Joint Estimation of Low-Rank Components and Connectivity Graph in High-Dimensional Graph Signals: Application to Brain Imaging

Abstract

This paper presents a graph signal processing algorithm to uncover the intrinsic low-rank components and the underlying graph of a high-dimensional, graph-smooth and grosslycorrupted dataset. In our problem formulation, we assume that the perturbation on the low-rank components is sparse and the signal is smooth on the graph. We propose an algorithm to estimate the low-rank components with the help of the graph and refine the graph with better estimated low-rank components. We propose to perform the low-rank estimation and graph refinement jointly so that low-rank estimation can benefit from the refined graph, and graph refinement can leverage the improved low-rank estimation. We propose to address the problem with an alternating optimization. Moreover, we perform a mathematical analysis to understand and quantify the impact of the inexact graph on the low-rank estimation, justifying our scheme with graph refinement as an integrated step in estimating low-rank components. We perform extensive experiments on the proposed algorithm and compare with state-of-the-art lowrank estimation and graph learning techniques. Our experiments use synthetic data and real brain imaging (MEG) data that is recorded when subjects are presented with different categories of visual stimuli. We observe that our proposed algorithm is competitive in estimating the low-rank components, adequately capturing the intrinsic task-related information in the reduced dimensional representation, and leading to better performance in a classification task. Furthermore, we notice that our estimated graph indicates compatible brain active regions for visual activity as neuroscientific findings.

Learning Graphs From Data: A Signal Representation Perspective

“…For example, [27] introduced a graph regression model for learning brain structural connectivity of patients with Alzheimer's disease, which is based on the signal smoothness model discussed in Section III-A. A similar framework [73],…”

“…Instead of the smoothness model adopted in [27], [73], the authors in [54] have utilized models on causal dependencies and proposed to infer effective connectivity networks of brain regions that may shed light on the understanding of the cause behind epilepsy. The signals that they use are electrocorticography (ECoG) time series data before and after ictal onset of seizures of epilepsy.…”