Rounding sum-of-squares relaxations

Abstract

We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-ofSquares proof system to transform a combining algorithman algorithm that maps a distribution over solutions into a (possibly weaker) solution-into a rounding algorithm that maps a solution of the relaxation to a solution of the original problem.Using this approach, we obtain algorithms that yield improved results for natural variants of several well-known problems:1. We give a quasipolynomial-time algorithm that approximates max x 2 =1 P (x) within an additive factor of ε P spectral , where ε > 0 is a constant, P is a degree d = O(1), n-variate polynomial with nonnegative coefficients, and P spectral is the spectral norm of a matrix corresponding to P 's coefficients. Beyond being of interest in its own right, obtaining such an approximation for general polynomials (with possibly negative coefficients) is a long-standing open question in quantum information theory, and our techniques have already led to improved results in this area (Brandão and Harrow, STOC '13).2. We give a polynomial-time algorithm that, given a subspace V ⊆ R n of dimension d that (almost) contains the characteristic function of a set of size n/k, finds a vector v ∈ V that satisfies Ei v 4i Ω(d −1/3 k(Ei v 2 i ) 2 ). This is a natural analytical relaxation of the problem of finding the sparsest element in a subspace, and it is also motivated by a connection to the Small-Set Expansion problem shown by Barak et al.of the previous best known algorithms for small-set expansion in a certain range of parameters.3. We use this notion of L4 vs. L2 sparsity to obtain a polynomial-time algorithm with substantially improved guarantees for recovering a planted sparse vector v in a random d-dimensional subspace of R n . If v has µn nonzero coordinates, we can recover it with high probability whenever µ O(min(1, n/d 2 )). In particular, when d √ n, this recovers a planted vector with up to Ω(n) nonzero coordinates. When d n 2/3 , our algorithm improves upon existing methods based on comparing the L1 and L∞ norms, which intrinsically require µ O 1/ √ d . We also show how this notion of L4 vs. L2 sparsity can be used to find a planted sparse vector in a random subspace, improving on a recent result of Demanet and Hand (2013).

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

“…Polynomial optimization over hyperspheres.-Theorem 1 also gives some improved results on the usefulness of a general semidefinite-programming relaxation method, called the sum-of-squares hierarchy [43,44], for polynomial optimization over hyperspheres (see, e.g., [10,45]). The relevance in physics is that pure states of a quantum system form exactly a hypersphere, and hence, some computational problems in quantum physics are, indeed, to optimize a polynomial over hyperspheres.…”

Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors

“…Subsequent works have found further applications and begun studying the problem in its own right [DH14,BKS14,QSW14]. In this problem, we are given a basis for a d-dimensional linear subspace of R n that is random except for one planted sparse direction, and the goal is to recover this sparse direction.…”

“…Sum-of-squares and non-convex methods do not share this limitation. They can recover planted vectors with constant relative sparsity even if the subspace has polynomial dimension (up to dimension O(n 1/2 ) for sum-of-squares [BKS14] and up to O(n 1/4 ) for non-convex methods [QSW14]). …”

“…Foremost, this work builds upon the SoS algorithms of [BKS14,BKS15,GM15,HSS15]. In each of these previous works, a machine learning decision problem is solved using an SDP relaxation for SoS.…”

“…In these works, the SDP value is large in the yes case and small in the no case, and the SDP value can be bounded using the spectrum of a specific matrix. This was implicit in [BKS14,BKS15], and in [HSS15] it was used to obtain a fast algorithm as well. In our work, we design spectral algorithms which use smaller matrices, inspired by the SoS certificates in previous works, to solve these machine-learning problems much faster, with almost matching guarantees.…”