5,380 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: Overview, Challenges and Applications
Research in Graph Signal Processing (GSP) aims to develop tools for
processing data defined on irregular graph domains. In this paper we first
provide an overview of core ideas in GSP and their connection to conventional
digital signal processing. We then summarize recent developments in developing
basic GSP tools, including methods for sampling, filtering or graph learning.
Next, we review progress in several application areas using GSP, including
processing and analysis of sensor network data, biological data, and
applications to image processing and machine learning. We finish by providing a
brief historical perspective to highlight how concepts recently developed in
GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE
Forecasting Time Series with VARMA Recursions on Graphs
Graph-based techniques emerged as a choice to deal with the dimensionality
issues in modeling multivariate time series. However, there is yet no complete
understanding of how the underlying structure could be exploited to ease this
task. This work provides contributions in this direction by considering the
forecasting of a process evolving over a graph. We make use of the
(approximate) time-vertex stationarity assumption, i.e., timevarying graph
signals whose first and second order statistical moments are invariant over
time and correlated to a known graph topology. The latter is combined with VAR
and VARMA models to tackle the dimensionality issues present in predicting the
temporal evolution of multivariate time series. We find out that by projecting
the data to the graph spectral domain: (i) the multivariate model estimation
reduces to that of fitting a number of uncorrelated univariate ARMA models and
(ii) an optimal low-rank data representation can be exploited so as to further
reduce the estimation costs. In the case that the multivariate process can be
observed at a subset of nodes, the proposed models extend naturally to Kalman
filtering on graphs allowing for optimal tracking. Numerical experiments with
both synthetic and real data validate the proposed approach and highlight its
benefits over state-of-the-art alternatives.Comment: submitted to the IEEE Transactions on Signal Processin
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