Related papers: The Sherrington-Kirkpatrick model: an overview
We consider a multi-layer Sherrington-Kirkpatrick spin-glass as a model for deep restricted Boltzmann machines and we solve for its quenched free energy, in the thermodynamic limit and allowing for a first step of replica symmetry breaking.…
In this note we study the zero temperature dynamics of the Sherrington Kirkppatric model and we investigate the statistical properties of the configurations that are obtained in the large time limit. We find that the replica symmetry is…
In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous-Hoover representation encoded by a function $\sigma:[0,1]^4\to\{-1,+1\}$, and one can…
This article is a discussion of some characteristic properties in connection with global models, particularly for the application of prediction, such as the approximation property, the interpolation property and the transmission property.
Latent space models are powerful statistical tools for modeling and understanding network data. While the importance of accounting for uncertainty in network analysis has been well recognized, the current literature predominantly focuses on…
This document collects several approaches attempted in proving the Chen-Raspaud Conjecture for k = 3. Each approach is detailed with full proofs and explanations of why it failed. This compilation aims to provide insights and a foundation…
We highlight overlap as one of the simplest inequalities in linear space that yields a number of useful results. One obtains the Cauchy-Schwarz inequality as a special case. More importantly, a variant of it is seen to work desirably in…
We extend to the anisotropic setting the existence of solutions for the Kirchhoff-Plateau problem and its dimensional reduction.
I review the classical theory of likelihood based inference and consider how it is being extended and developed for use in complex models and sampling schemes.
A comprehensive review will be given about the rich mathematical structure of mean field spin glass theory, mostly developed, until now, in the frame of the methods of theoretical physics, based on deep physical intuition and hints coming…
Modern applications routinely collect high-dimensional data, leading to statistical models having more parameters than there are samples available. A common solution is to impose sparsity in parameter estimation, often using penalized…
We give a general overview of the influence of William Thurston on the French mathematical school and we show how some of the major problems he solved are rooted in the French mathematical tradition. At the same time, we survey some of…
A review of the replica symmetric solution of the classical and quantum, infinite-range, Sherrington-Kirkpatrick spin glass is presented.
This report introduces general ideas and some basic methods of the Bayesian probability theory applied to physics measurements. Our aim is to make the reader familiar, through examples rather than rigorous formalism, with concepts such as:…
Beginning in the 1970s, statistician-cum-logician Per Martin-L\"of wrote a series of papers developing what became Martin-L\"of type theory, realizing a system where the distinction between mathematics and programming disappears. Inspired…
Michel Talagrand played a decisive role in the transformation of mean-field spin glass theory into a rigorous mathematical subject. This chapter offers a narrative account of that development. We begin with the physical origins of the…
While the Gibbs states of spin glass models have been noted to have an erratic dependence on temperature, one may expect the mean over the disorder to produce a continuously varying ``quenched state''. The assumption of such continuity in…
We calculate the probability distribution of the overlap between a spin glass and a copy of itself in which the bonds are randomly perturbed in varying degrees. The overlap distribution is shown to go to a delta distribution in the…
In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient…
Probabilistic modeling is cyclical: we specify a model, infer its posterior, and evaluate its performance. Evaluation drives the cycle, as we revise our model based on how it performs. This requires a metric. Traditionally, predictive…