Exploring high-dimensional random landscapes: from spin glasses to random matrices, passing through simple chaotic systems

TL;DR

This paper combines landscape and dynamical approaches to analyze high-dimensional random landscapes, revealing new insights into their structure, stability, and complexity across various models including spin glasses and neural networks.

Contribution

It introduces exact dynamic-static comparisons, stability-based complexity calculations, and new methods to analyze minima and eigenvector overlaps in high-dimensional random landscapes.

Findings

Exact dynamic-static comparison in non-reciprocal models

Calculation of annealed complexity for neural networks

Probing barriers and minima distribution in p-spin landscapes

Abstract

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 understanding of that behavior, by combining landscape-based approaches, via the Kac-Rice formalism, with dynamical approaches, paying special attention to both systems with reciprocal and with non-reciprocal interactions. After surveying core techniques and results through the spherical p-spin model, this thesis delivers three main advances: (i) exact dynamic-static comparison in a solvable class of models with non-reciprocal interactions, pinpointing differences and similarities of the two approaches; (ii) a stability-based calculation of the mean number of fixed points (i.e., annealed complexity) of the Sompolinsky-Crisanti-Sommers random neural network, for any level of non-reciprocity; (iii) two approaches to probe the barriers and the distribution of deep local minima in the landscape of the p-spin model; (iv) some results on the overlaps among eigenvectors of spiked, correlated random matrices, which are useful to explore the geometry of energy landscapes. Together, these results sharpen our understanding of these systems, while providing new tools and opening new doors for future research directions.