Complex energy landscapes in spiked-tensor and simple glassy models: ruggedness, arrangements of local minima and phase transitions

TL;DR

This paper analyzes high-dimensional energy landscapes in glassy systems and inference models, revealing phase transitions and landscape complexity using a novel Kac-Rice framework that improves upon previous methods.

Contribution

It introduces a new Kac-Rice based framework for computing landscape complexity, surpassing the limitations of the replica method in glassy and inference models.

Findings

Characterization of phase transitions in energy landscapes

Development of a generalized Kac-Rice complexity computation

Identification of limitations in the replica method

Abstract

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 on the spiked-tensor model and generalizations. We thoroughly analyze the statistical properties of the corresponding landscapes and characterize the associated geometrical phase transitions. In order to perform our study, we develop a framework based on the Kac-Rice method that allows to compute the complexity of the landscape, i.e. the logarithm of the typical number of stationary points and their Hessian. This approach generalizes the one used to compute rigorously the annealed complexity of mean-field glass models. We discuss its advantages with respect to previous frameworks, in particular the thermodynamical replica method which is shown to lead to partially incorrect predictions.