An Argmax Principle for Sum-of-Squares Relaxations on the Sphere

We develop an argmax principle for analyzing sum-of-squares relaxations of optimization problems over the unit sphere. Given a feasible pseudo-expectation, we form a polynomial of high-order pseudo-moments, such as Phi_k(u)=widetilde{mathbb E}langle x,urangle^2k. Our guiding principle is that its maximizers are rounding candidates: their local and global optimality conditions reveal the reweighed pseudo-expectation inequalities governing SoS convergence. This viewpoint unifies several problems previously analyzed by rather different techniques. We obtain three results. First, for Best Separable State, we give a degree-O\(sqrt{n/epsilon}\) SoS analysis for approximating h_sep(P) in the perfect-completeness regime, improving and simplifying Barak, Kothari and Steurer (STOC'17). The dependence is essentially tight for inverse-linear gap under the Exponential-Time Hypothesis, matching hardness from QMA(2) protocols. Second, for the matrix 2→4 norm, degree-O\(sqrt n/epsilon\) SoS gives a multiplicative \(1+epsilon\) approximation. Barak et al. (STOC'12) previously gave a comparable-time constant-gap decision algorithm; our result gives a multiplicative guarantee and extends to a family of p→ q norms with even q. Finally, for degree-d polynomial optimization, we recover the convergence theorem of Bhattiprolu et al. (FOCS'17) with a shorter, more direct proof: degree-k SoS gives approximation ratio O_d((n/k)^d/2-1). The paper introduces no new relaxation. Instead, the high-moment argmax gives a common way to read an SoS solution, unifying previously separate convergence analyses and yielding sharper bounds or simpler proofs.