  Learning a graph topology to reveal the underlying relationship between data
entities plays an important role in various machine learning and data analysis
tasks. Under the assumption that structured data vary smoothly over a graph,
the problem can be formulated as a regularised convex optimisation over a
positive semidefinite cone and solved by iterative algorithms. Classic methods
require an explicit convex function to reflect generic topological priors, e.g.
the $\ell_1$ penalty for enforcing sparsity, which limits the flexibility and
expressiveness in learning rich topological structures. We propose to learn a
mapping from node data to the graph structure based on the idea of learning to
optimise (L2O). Specifically, our model first unrolls an iterative primal-dual
splitting algorithm into a neural network. The key structural proximal
projection is replaced with a variational autoencoder that refines the
estimated graph with enhanced topological properties. The model is trained in
an end-to-end fashion with pairs of node data and graph samples. Experiments on
both synthetic and real-world data demonstrate that our model is more efficient
than classic iterative algorithms in learning a graph with specific topological
properties.
