Semi-supervised Learning for Aggregated Multilayer Graphs Using Diffuse Interface Methods and Fast Matrix Vector Products

TL;DR

This paper introduces a scalable semi-supervised classification method for multilayer graphs using diffuse interface techniques and fast matrix-vector products, effectively handling large, high-dimensional data such as image segmentation.

Contribution

It extends diffuse interface methods to multilayer graphs and develops efficient numerical algorithms for large-scale, high-dimensional data analysis.

Findings

Able to process graphs with up to 10 million nodes per layer

Achieves rapid computation on high-dimensional data sets

Method is scalable and runs efficiently on standard laptops

Abstract

We generalize a graph-based multiclass semi-supervised classification technique based on diffuse interface methods to multilayer graphs. Besides the treatment of various applications with an inherent multilayer structure, we present a very flexible approach that interprets high-dimensional data in a low-dimensional multilayer graph representation. Highly efficient numerical methods involving the spectral decomposition of the corresponding differential graph operators as well as fast matrix-vector products based on the nonequispaced fast Fourier transform (NFFT) enable the rapid treatment of large and high-dimensional data sets. We perform various numerical tests putting a special focus on image segmentation. In particular, we test the performance of our method on data sets with up to 10 million nodes per layer as well as up to 104 dimensions resulting in graphs with up to 52 layers. While all presented numerical experiments can be run on an average laptop computer, the linear dependence per iteration step of the runtime on the network size in all stages of our algorithm makes it scalable to even larger and higher-dimensional problems.