Project Description
Spatio-temporal data, such as data recorded by weather stations in different locations over time, is
typically correlated. Such correlation can be described by a sparse graph, where data resides on nodes,
and edges between nodes characterise the similarity of the data. A graph filter is an operator that allows
to perform spatial and temporal filtering of the data on a graph, and can also be used to describe the
spatio-temporal structure — e.g. through explaining how one set of graph data is generated from another
set.
The identification of such a graph filter requires two components to be established — (i) a graph shift
operator, and (ii) the filter coefficients. While the identification of the filter coefficients is a quadratic
problem and hence straightforward to solve in isolation, the graph shift operator requires to solve a highly
non-linear optimisation problem. There are methods that have been suggested in the literature to
address this, see e.g. [1], with typically modest results.
A different approach to non-linear optimisation is to utilise a technique refered to as reproducing Kernel
Hilbert spaces (RKHS). In RKHS, a problem that is non-linear in a few parameters is projected into a
higher-dimensional spaces, where the problem can be described as a convex one (se
e e.g. [2]). A
solution is then transfered back into the lower-dimensional space. Initial results with RKHS provide very
promising results, but need to be further investigated.
[1] A. Natali, M. Coutino and G. Leus, “Topology-Aware Joint Graph Filter and Edge Weight Identification
for Network Processes,” 2020 IEEE 30th International Workshop on Machine Learning for Signal
Processing (MLSP), 2020, pp. 1-6, doi: 10.1109/MLSP49062.2020.9231913.
[2] M. Sun, M. E. Davies, J. R. Hopgood and I. Proudler, “Adaptive Kernel Kalman Filter Multi-Sensor
Fusion,” 2021 IEEE 24th International Conference on Information Fusion (FUSION), 2021, pp. 1-8.
Key Objectives
– to understand the overall problem is non-linear system identification from graph data;
– to review relevant literature on graph signal processing, and reproducing Kernel Hilbert spaces;
– to implement graphs and generate data with known ground truth;
– to implement and test algorithms in [1] and on reproducing Kernel Hilber spaces;
– to characterise the performance of these algorithms on data with known ground truth (i.e. where the
graph filter to be identified is known);
– to investigate and make sense of running the algorithms on known data sets without ground truth (such
as meteorological data, e.g. temperature or wind speed).