Learning Heat Diffusion Graphs

Abstract

Abstract-Effective information analysis generally boils down to properly identifying the structure or geometry of the data, which is often represented by a graph. In some applications, this structure may be partly determined by design constraints or predetermined sensing arrangements, like in road transportation networks for example. In general though, the data structure is not readily available and becomes pretty difficult to define. In particular, the global smoothness assumptions, that most of the existing works adopt, are often too general and unable to properly capture localized properties of data. In this paper, we go beyond this classical data model and rather propose to represent information as a sparse combination of localized functions that live on a data structure represented by a graph. Based on this model, we focus on the problem of inferring the connectivity that best explains the data samples at different vertices of a graph that is a priori unknown. We concentrate on the case where the observed data is actually the sum of heat diffusion processes, which is a quite common model for data on networks or other irregular structures. We cast a new graph learning problem and solve it with an efficient nonconvex optimization algorithm. Experiments on both synthetic and real world data finally illustrate the benefits of the proposed graph learning framework and confirm that the data structure can be efficiently learned from data observations only. We believe that our algorithm will help solving key questions in diverse application domains such as social and biological network analysis where it is crucial to unveil proper geometry for data understanding and inference.

State-Space Network Topology Identification From Partial Observations

“…Here, the size of the circle representing a vertex is proportional to the degree of the node. From the estimated graph, we can observe that the region of Berlin presents the highest degree which is consistent with the concentration results in [33]. Further, the strong connectivity along the France-Germany region correlates with the spreading pattern of the agent.…”

“…(b) Learned graph from dataset using the proposed one-shot state graph estimator. (c) Learned graph in [33] using the ETEX dataset. Alternatively, we could solve for Ł i and A by means of alternative minimization [36].…”

Network Inference From Consensus Dynamics With Unknown Parameters

“…This allows us to obtain better network recovery performance. On the other hand, the formulations with unknown parameters allow our model to encompass a wider range of settings as opposed to those assuming a specific dynamics (represented via a graph filter) [23], [24].…”

Time-varying Graph Learning Based on Sparseness of Temporal Variation

“…In order to make the optimization problem tractable, we rewrite (22) and (21) in vector form, with W t and Z t replaced by w t ∈ R N (N −1)/2 + and z t ∈ R N (N −1)/2 + respectively. Only the upper-triangular parts of W t and Z t are considered given that the graph is undirected.…”