Learning Heat Diffusion Graphs

Information analysis of data often boils down to properly identifying their hidden structure. In many cases, the data structure can be described by a graph representation that supports signals in the dataset. In some applications, this graph may be partly determined by design constraints or predetermined sensing arrangements. In general though, the data structure is not readily available nor easily defined. In this paper, we propose to represent structured data as a sparse combination of localized functions that live on a graph. This model is more appropriate to represent local data arrangements than the classical global smoothness prior. 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 are actually the sum of heat diffusion processes, which is a widely used 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 data structure for understanding and inference.