Distributed Cartesian Power Graph Segmentation for Graphon Estimation

TL;DR

This paper extends total variation denoising to Cartesian power graphs, introducing PGFL for network segmentation and graphon estimation, achieving optimal error rates and outperforming existing methods in non-parametric network modeling.

Contribution

It introduces the Power Graph Fused Lasso (PGFL) for segmentation on Cartesian power graphs and applies it to graphon estimation, providing theoretical guarantees and empirical improvements.

Findings

PGFL achieves optimal mean-square error rates for connected graphs.

PGFL effectively denoises networks by learning underlying graph structures.

Empirical results show PGFL outperforms existing graphon estimation methods.

Abstract

We study an extention of total variation denoising over images to over Cartesian power graphs and its applications to estimating non-parametric network models. The power graph fused lasso (PGFL) segments a matrix by exploiting a known graphical structure, $G$, over the rows and columns. Our main results shows that for any connected graph, under subGaussian noise, the PGFL achieves the same mean-square error rate as 2D total variation denoising for signals of bounded variation. We study the use of the PGFL for denoising an observed network $H$, where we learn the graph $G$ as the $K$-nearest neighborhood graph of an estimated metric over the vertices. We provide theoretical and empirical results for estimating graphons, a non-parametric exchangeable network model, and compare to the state of the art graphon estimation methods.