Grothendeick Lp Problem for Gaussian Matrices

The Grothendieck Lp problem is defined as an optimization problem that maximizes the quadratic form of a Gaussian matrix over the unit Lp ball. The p=2 case corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble, while for p=∞ this problem is known as the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In this talk, I will describe the limit of this optimization problem for general p and discuss some results on the behavior of the near optimizers along with some open problems. This is based on a joint work with Arnab Sen.