Abstract

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.

Highlights

  • The classical Grothendieck inequality has applications to the design of approximation algorithms for NP-hard optimization problems

  • A simple transformation [AN04] relates the Grothendieck problem to the Frieze-Kannan Cut Norm problem [FK99], and as such the constant-factor approximation algorithm for the Grothendieck problem has found a variety of applications in combinatorial optimization; see the survey [KN12] for much more on this topic

  • A rigorous analysis of a polynomial-time approximation algorithm for this problem appears in the work of Nemirovski [Nem07], where the generalized orthogonal Procrustes problem is treated as an important special case of a more general family of prob√lems called “quadratic optimization under orthogonality constraints”, for which he obtains a O( 3 n + d + log K) approximation algorithm

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Summary

Introduction

There exists a polynomial time algorithm that takes as input M ∈ Mn(Mn(R)) and outputs. There exists a polynomial time algorithm that takes as input M ∈ Mn(Mn(C)) and outputs. Th√e implied constants in the O(1) terms of Theorem 1 can be taken to be any number greater than 2 2 in the real case, an√d any number greater than 2 in the complex case. There is no reason to believe that the factor 2 2 in the real case is optimal, but the factor 2 in the complex case is sharp in a certain natural sense that will become clear later. The novelty of the applications to combinatorial optimization that are described below is the mere existence of a constant-factor approximation algorithm

Applications of Theorem 1
The Grothendieck problem
Robust PCA
The orthogonal Procrustes problem
A Frieze-Kannan decomposition for 4-tensors
Quantum XOR games
The noncommutative Grothendieck inequality
The complex case
The rounding algorithm
Rounding procedure
An intuitive description of the rounding procedure in the commutative case
Proof of Theorem 4
Analysis of the rounding procedure
Derandomized rounding
Two-dimensional rounding
Rounding in the Hermitian case
Output
Proof of Theorem 10
A2 A3 A4
Direct rounding in the real case
Some applications
Constant-factor algorithm for robust PCA problems
A constant-factor algorithm for the orthogonal Procrustes problem
An algorithmic noncommutative dense regularity lemma
Full Text
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