  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.
