LIDS and Stats Tea Talk

“Nonparametric Regression in Dirichlet Spaces: A Random Obstacle Approach”

In this paper, we consider nonparametric estimation over general Dirichlet metric measure spaces. Unlike the more commonly studied reproducing kernel Hilbert space, whose elements may be defined pointwise, a Dirichlet space typically only contain equivalence classes, i.e. its elements are only unique almost everywhere. This lack of pointwise definition presents significant challenges in the context of nonparametric estimation, for example the classical ridge regression problem is ill-posed. In this paper, we develop a new technique for regularizing the ridge loss by replacing pointwise evaluations with certain local means around the boundaries of obstacles centered at each data point. The resulting regularized empirical risk functional is well-posed and even admits a representer theorem in terms of certain equilibrium potentials, which are truncated versions of the associated Green function, cut-off at a data-driven threshold. We study the global, out-of-sample consistency of the sample minimizer, and derive an adaptive upper bound on its convergence rate that highlights the interplay of the analytic, geometric, and probabilistic properties of the Dirichlet form. We also construct a simple Nadaraya-Watson type estimator that achieves the minimax optimal estimation rate with some knowledge of the geometry of the metric measure space. Our framework notably does not require the smoothness of the underlying space and is applicable to both manifold and fractal settings. To the best of our knowledge, this is the first paper to obtain out-of-sample convergence guarantees in the frame-work of general metric measure Dirichlet spaces.