, such as the COVID-19 pandemic. In order to achieve efficient and scalable contact tracing in real time, digital devices can play an important role. While a lot of attention has been paid to analyzing the privacy and ethical risks of the associated mobile applications, so far much less research has been devoted to optimizing their performance and assessing their impact on the mitigation of the epidemic. We develop Bayesian inference methods to estimate the risk that an individual is infected. This inference is based on the list of his recent contacts and their own risk levels, as well as personal information such as results of tests or presence of syndromes. We propose to use probabilistic risk estimation to optimize testing and quarantining strategies for the control of an epidemic. Our results show that in some range of epidemic spreading (typically when the manual tracing of all contacts of infected people becomes practically impossible but before the fraction of infected people reaches the scale where a lockdown becomes unavoidable), this inference of individuals at risk could be an efficient way to mitigate the epidemic. Our approaches translate into fully distributed algorithms that only require communication between individuals who have recently been in contact. Such communication may be encrypted and anonymized, and thus, it is compatible with privacy-preserving standards. We conclude that probabilistic risk estimation is capable of enhancing the performance of digital contact tracing and should be considered in the mobile applications.

Compressed sensing has triggered a major evolution in signal acquisition. It consists of sampling a sparse signal at low rate and later using computational power for the exact reconstruction of the signal, so that only the necessary information is measured. Current reconstruction techniques are limited, however, to acquisition rates larger than the true density of the signal. We design a new procedure that is able to reconstruct the signal exactly with a number of measurements that approaches the theoretical limit, i.e., the number of nonzero components of the signal, in the limit of large systems. The design is based on the joint use of three essential ingredients: a probabilistic approach to signal reconstruction, a message-passing algorithm adapted from belief propagation, and a careful design of the measurement matrix inspired by the theory of crystal nucleation. The performance of this new algorithm is analyzed by statistical-physics methods. The obtained improvement is confirmed by numerical studies of several cases.

In the 17th century, Chinese medicine appeared in France; since then, it never stopped evolving and is applied by French practitioners. Today, acupuncture is widely used in clinic treatment in France.

This paper first describes, from a high level viewpoint, the main challenges that had to be solved in order to develop a theory of spin glasses in the last fifty years. It then explains how important inference problems, notably those occurring in machine learning, can be formulated as problems in statistical physics of disordered systems. However, the main questions that we face in the analysis of deep networks require to develop a new chapter of spin glass theory, which will address the challenge of structured data.