Convergence of graph Dirichlet energies and graph Laplacians on intersecting manifolds of varying dimensions

Abstract: We study $\Gamma$-convergence of graph Dirichlet energies and spectral convergence of graph Laplacians on unions of intersecting manifolds of potentially different dimensions. Our investigation is motivated by problems of machine learning, as real-world data often consist of parts or classes with different intrinsic dimensions. An important challenge is to understand which machine learning methods adapt to such varied dimensionalities. We investigate the standard unnormalized and the normalized graph Dirichlet energies. We show that the unnormalized energy and its associated graph Laplacian asymptotically only sees the variations within the manifold of the highest dimension. On the other hand, we prove that the normalized Dirichlet energy converges to a (tensorized) Dirichlet energy on the union of manifolds that adapts to all dimensions simultaneously. We also establish the related spectral convergence and present a few numerical experiments to illustrate our findings.