On l(p)-Gaussian-Grothendieck Problem

For p >= 1 and (g(ij))(1 <= i,j <= n) being a matrix of i.i.d. standard Gaussian entries, we study the n-limit of the l(p)-Gaussian-Grothendieck problem defined as max{Sigma(n)(i,j=1) g(ij)x(i)x(j) : x is an element of R-n, Sigma(n)(i=1)vertical bar x(i)vertical bar(p) = 1}. The case p = 2 corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when p = infinity, the maximum value is essentially 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 the present work, we focus on the cases 1 <= p < 2 and 2 < p < infinity. For the former, we compute the limit of the l(p)-Gaussian-Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order of n(-)(1).