Complex Energy Landscapes in Spiked-Tensor and Simple Glassy Models: Ruggedness, Arrangements of Local Minima, and Phase Transitions

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 us 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.