Related papers: High-Dimensional Random Fields and Random Matrix T…
High-dimensional random landscapes underlie phenomena as diverse as glassy physics and optimization in machine learning, and even their simplest toy models already display extraordinarily rich behavior. This thesis aims to deepen our…
We study rough high-dimensional landscapes in which an increasingly stronger preference for a given configuration emerges. Such energy landscapes arise in glass physics and inference. In particular we focus on random Gaussian functions, and…
We start with a rather detailed, general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random $N (\gg 1)$-dimensional Gaussian landscape and confined by a…
Finding the mean of the total number of stationary points for N-dimensional random Gaussian landscapes can be reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. First such a reduction is…
We give an asymptotic evaluation of the complexity of spherical p-spin spin-glass models via random matrix theory. This study enables us to obtain detailed information about the bottom of the energy landscape, including the absolute minimum…
We present a null model for single- and multi-layered complex systems constructed using homogeneous and isotropic random Gaussian maps. By means of a Kac-Rice formalism, we show that the mean number of fixed points can be calculated as the…
Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite size $2$-spin spherical model using both…
We study the energy landscape near the ground state of a model of a single particle in a random potential with trivial topology. More precisely, we find the large dimensional limit of the Hessian spectrum at the global minimum of the…
We calculate the mean total number of equilibrium points in a system of $N$ random autonomous ODE's introduced by Cugliandolo et al. in 1997 to describe non-relaxational glassy dynamics on the high-dimensional sphere. In doing it we suggest…
Marginal optima are minima or maxima of a function with many nearly flat directions. In settings with many competing optima, marginal ones tend to attract algorithms and physical dynamics. Often, the important family of marginal attractors…
We study isotropic Gaussian random fields on the high-dimensional sphere with an added deterministic linear term, also known as mixed p-spin Hamiltonians with external field. We prove that if the external field is sufficiently strong, then…
We investigate thermodynamic phase transitions of the joint presence of spin glass (SG) and random field (RF) using a random graph model that allows us to deal with the quenched disorder. Therefore, the connectivity becomes a controllable…
Tensor models play an increasingly prominent role in many fields, notably in machine learning. In several applications, such as community detection, topic modeling and Gaussian mixture learning, one must estimate a low-rank signal from a…
We study the 3-spin spherical model with mean-field interactions and Gaussian random couplings. For moderate system sizes of up to 20 spins, we obtain all stationary points of the energy landscape by means of the numerical polynomial…
We calculate the density of stationary points and minima of a $N\gg 1$ dimensional Gaussian energy landscape. We use it to show that the point of zero-temperature replica symmetry breaking in the equilibrium statistical mechanics of a…
In this PhD Thesis we investigate the geometry of random fields on compact Riemannian manifolds, in particular the two-dimensional sphere. In the first part, we characterize isotropic Gaussian fields on homogeneous spaces of a compact group…
Finding the mean of the total number N_{tot} of critical points for N-dimensional random energy landscapes is reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. For any finite N we provide the…
We study the annealed complexity of a random Gaussian homogeneous polynomial on the $N$-dimensional unit sphere in the presence of deterministic polynomials that depend on fixed unit vectors and external parameters. In particular, we…
We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense…
We study the spectral properties of and spectral-crossovers between different random matrix ensembles (Poissonian, GOE, GUE) in correlated spin-chain systems, in the presence of random magnetic fields, and the scalar spin-chirality term,…