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.

研究高维能量景观，并使用 Kac-Rice 方法框架进行分析，以探索平均场玻璃模型的退火复杂性。