Geometric structures can aid statistics in several ways. In high dimensional statistics, geometric structures can be used to reduce dimensionality. High dimensional data entails the curse of dimensionality, which can be avoided by if there are low dimensional geometric structures. On the other hand, geometric structures also provide useful information. Structures may carry scientific meaning about the data and can be used as features to enhance supervised or unsupervised learning.
In this defense, I will explore how statistical inference can be done on geometric structures. First, I will explore the minimax rates of dimension estimator and reach estimator. Second, I will investigate inference on cluster trees and persistent homology of density filtration on rips complex. Third, I will extend and improve R package TDA for computing topological data analysis.