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start [2025/04/29 19:49]
Djalil Chafaï 		start [2025/05/08 18:03] (Version actuelle)
Djalil Chafaï
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       * 28 avril 2025. [[|Benjamin Gess (Berlin & Leipzig)]]. **Gradient flow structures and large deviations for porous media equations.**\\ //While the derivation of nonlinear but uniformly parabolic equations from microscopic dynamics, fluctuations around these limits, and the corresponding canonical choice of a gradient flow structure are now well-understood, less is known for equations with either degenerate, or unbounded diffusivity. Specifically, for the model case of the porous medium equation (PME), multiple gradient flow structures have been identified since the works of Brézis and Otto; however, it remains unclear which, if any, are thermodynamic in nature, meaning that they arise through the large deviations of a microscopic model. In this talk, to demonstrate that the (formal) geometric picture we obtain is thermodynamic, we examine a rescaling of the zero-range process (ZRP) that converges to the PME and prove a full large deviations principle. The proof of this result is complicated by the degeneracy and unboundedness of the diffusivity. We then discuss how the large deviations rigorously identify a gradient flow structure for the PME.//       * 28 avril 2025. [[|Benjamin Gess (Berlin & Leipzig)]]. **Gradient flow structures and large deviations for porous media equations.**\\ //While the derivation of nonlinear but uniformly parabolic equations from microscopic dynamics, fluctuations around these limits, and the corresponding canonical choice of a gradient flow structure are now well-understood, less is known for equations with either degenerate, or unbounded diffusivity. Specifically, for the model case of the porous medium equation (PME), multiple gradient flow structures have been identified since the works of Brézis and Otto; however, it remains unclear which, if any, are thermodynamic in nature, meaning that they arise through the large deviations of a microscopic model. In this talk, to demonstrate that the (formal) geometric picture we obtain is thermodynamic, we examine a rescaling of the zero-range process (ZRP) that converges to the PME and prove a full large deviations principle. The proof of this result is complicated by the degeneracy and unboundedness of the diffusivity. We then discuss how the large deviations rigorously identify a gradient flow structure for the PME.//
       * 5 mai 2025. [[|Giulio Biroli (LPENS)]]. **Random Energy Model, Kernel Density Estimation and Diffusion Models for Generative AI.**\\ //The Random Energy Model (REM), introduced by Derrida in 1981, has become a cornerstone in the study of disordered systems, with profound implications in both physics and probability theory. Kernel density estimation (KDE) is a fundamental non-parametric tool in statistics for estimating probability densities from data. Diffusion models have recently emerged as powerful generative methods in artificial intelligence, enabling the creation of images, audio, and video.\\       * 5 mai 2025. [[|Giulio Biroli (LPENS)]]. **Random Energy Model, Kernel Density Estimation and Diffusion Models for Generative AI.**\\ //The Random Energy Model (REM), introduced by Derrida in 1981, has become a cornerstone in the study of disordered systems, with profound implications in both physics and probability theory. Kernel density estimation (KDE) is a fundamental non-parametric tool in statistics for estimating probability densities from data. Diffusion models have recently emerged as powerful generative methods in artificial intelligence, enabling the creation of images, audio, and video.\\
-In this talk, I will present a connection between these three distinct problems. By focusing on the asymptotic regime of high data dimensionality and large sample sizes, I will show how analytical techniques and theoretical insights from the REM can be leveraged to study both KDE and diffusion models. In particular, I will highlight how phase transitions observed in the REM, and their connection to extreme value theory and limiting laws of sums of i.i.d. random variables, have remarkable counterparts and consequences for these other problems in statistics and machine learning. +In this talk, I will present a connection between these three distinct problems. By focusing on the asymptotic regime of high data dimensionality and large sample sizes, I will show how analytical techniques and theoretical insights from the REM can be leveraged to study both KDE and diffusion models. In particular, I will highlight how phase transitions observed in the REM, and their connection to extreme value theory and limiting laws of sums of i.i.d. random variables, have remarkable counterparts and consequences for these other problems in statistics and machine learning.// 
- +      * 12 mai 2025. [[|Gérard Ben Arous (NYU & IHES)]]. **Local geometry of high-dimensional mixture models: Effective spectral theory and dynamical transitions.**\\ //I will survey recent progress in the understanding of the optimization dynamics for high dimensional ML tasks, and in particular the local geometry of empirical risks in high dimensions in the case of classification tasks, via the spectral theory of their Hessian and information matrices. Joint work with Aukosh Jagannath, Jiaoyang Huang, Reza Gheissari.//
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-      * 12 mai 2025. [[|Gérard Ben Arous (NYU)]]. **Titre à préciser.**\\ ////+
       * 2 juin 2025. Assemblée générale du projet PSL [[https://www.ceremade.dauphine.fr/dokuwiki/psl-spm:start|Statistical Physics and Mathematics]]\\ [[|Pierre Le Doussal (LPENS)]]. **Titre à préciser.**\\ [[|Mathieu Lewin (CEREMADE)]]. **Titre à préciser.**\\ ////         * 2 juin 2025. Assemblée générale du projet PSL [[https://www.ceremade.dauphine.fr/dokuwiki/psl-spm:start|Statistical Physics and Mathematics]]\\ [[|Pierre Le Doussal (LPENS)]]. **Titre à préciser.**\\ [[|Mathieu Lewin (CEREMADE)]]. **Titre à préciser.**\\ ////  
  
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