Embedded Files
Francesco Guerra
Francesco Guerra
The replica trick on interpolating replicas
The replica trick on interpolating replicas
We review the historical development of the replica trick, and point out the great power of the method. The traditional interpretation is based on analytic continuation with respect to the number of replicas. From this point of view, replica symmetry and replica symmetry breaking are connected to a different choice of analytic continuations starting from an integer number of replicas. A different possible interpretation is based on direct interpolation on the number of replicas. Then replica symmetry breaking corresponds to a phase transition. Derrida Random Energy Model gives the simplest example for the interpolating replica trick. We give also some recent results on the rigorous proof that the Almeida-Thouless line is the line of replica symmetry breaking for the Sherrington-Kirkpatrick model in random external field.
Dmitry Panchenko
Dmitry Panchenko
The generalized TAP free energy
The generalized TAP free energy
I will start with the definition of the generalized TAP correction motivated by the infinitary nature of the tree of states in the Parisi ansatz (this definition was first introduced by Eliran Subag in the setting of spherical models) and give a formula for the correction in the setting of mixed p-spin models with Ising spins. I will then describe various properties of the generalized TAP states (the maximizers in the generalized TAP free energy representation):1. give a formula for the energy of TAP states as a function of their self-overlap;2. write down critical point equations;3. give a condition for the TAP correction to be classical (for which Plefka's condition is necessary);4. mention that the generalized TAP states include ancestor states by definition, but all TAP states have the same order parameter as ancestor states;5. mention that almost all of these results have direct analogues at zero temperature.This is joint work with Wei-Kuo Chen and Eliran Subag; see arXiv:1812.05066 and arXiv:1903.01030.
Véronique Gayrard
Véronique Gayrard
I will review the main rigorous results on aging of activated dynamics of mean-field spin glasses obtained over the last decade, including the most recent ones.
Using field-theoretic perturbative renormalization group methods on the Bray-Roberts reduced Landau-Ginzburg-type theory for a short-range Ising spin glass in space of dimension d, we show that at nonzero magnetic field the nature of the corresponding transition is modified as follows: a) for dimensions 6<d≤8 and sufficiently weak magnetic field, the phase just below the transition is the so-called one-step RSB phase, instead of the full RSB that occurs in mean-field theory; the transition remains continuous with a diverging correlation length. Further, b) for d-6 small and positive, there is a tricritical point on the transition line at a magnetic field value that tends to zero as d→6^+, such that the transition is continuous at field strengths lower than the tricritical value, but at higher field values it becomes quasi-first-order, that is the correlation length does not diverge, and there is a jump in part of the order parameter, but no latent heat. Finally, c) for d≤6, we argue that the quasi-first-order transition to the one-step RSB phase persists down to arbitrarily small nonzero fields.
Antonio Auffinger
Antonio Auffinger
Quenched complexity of saddles in the spherical p-spin model
Quenched complexity of saddles in the spherical p-spin model
I will discuss the quenched complexity of saddles in the spherical p spin-glass model. Our main result confirms the almost surely existence of a layered structure of critical points of the energy landscape. I will then relate this computation to a detailed information about the landscape around the ground state energy and the structure of the Parisi measure at zero temperature.Based on a joint work with Julian Gold (Northwestern University) and Yi Gu (Northwestern University).
The quest for understanding replica symmetry breaking led to the cavity method. The single-sample cavity equations, appropriately iterated, become efficient message-passing algorithms. Understanding the performance of these algorithms on large size system is best done using... replicas. This talk will explore some aspects of this circular construction which was elaborated gradually in the last 40 years.
Andrea Montanari
Andrea Montanari
Full replica symmetry breaking and computational complexity
Full replica symmetry breaking and computational complexity
We consider the problem of finding an approximate ground state of the Sherrington-Kirkpatrick model. Namely, we seek an algorithm that takes as input a realization of the couplings matrix J and returns as output a spin configuration σ, such that 〈σ,Jσ〉≥(1−ε) max〈σ,Jσ〉with high probability (over the random choice of J). This problem in NP hard in worst case, and indeed hard to approximate within a logarithmic factor. We show that the full replica symmetry breaking structure of the low-energy states can be leveraged to construct an algorithm to achieve the stated objective, with complexity at most C(ε)N²logN.
Understanding the geometrical properties of high-dimensional, random energy landscapes is an important problem in the physics of glassy systems, with plenty of interdisciplinary applications. In this talk I will focus on the energy landscape of the p-spin model with spherical spins, and present a framework to compute the statistical properties of the saddle points surrounding local minima of the landscape. I will discuss how this computation allows to extract information on the distribution of energy barriers surrounding the minimum, as well as on its connectivity in configuration space. I will comment on the dynamical implications on these results, especially for the activated regime of the dynamics, relevant when the dimension of configuration space is large but finite.
Lenka Zdeborová
Lenka Zdeborová
The successes and the multitude of applications of deep learning methods have spurred efforts towards quantitative modeling of the performance of deep neural networks. In particular, an information-theoretic approach linking generalization capabilities to compression has been receiving increasing interest. Nevertheless, it is in practice computationally intractable to compute entropies and mutual informations in industry-sized neural networks. In this talk, we will consider instead a class of models of deep neural networks, for which an expression for these information-theoretic quantities can be derived from the replica method. We will examine how mutual informations between hidden and input variables can be reported along the training of such neural networks on synthetic datasets. Finally we will discuss the numerical results of a few training experiments.This work was done in collaboration with Andre Manoel (Owkin), Clément Luneau (EPFL), Jean Barbier (EPFL), Nicolas Macris (EPFL), Florent Krzakala (LPS ENS) and Lenka Zdeborova (IPHT CEA).
Amin Coja-Oghlan
Amin Coja-Oghlan
We determine the rank of a random matrix A over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula vindicates a conjecture of Lelarge (2013). The proofs are based on the Aizenman-Sims-Starr scheme and a novel random perturbation, applicable to any matrix, that likely diminishes the number of short linear relations.
Google Sites
Report abuse