For p ≥ 1 and (gij)1≤i,j≤n being a matrix of i.i.d. standard Gaussian entries, we study the n-limit of the ℓp-Gaussian-Grothendieck problem defined as n max (Equation presented). The case p = 2 corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when p = ∞, 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 < ∞. For the former, we compute the limit of the ℓp-Gaussian-Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates.
Bibliographical noteFunding Information:
This work was supported by the National Science Foundation [DMS-17-52184].
This work was supported by the National Science Foundation [DMS-17-52184]. WKC and AS would like to thank Souvik Dhara for explaining the results of  and for a helpful discussion.
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