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

Rui Liu, Hossein Nejati+1 more|arXiv (Cornell University)|Jan 8, 2018

Functional Brain Connectivity Studies37 references3 citations

TL;DR

This paper proposes a joint low-rank component and graph structure estimation framework for high-dimensional, graph-smooth, and grossly corrupted data, particularly applied to MEG brain imaging. By iteratively refining the graph and low-rank components using alternating optimization, the method improves both dimensionality reduction and connectivity inference, achieving state-of-the-art classification accuracy and neuroscientifically plausible brain network maps.

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 grossly-corrupted 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 low-rank 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.

Motivation & Objective

• Address the challenge of low-rank component estimation in high-dimensional, graph-smooth, and grossly corrupted data where the underlying graph is unknown or inexact.
• Simultaneously estimate low-rank components and refine the graph structure to improve both estimation accuracy and network inference.
• Justify the integration of graph refinement into low-rank estimation by analyzing the impact of inexact graphs on low-rank recovery.
• Demonstrate superior performance on real MEG data, achieving higher classification accuracy and producing brain connectivity maps consistent with neuroscientific findings.

Proposed method

• Formulate the problem as a joint optimization of low-rank components and graph Laplacian, assuming sparse perturbations and graph-smoothness.
• Use alternating optimization to iteratively update the low-rank component estimate and refine the graph structure.
• Leverage graph Laplacian regularization to enforce smoothness of the low-rank signal on the estimated graph.
• Apply spectral graph regularization and incorporate sparsity constraints to model perturbations and ensure robustness to noise.
• Perform mathematical analysis to quantify the effect of graph inaccuracy on low-rank estimation, justifying the need for joint refinement.
• Use a coherence-based initial graph derived from resting-state MEG data, which is then refined through iterative learning.

Experimental results

Research questions

• RQ1 How does the inaccuracy of an initial graph affect the estimation of low-rank components in high-dimensional, corrupted data?
• RQ2 Can joint estimation of low-rank components and graph structure lead to improved performance compared to sequential or independent approaches?
• RQ3 Does the proposed method produce graph connectivity patterns that are consistent with known neuroscientific findings in brain imaging?
• RQ4 How does the joint learning framework compare to state-of-the-art methods in terms of classification accuracy on MEG data?
• RQ5 Can the refined graph structure better capture task-related neural activity patterns, such as those associated with face perception?

Key findings

• The proposed LGE method achieved the highest classification accuracy on MEG data—64.26% at 96–105ms and 79.53% at 141–150ms—significantly outperforming PCA, RPCA, RPCAG, and GL-SigRep.
• Statistical tests confirmed the superiority of LGE, with p-values < 0.05 for all competing methods across both time windows.
• At 105ms post-stimulus, LGE estimated graph connectivity primarily in the occipital and left occipitotemporal regions, aligning with early visual processing in neuroscientific literature.
• At 150ms, LGE’s graph highlighted the right occipitotemporal region, consistent with the N170 face-processing marker, while GL-SigRep and SGDict showed less biologically plausible connectivity patterns.
• The estimated graph from LGE showed stronger correspondence with known functional brain networks than GL-SigRep and SGDict, especially in capturing task-specific activation during face perception.
• Mathematical analysis confirmed that graph inaccuracy degrades low-rank estimation, validating the necessity of iterative graph refinement in the joint framework.