46 research outputs found
Efficient Rounding for the Noncommutative Grothendieck Inequality
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
We give a counterexample to a trilinear version of the operator space
Grothendieck theorem. In particular, we show that for trilinear forms on
, 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
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
Disentangling Orthogonal Matrices
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
We prove that for any it is NP-hard to approximate the
non-commutative Grothendieck problem to within a factor ,
which matches the approximation ratio of the algorithm of Naor, Regev, and
Vidick (STOC'13). Our proof uses an embedding of 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
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