Graph Learning from Filtered Signals: Graph System and Diffusion Kernel Identification

TL;DR

This paper presents a new framework for graph system identification from filtered signals, enabling the learning of graph structures and filters, including diffusion kernels, with demonstrated superior performance on synthetic and real climate data.

Contribution

It introduces a joint algorithm for identifying graphs and graph-based filters from data, applicable to diffusion kernels and reducing to Laplacian estimation in special cases.

Findings

The algorithm outperforms existing methods in experiments.

It successfully models temperature signals in climate data.

The framework generalizes graph Laplacian estimation.

Abstract

This paper introduces a novel graph signal processing framework for building graph-based models from classes of filtered signals. In our framework, graph-based modeling is formulated as a graph system identification problem, where the goal is to learn a weighted graph (a graph Laplacian matrix) and a graph-based filter (a function of graph Laplacian matrices). In order to solve the proposed problem, an algorithm is developed to jointly identify a graph and a graph-based filter (GBF) from multiple signal/data observations. Our algorithm is valid under the assumption that GBFs are one-to-one functions. The proposed approach can be applied to learn diffusion (heat) kernels, which are popular in various fields for modeling diffusion processes. In addition, for specific choices of graph-based filters, the proposed problem reduces to a graph Laplacian estimation problem. Our experimental results demonstrate that the proposed algorithm outperforms the current state-of-the-art methods. We also implement our framework on a real climate dataset for modeling of temperature signals.