A Cheeger Inequality for the Graph Connection Laplacian

TL;DR

This paper establishes a Cheeger inequality linking the solvability of the O(d) synchronization problem with the spectral properties of the graph Connection Laplacian, providing theoretical performance guarantees for spectral algorithms.

Contribution

It formulates and proves a Cheeger-type inequality for the graph Connection Laplacian in the context of O(d) synchronization, connecting spectral properties to problem solvability.

Findings

Proves a Cheeger inequality for the graph Connection Laplacian.

Provides worst-case performance guarantees for spectral synchronization methods.

Links spectral gap to the accuracy of estimating orthogonal transformations.

Abstract

The O(d) Synchronization problem consists of estimating a set of unknown orthogonal transformations O_i from noisy measurements of a subset of the pairwise ratios O_iO_j^{-1}. We formulate and prove a Cheeger-type inequality that relates a measure of how well it is possible to solve the O(d) synchronization problem with the spectra of an operator, the graph Connection Laplacian. We also show how this inequality provides a worst case performance guarantee for a spectral method to solve this problem.