5 research outputs found
Characterization and Inference of Graph Diffusion Processes from Observations of Stationary Signals
Many tools from the field of graph signal processing exploit knowledge of the
underlying graph's structure (e.g., as encoded in the Laplacian matrix) to
process signals on the graph. Therefore, in the case when no graph is
available, graph signal processing tools cannot be used anymore. Researchers
have proposed approaches to infer a graph topology from observations of signals
on its nodes. Since the problem is ill-posed, these approaches make
assumptions, such as smoothness of the signals on the graph, or sparsity
priors. In this paper, we propose a characterization of the space of valid
graphs, in the sense that they can explain stationary signals. To simplify the
exposition in this paper, we focus here on the case where signals were i.i.d.
at some point back in time and were observed after diffusion on a graph. We
show that the set of graphs verifying this assumption has a strong connection
with the eigenvectors of the covariance matrix, and forms a convex set. Along
with a theoretical study in which these eigenvectors are assumed to be known,
we consider the practical case when the observations are noisy, and
experimentally observe how fast the set of valid graphs converges to the set
obtained when the exact eigenvectors are known, as the number of observations
grows. To illustrate how this characterization can be used for graph recovery,
we present two methods for selecting a particular point in this set under
chosen criteria, namely graph simplicity and sparsity. Additionally, we
introduce a measure to evaluate how much a graph is adapted to signals under a
stationarity assumption. Finally, we evaluate how state-of-the-art methods
relate to this framework through experiments on a dataset of temperatures.Comment: Submitted to IEEE Transactions on Signal and Information Processing
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Graph signal processing for machine learning: A review and new perspectives
The effective representation, processing, analysis, and visualization of
large-scale structured data, especially those related to complex domains such
as networks and graphs, are one of the key questions in modern machine
learning. Graph signal processing (GSP), a vibrant branch of signal processing
models and algorithms that aims at handling data supported on graphs, opens new
paths of research to address this challenge. In this article, we review a few
important contributions made by GSP concepts and tools, such as graph filters
and transforms, to the development of novel machine learning algorithms. In
particular, our discussion focuses on the following three aspects: exploiting
data structure and relational priors, improving data and computational
efficiency, and enhancing model interpretability. Furthermore, we provide new
perspectives on future development of GSP techniques that may serve as a bridge
between applied mathematics and signal processing on one side, and machine
learning and network science on the other. Cross-fertilization across these
different disciplines may help unlock the numerous challenges of complex data
analysis in the modern age