Learning General Differential Operators on Unknown Manifolds
Restricted (Penn State Only)
- Author:
- Peoples, Wilson
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- August 06, 2024
- Committee Members:
- Pierre-Emmanuel Jabin, Program Head/Chair
John Harlim, Chair & Dissertation Advisor
Bharath Kumar Sriperumbudur, Outside Unit & Field Member
Alexei Novikov, Major Field Member
Jeffrey Case, Major Field Member - Keywords:
- Manifold Learning
Graph Laplacian
Operator Estimation
Radial Basis Functions
Kernel Methods - Abstract:
- In this dissertation, we develop multiple frameworks for generalizing the typical setting of manifold learning methods. First, we propose a method for estimating arbitrary differential operators defined on tensor fields over unknown manifolds, overcoming the limited paradigm of approximating the Laplace-Beltrami operator on functions standard to various popular graph-based manifold learning methods. This method is an extension of the Radial Basis Function (RBF) method for approximating differential operators and uses estimated local tangent spaces to relate differentiation in the ambient space to differentiation on the unknown manifold. Secondly, to overcome various limitations of the RBF approach, such as memory considerations for high dimensional ambient space as well as numerical and theoretical difficulties arising from non-symmetric approximations, we introduce an additional method for approximating arbitrary differential operators on tensor fields. This method constructs low-dimensional local meshes from point cloud data projected to estimated tangent spaces. Higher-order information is incorporated into the local meshes, enabling estimations of operators involving higher-order curvature terms (e.g., the Bochner and Hodge Laplacians). Finally, we revisit a more standard graph-based approach in the general setting of data sampled from a compact manifold with boundary. In particular, we propose the truncated graph-Laplacian, a slight modification to the Diffusion Maps (DM) algorithm, as an approximation to the Dirichlet Laplacian. We study its convergence, as well as the convergence properties of the unaltered DM algorithm applied to manifolds with boundary. For each of these generalizations, we provide various theoretical convergence guarantees, as well as detailed numerical investigations.