A Connectedness Constraint for Learning Sparse Graphs

TL;DR

This paper introduces a convex constraint to enforce connectedness in sparse graph learning, facilitating the creation of meaningful networks from data in a computationally tractable way.

Contribution

It formulates connectedness as an analytical, convex constraint, enabling its integration into sparse graph learning methods.

Findings

Convex connectedness constraint effectively enforces connectivity.

Method successfully learns sparse, connected graphs from real and simulated data.

Approach relates to distributed consensus and graph Laplacian learning.

Abstract

Graphs are naturally sparse objects that are used to study many problems involving networks, for example, distributed learning and graph signal processing. In some cases, the graph is not given, but must be learned from the problem and available data. Often it is desirable to learn sparse graphs. However, making a graph highly sparse can split the graph into several disconnected components, leading to several separate networks. The main difficulty is that connectedness is often treated as a combinatorial property, making it hard to enforce in e.g. convex optimization problems. In this article, we show how connectedness of undirected graphs can be formulated as an analytical property and can be enforced as a convex constraint. We especially show how the constraint relates to the distributed consensus problem and graph Laplacian learning. Using simulated and real data, we perform experiments to learn sparse and connected graphs from data.