High-dimensional random landscapes and random matrices

Most of optimization problems can be formulated as search of the global minimum of a cost function which is convenient to think of as a landscape in configuration space. When landscapes are high-dimensional and random the search is difficult and one would like to understand generic features of such landscapes. Simple, yet rich and non-trivial models of random landscapes are provided by mean-field spin glasses and related systems. I am going to present a picture of the “topology trivialization transition” (in the sense of an abrupt reduction of the number of stationary points and minima of the underlying energy landscape) which takes place in the vicinity of the zero-temperature glass transition of p-spin spherical model of spin glasses. In particular, I will emphasize the role of the “edge scaling” and the Tracy–Widom distribution of the largest eigenvalues of random matrices for providing some universal features of the above transition. I also discuss how similar methods can be used for counting equilibria in a system of autonomous random differential equations and for getting bounds on the number of connected domains of random algebraic varieties. The results to be presented in the talk were obtained in recent joint works with C. Nadal, P. Le Doussal, B. Khoruzhenko, A. Lerrio, and E. Lundberg.