Towards Metric Measure Space Learning

Density estimation is pivotal in manifold learning and data analysis, providing insightinto the underlying geometric structure of data. In the first part of this thesis, we link density estimation to the concept of weight from metric geometry. Specifically, we show that the weight function estimates the reciprocal of the data density function on an embedded Riemannian manifold. This leads to a novel normalization method for estimating the Laplace-Beltrami operator. In the second part, we develop the theory to generalize density estimation to metric measure spaces beyond smooth manifolds. By extending the Lebesgue Differentiation Theorem to kernel-weighted integrals, we accommodate a broader class of kernel functions than previously used in manifold learning. The final chapter shows that many aspects of our work fit into enriched category theory. We demonstrate that generalized metric spaces form an enriched category, following Lawvere’s results. This perspective allows us to define generalized kernel spaces and show that the outer measure defines a functor. This section is speculative and aims to inspire future research directions in data science.

Author

• Ayub, Yemeen

Identifier

hdl:1920/14458

OAI identifier oai:identifier

oai:MARS:1920/14458