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We study how well functions over the boolean hypercube of the form $f_k(x)=(|x|-k)(|x|-k-1)$ can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in $ell_{infty}$-norm as well as in $ell_1$-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on $ell_1$-approximation of $f_k$; (2) a proof that Grigorievs lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from his work; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.
The degree-$4$ Sum-of-Squares (SoS) SDP relaxation is a powerful algorithm that captures the best known polynomial time algorithms for a broad range of problems including MaxCut, Sparsest Cut, all MaxCSPs and tensor PCA. Despite being an explicit alg
We characterize the approximate monomial complexity, sign monomial complexity , and the approximate L 1 norm of symmetric functions in terms of simple combinatorial measures of the functions. Our characterization of the approximate L 1 norm solves th
We study the computational complexity of approximating the 2->q norm of linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as connections between this question and issues arising in quantum information theory and the study o
We prove that for every $epsilon>0$ and predicate $P:{0,1}^krightarrow {0,1}$ that supports a pairwise independent distribution, there exists an instance $mathcal{I}$ of the $mathsf{Max}P$ constraint satisfaction problem on $n$ variables such that no
We prove that with high probability over the choice of a random graph $G$ from the ErdH{o}s-Renyi distribution $G(n,1/2)$, a natural $n^{O(varepsilon^2 log n)}$-time, degree $O(varepsilon^2 log n)$ sum-of-squares semidefinite program cannot refute th