201 research outputs found
The positive semidefinite Grothendieck problem with rank constraint
Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of
size m x m, the positive semidefinite Grothendieck problem with
rank-n-constraint (SDP_n) is
maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m
\in S^{n-1}.
In this paper we design a polynomial time approximation algorithm for SDP_n
achieving an approximation ratio of
\gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 -
\Theta(1/n).
We show that under the assumption of the unique games conjecture the achieved
approximation ratio is optimal: There is no polynomial time algorithm which
approximates SDP_n with a ratio greater than \gamma(n). We improve the
approximation ratio of the best known polynomial time algorithm for SDP_1 from
2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter
approximation ratio for SDP_n when A is the Laplacian matrix of a graph with
nonnegative edge weights.Comment: (v3) to appear in Proceedings of the 37th International Colloquium on
Automata, Languages and Programming, 12 page
Grothendieck inequalities for semidefinite programs with rank constraint
Grothendieck inequalities are fundamental inequalities which are frequently
used in many areas of mathematics and computer science. They can be interpreted
as upper bounds for the integrality gap between two optimization problems: a
difficult semidefinite program with rank-1 constraint and its easy semidefinite
relaxation where the rank constrained is dropped. For instance, the integrality
gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen
as a Grothendieck inequality. In this paper we consider Grothendieck
inequalities for ranks greater than 1 and we give two applications:
approximating ground states in the n-vector model in statistical mechanics and
XOR games in quantum information theory.Comment: 22 page
Tightness of the maximum likelihood semidefinite relaxation for angular synchronization
Maximum likelihood estimation problems are, in general, intractable
optimization problems. As a result, it is common to approximate the maximum
likelihood estimator (MLE) using convex relaxations. In some cases, the
relaxation is tight: it recovers the true MLE. Most tightness proofs only apply
to situations where the MLE exactly recovers a planted solution (known to the
analyst). It is then sufficient to establish that the optimality conditions
hold at the planted signal. In this paper, we study an estimation problem
(angular synchronization) for which the MLE is not a simple function of the
planted solution, yet for which the convex relaxation is tight. To establish
tightness in this context, the proof is less direct because the point at which
to verify optimality conditions is not known explicitly.
Angular synchronization consists in estimating a collection of phases,
given noisy measurements of the pairwise relative phases. The MLE for angular
synchronization is the solution of a (hard) non-bipartite Grothendieck problem
over the complex numbers. We consider a stochastic model for the data: a
planted signal (that is, a ground truth set of phases) is corrupted with
non-adversarial random noise. Even though the MLE does not coincide with the
planted signal, we show that the classical semidefinite relaxation for it is
tight, with high probability. This holds even for high levels of noise.Comment: 2 figure
Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups
The little Grothendieck problem consists of maximizing
over binary variables , 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 restricting to take
values in the group of orthogonal matrices, where 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 , we show
a constant approximation ratio of , where is
the expected average singular value of a d x d matrix with random Gaussian
i.i.d. entries. For d=1 we recover the known
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~. 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
Quantum XOR Games
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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