Graph connection Laplacian and random matrices with random blocks

TL;DR

This paper develops a theoretical framework for understanding the spectral distribution of large random matrices with block entries, aiding the interpretation of Graph Connection Laplacian outputs in high-dimensional data analysis.

Contribution

It introduces a new theory for the spectral behavior of block-structured random matrices, including cases with dependent blocks, relevant for GCL applications.

Findings

The spectral distribution matches theoretical predictions in simulations.

The theory covers matrices with dependent and independent blocks.

Numerical results validate the theoretical framework.

Abstract

Graph connection Laplacian (GCL) is a modern data analysis technique that is starting to be applied for the analysis of high dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones appearing in the null case of GCL, i.e the case where there is no structure in the dataset under investigation. Developing this understanding is important in making sense of the output of the algorithms based on GCL. We hence develop a theory explaining the behavior of the spectral distribution of a large class of random matrices, in particular random matrices with random block entries of fixed size. Part of the theory covers the case where there is significant dependence between the blocks. Numerical work shows that the agreement between our theoretical predictions and numerical simulations is generally very good.