19 research outputs found
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
Tight hardness of the non-commutative Grothendieck problem
We prove that for any ε > 0 it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor 1=2+ε, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC’13). Our proof uses an embedding of ℓ2 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. We also observe that one can obtain a tight NP-hardness result for the commutative Little Grothendieck problem; previously, this was only known based on the Unique Games Conjecture (Khot and Naor, Mathematika 2009)
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
A Greedy Algorithm for Subspace Approximation Problem
In the subspace approximation problem, given m points in R^{n} and an integer k <= n, the goal is to find a k-dimension subspace of R^{n} that minimizes the l_{p}-norm of the Euclidean distances to the given points. This problem generalizes several subspace approximation problems and has applications from statistics, machine learning, signal processing to biology. Deshpande et al. [Deshpande et al., 2011] gave a randomized O(sqrt{p})-approximation and this bound is proved to be tight assuming NP != P by Guruswami et al. [Guruswami et al., 2016]. It is an intriguing question of determining the performance guarantee of deterministic algorithms for the problem. In this paper, we present a simple deterministic O(sqrt{p})-approximation algorithm with also a simple analysis. That definitely settles the status of the problem in term of approximation up to a constant factor. Besides, the simplicity of the algorithm makes it practically appealing