Tight size-degree bounds for sums-of-squares proofs

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size n^Ω(d) for values of d = d(n) from constant all the way up to n^δ for some universal constant δ. This shows that the n^O(d) running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Krajícek'04] and [Dantchev and Riis'03], and then applying a restriction argument as in [Atserias, Müller, and Oliva'13] and [Atserias, Lauria, and Nordström'14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively.