44 research outputs found

    Efficient Rounding for the Noncommutative Grothendieck Inequality

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    The classical Grothendieck inequality has applications to the design of approximation algorithms for NP-hard optimization problems. We show that an algorithmic interpretation may also be given for a noncommutative generalization of the Grothendieck inequality due to Pisier and Haagerup. Our main result, an efficient rounding procedure for this inequality, leads to a constant-factor polynomial time approximation algorithm for an optimization problem which generalizes the Cut Norm problem of Frieze and Kannan, and is shown here to have additional applications to robust principle component analysis and the orthogonal Procrustes problem

    Failure of the trilinear operator space Grothendieck theorem

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    We give a counterexample to a trilinear version of the operator space Grothendieck theorem. In particular, we show that for trilinear forms on \ell_\infty, the ratio of the symmetrized completely bounded norm and the jointly completely bounded norm is in general unbounded, answering a question of Pisier. The proof is based on a non-commutative version of the generalized von Neumann inequality from additive combinatorics.Comment: Reformatted for Discrete Analysi

    Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups

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    The little Grothendieck problem consists of maximizing ijCijxixj\sum_{ij}C_{ij}x_ix_j over binary variables xi{±1}x_i\in\{\pm1\}, where C is a positive semidefinite matrix. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C a dn x dn positive semidefinite matrix, the objective is to maximize ijTr(CijTOiOjT)\sum_{ij}Tr (C_{ij}^TO_iO_j^T) restricting OiO_i to take values in the group of orthogonal matrices, where CijC_{ij} denotes the (ij)-th d x d block of C. We propose an approximation algorithm, which we refer to as Orthogonal-Cut, to solve this problem and show a constant approximation ratio. Our method is based on semidefinite programming. For a given d1d\geq 1, we show a constant approximation ratio of αR(d)2\alpha_{R}(d)^2, where αR(d)\alpha_{R}(d) is the expected average singular value of a d x d matrix with random Gaussian N(0,1/d)N(0,1/d) i.i.d. entries. For d=1 we recover the known αR(1)2=2/π\alpha_{R}(1)^2=2/\pi approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous problem over the Unitary Group. Orthogonal-Cut also serves as an approximation algorithm for several applications, including the Procrustes problem where it improves over the best previously known approximation ratio of~122\frac1{2\sqrt{2}}. The little Grothendieck problem falls under the class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and it provides a more efficient algorithm with better approximation ratios and matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints.Comment: Updates in version 2: extension to the complex valued (unitary group) case, sharper lower bounds on the approximation ratios, matching integrality gap, and a generalized rank constrained version of the problem. Updates in version 3: Improvement on the expositio

    Disentangling Orthogonal Matrices

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    Motivated by a certain molecular reconstruction methodology in cryo-electron microscopy, we consider the problem of solving a linear system with two unknown orthogonal matrices, which is a generalization of the well-known orthogonal Procrustes problem. We propose an algorithm based on a semi-definite programming (SDP) relaxation, and give a theoretical guarantee for its performance. Both theoretically and empirically, the proposed algorithm performs better than the na\"{i}ve approach of solving the linear system directly without the orthogonal constraints. We also consider the generalization to linear systems with more than two unknown orthogonal matrices

    Tight hardness of the non-commutative Grothendieck problem

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    We prove that for any ε>0\varepsilon > 0 it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor 1/2+ε1/2 + \varepsilon, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of 2\ell_2 into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates

    Quantum XOR Games

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    We introduce quantum XOR games, a model of two-player one-round games that extends the model of XOR games by allowing the referee's questions to the players to be quantum states. We give examples showing that quantum XOR games exhibit a wide range of behaviors that are known not to exist for standard XOR games, such as cases in which the use of entanglement leads to an arbitrarily large advantage over the use of no entanglement. By invoking two deep extensions of Grothendieck's inequality, we present an efficient algorithm that gives a constant-factor approximation to the best performance players can obtain in a given game, both in case they have no shared entanglement and in case they share unlimited entanglement. As a byproduct of the algorithm we prove some additional interesting properties of quantum XOR games, such as the fact that sharing a maximally entangled state of arbitrary dimension gives only a small advantage over having no entanglement at all.Comment: 43 page
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