Utilizing Geometry in Some Statistical and Machine Learning Problems

The goal of this thesis is to explore the fundamental role of geometry in learning and inference across various statistical and machine learning problems. As complex data becomes increasingly prevalent in modern applications, geometry is inherently embedded within it, either known or to be discovered. Efficient and reliable learning and inference should consider this geometry. In this thesis, we demonstrate how geometry can be utilized to address crucial issues in statistics and machine learning for effective learning and inference. Specifically, we first present both intrinsic and extrinsic deep neural network (DNN) architectures as versatile deep learning frameworks for manifold-valued data. These frameworks harness the geometry of the underlying manifolds, and we derive convergence rates for estimators based on the proposed DNN models. We also establish an extrinsic Bayesian optimization framework for addressing general optimization problems on manifolds. Additionally, we propose neural-network-based numerical schemes that preserve variational structures when solving surface PDEs and geometric flows. Lastly, we investigate high-dimensional data distribution estimation using adaptive Bayesian deep generative models, emphasizing lower-dimensional manifold-supported distributions of uncertain smoothness. Each chapter is devoted to a separate topic.