Suppose W is a symmetric matrix with i.i.d. Gaussian entries (GOE). We consider the problem of certifying an upper bound on the maximum value of the quadratic form x'Wx over all vectors x in the hypercube {+/- 1/sqrt(n)}^n. Under certain hardness assumptions, we show that no polynomial-time algorithm can certify a better upper bound than the trivial spectral bound (the top eigenvalue of W). One consequence of this is that no efficiently-computable convex relaxation (e.g. sum-of-squares) can improve on the spectral bound. In contrast, the true maximum value of x'Wx over the hypercube is known to be smaller than the spectral bound (with high probability), given by the Parisi constant from statistical physics.

Our proof proceeds in two steps. First, we give a reduction from a certain detection problem in the negatively-spiked Wishart model to the above certification problem. We then give formal evidence that this Wishart problem is computationally hard, using a method of Hopkins and Steurer based on low-degree truncation of the likelihood ratio. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over the hypercube is much larger than that of a GOE matrix.

Based on joint work with Afonso Bandeira and Tim Kunisky, available at https://arxiv.org/abs/1902.07324.

Alex Wein is a Courant Instructor (postdoc) in the math department at the Courant Institute at New York University. Previously he received his PhD from the math department at MIT, advised by Ankur Moitra. Homepage: https://cims.nyu.edu/~aw128/