Martin Andersson: Through a Gaussian, Darkly

Abstract

This thesis studies learning problems where a unifying theme is Gaussian or exponential decay, either as a designed similarity measure between points or as a structural property of the underlying stochastic processes.

In Papers I–II, data takes the form of a point cloud with support on an underlying Riemannian manifold Ω of lower dimension than the ambient Euclidean space. In Paper I, we study the graph Laplacian applied to linear functions near singularities in Ω, such as intersections of smooth manifolds. We prove non-asymptotic results showing how the graph Laplacian acts as a first-order rather than second-order operator near singularities, and use finite-sample concentration to develop a hypothesis test for their presence. In Paper II, we study a dimension estimator based on Gaussian kernel sums and prove concentration bounds with explicit dependence on geometric and distributional parameters. We also prove anti-concentration bounds that give a lower limit on the achievable estimation precision.

In Paper III, we study optimal switching problems using a simulation and regression framework. We prove concentration bounds for k-nearest neighbor regression under sub-Gaussian and sub-exponential transition density bounds on the underlying process, and carry out an empirical comparison on benchmark problems in dimensions up to 50. Classical methods such as k-NN and random forests perform well compared to feedforward neural networks, including on high-dimensional problems.

Lastly, in Paper IV, we consider non-parametric estimation of a solution-dependent diffusion coefficient in a parabolic stochastic partial differential equation. To our knowledge, this is the first non-parametric predictor for a solution-dependent diffusion coefficient in this setting. The predictor is based on integrating a finite-difference approximation of the differential operator over a space-time window, and we prove consistency with explicit convergence rates in the discretization parameter. The convergence analysis relies on Gaussian bounds on the fundamental solution.