SciPost Phys. Lect. Notes 102 (2025) ·
published 13 October 2025
|
· pdf
We discuss tools and concepts that emerge when studying high-dimensional random landscapes, i.e., random functions on high-dimensional spaces. As an illustrative example, we consider an inference problem in two forms: low-rank matrix estimation (case 1) and low-rank tensor estimation (case 2). We show how to map the inference problem onto the optimization problem of a high-dimensional landscape, which exhibits distinct geometrical properties in the two cases. We discuss methods for characterizing typical realizations of these landscapes and their optimization through local dynamics. We conclude by highlighting connections between the landscape problem and large deviation theory.
SciPost Phys. 10, 002 (2021) ·
published 4 January 2021
|
· pdf
We focus on the energy landscape of a simple mean-field model of glasses and analyze activated barrier-crossing by combining the Kac-Rice method for high-dimensional Gaussian landscapes with dynamical field theory. In particular, we consider Langevin dynamics at low temperature in the energy landscape of the pure spherical $p$-spin model. We select as initial condition for the dynamics one of the many unstable index-1 saddles in the vicinity of a reference local minimum. We show that the associated dynamical mean-field equations admit two solutions: one corresponds to falling back to the original reference minimum, and the other to reaching a new minimum past the barrier. By varying the saddle we scan and characterize the properties of such minima reachable by activated barrier-crossing. Finally, using time-reversal transformations, we construct the two-point function dynamical instanton of the corresponding activated process.
Dr Ros: "I would like to thank the Refe..."
in Submissions | report on High-dimensional random landscapes: from typical to large deviations