A long-standing challenge in multimodal brain network analyses is to integrate topologically different brain networks obtained from diffusion and functional MRI in a coherent statistical framework. Existing multimodal frameworks will inevitably destroy the topological difference of the networks. In this paper, we propose a novel topological learning framework that integrates networks of different topology through persistent homology. Such challenging task is made possible through the introduction of a new topological loss that bypasses intrinsic computational bottlenecks and thus enables us to perform various topological computations and optimizations with ease. We validate the topological loss in extensive statistical simulations with ground truth to assess its effectiveness of discriminating networks. Among many possible applications, we demonstrate the versatility of topological loss in the twin imaging study where we determine the extend to which brain networks are genetically heritable.
Keywords: Multimodal brain networks; Persistent homology; Topological data analysis; Twin brain imaging study; Wasserstein distance.