High-dimensional random landscapes: from typical to large deviations

In these notes 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.