Efficient Rounding for the Noncommutative Grothendieck Inequality

TL;DR

This paper introduces an efficient rounding algorithm for the noncommutative Grothendieck inequality, enabling polynomial-time approximation algorithms for complex optimization problems with applications in data analysis and matrix alignment.

Contribution

It provides the first efficient rounding procedure for the noncommutative Grothendieck inequality, leading to new approximation algorithms for related optimization problems.

Findings

Polynomial-time constant-factor approximation for a generalized Cut Norm problem

Applications to robust principal component analysis

Applications to the orthogonal Procrustes problem

Abstract

$ \newcommand{\cclass}[1]{{\textsf{#1}}} $The classical Grothendieck inequality has applications to the design of approximation algorithms for $\cclass{NP}$-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 polynomial-time constant-factor 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 principal component analysis and the orthogonal Procrustes problem.