  The problem of graph learning concerns the construction of an explicit
topological structure revealing the relationship between nodes representing
data entities, which plays an increasingly important role in the success of
many graph-based representations and algorithms in the field of machine
learning and graph signal processing. In this paper, we propose a novel graph
learning framework that incorporates the node-side and observation-side
information, and in particular the covariates that help to explain the
dependency structures in graph signals. To this end, we consider graph signals
as functions in the reproducing kernel Hilbert space associated with a
Kronecker product kernel, and integrate functional learning with
smoothness-promoting graph learning to learn a graph representing the
relationship between nodes. The functional learning increases the robustness of
graph learning against missing and incomplete information in the graph signals.
In addition, we develop a novel graph-based regularisation method which, when
combined with the Kronecker product kernel, enables our model to capture both
the dependency explained by the graph and the dependency due to graph signals
observed under different but related circumstances, e.g. different points in
time. The latter means the graph signals are free from the i.i.d. assumptions
required by the classical graph learning models. Experiments on both synthetic
and real-world data show that our methods outperform the state-of-the-art
models in learning a meaningful graph topology from graph signals, in
particular under heavy noise, missing values, and multiple dependency.
