Topologically penalized regression on manifolds
This work addresses regression problems for data on manifolds, offering a method that leverages geometry and topology, but it appears incremental as it builds on existing manifold learning techniques.
The authors tackled regression on manifolds by using eigenfunctions of the Laplace-Beltrami operator with topological penalties based on sub-level sets, resulting in competitive performance on synthetic and real datasets and providing theoretical guarantees on prediction error and smoothness.
We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is ''topologically smooth''.