A substantial amount of mathematical work has been devoted to studying structural
properties of mean-field spin glasses, and in particular, geometric properties of the Gibbs measure. Over the last ten years, ideas from spin glass theory have spurred dramatic advances in the field of random combinatorial optimization and random constraint satisfaction problems (CSPs), allowing to characterize...
A new approach to the identification of the limit free energy of
mean-field disordered systems, based on Hamilton-Jacobi equations, has
started to emerge. The goal of this talk will be to review the results
obtained so far with this method, as well as the challenges ahead. The
models considered will be either fully connected spin glasses, or models
from statistical inference,...
I will present recent results concerning the Bayesian estimation of low-rank matrices corrupted by structured noise, namely rotational invariant noise with generic spectrum. Using the replica method we derive the optimal performance limit. This is possible by exploiting the low-rank structure of the matrix signal implying that we can reduce the model to an effective quadratic model of the...
A substantial amount of mathematical work has been devoted to studying structural
properties of mean-field spin glasses, and in particular, geometric properties of the Gibbs measure. Over the last ten years, ideas from spin glass theory have spurred dramatic advances in the field of random combinatorial optimization and random constraint satisfaction problems (CSPs), allowing to characterize...
I will consider a recently introduced soft spin glass model, named the KHGPS model, in which soft spins are subjected to a local random anharmonic quartic potential and an external magnetic field, and interact through the usual SK-like random pairwise term. Depending on the control parameters, at zero temperature the model undergoes to a spin glass transition that can be in two different...
Consider n items, each of which is characterized by one of d+1 possible features in {0,...,d}. We study the inference task of learning these types by queries on subsets, or pools, of the items that only reveal a form of coarsened information on the features - in our case, the sum of all the features in the pool. Related prominent problems are the quantitative group testing problem, of which it...
For the standard vertex cover problem, linear relaxation has integrality gap 2. In our work, we explore an extension of this problem by considering i) random hyperedges and ii) low degree vertices. We conjecture the value of the integrality gap and prove almost tight upper and lower bounds. Based on joint work with N. Grometto, G. Arpino, R. Barboni and A. Bandeira.
We consider the well-posedness of inferring the input of a
randomly-initialized large ReLU neural network from its output, i.e.
characterizing injectivity.
Focusing on layerwise injectivity properties, we discuss recent
work connecting this question to spherical integral geometry, and
present a conjecture
for a sharp injectivity threshold (in terms of the expansivity of
the layer) based...
Spin glass models involving multiple replicas with constrained overlaps have been studied by (among others) Franz, Parisi, Talagrand, Panchenko and Ko. The latter three authors have shown that the limiting free energy is given by a Parisi type minimization. In this talk we will discuss how for the spherical Sherrington-Kirkpatrick (SSK, i.e. 2−spin) model it can also be expressed in terms of a...
The Kac-Rice formula allows one to study the complexity of high-dimensional Gaussian random functions (meaning asymptotic counts of critical points) via the determinants of large random matrices. We present new results on determinant asymptotics for non-invariant random matrices, and use them to compute the (annealed) complexity of the elastic manifold. This is a classical disordered elastic...
We study the (weak) triviality of the Ising Sherrington-Kirkpatrick TAP energy in the high temperature regime, up to the AT line. Applying the Kac-Rice formula and large deviation techniques, we obtain a variational formula for the TAP complexity of the form sup(q,A)∈[0,1]×R G(q, A) for certain function G(q, A).
For any β and h, we consider the solution (q∗, A∗) of the usual fixed point equa-...
In these lectures, we will overview the tools used to analyze the functional ordered parameter of mean field spin glasses at positive and zero temperature. We will also describe recent results on the p+s spherical model.
In their seminal `77 paper, Thouless, Anderson and Palmer (TAP) proposed their famous free energy for the SK model. The main focus of these talks is a related but different free energy, whose definition is guided in a natural way by properties of the Gibbs measure. It was introduced for spherical mixed p-spin models (arXiv:1804.10576) and in a joint work with Wei-Kuo Chen and Dmitry Panchenko...
I will present an algorithm which efficiently samples from the Sherrington-Kirkpatrick measure with no external field at high temperature.
The approach uses a discretized version of the stochastic localization process of Eldan, together with a subroutine for computing the mean vector, or magnetization, of a family of SK measures tilted by an appropriate external field. Our analysis shows...
The statics and dynamics of mean-field models of spin glasses have been studied in-depth by the physics community since the '70s. At the heart of this is the trade-off between the notions of replica symmetry breaking, shattering, and metastability. I will survey the current mathematical understanding of these ideas in the “simple” case of the spherical p-spin model. I will start by recalling...
In these lectures, we will overview the tools used to analyze the functional ordered parameter of mean field spin glasses at positive and zero temperature. We will also describe recent results on the p+s spherical model.
In their seminal `77 paper, Thouless, Anderson and Palmer (TAP) proposed their famous free energy for the SK model. The main focus of these talks is a related but different free energy, whose definition is guided in a natural way by properties of the Gibbs measure. It was introduced for spherical mixed p-spin models (arXiv:1804.10576) and in a joint work with Wei-Kuo Chen and Dmitry Panchenko...
