Bayesian nonparametric estimation under the manifold hypothesis

Thesis

Developing methodologies and theoretical understanding of statistical methods at large scales is a pressing matter given the recent growth in the high volume of data available, both in terms of sample sizes (large n) and number of predictors (large p). While the first regime leads mainly to computational issues, the second one also comes with specific statistical questions, as a consequence of the curse of dimensionality. In this thesis, we investigate how the manifold hypothesis, the idea that high dimensional predictors often possesses unknown low dimensional structures, can improve statistical convergence results for Bayesian nonparametric estimation procedures in this context.

In the introduction of this thesis, we will introduce the mathematical tools used in the subsequent chapters to tackle the aforementioned questions. We begin by reviewing in Section 1.1 basic concepts of differential and Riemannian geometry, focusing on submanifolds of Euclidean spaces. We will then in Section 1.2 present fundamental results of spectral theory of compact submanifolds and finite graphs, and show how random graphs can provide meaningful approximations of unknown submanifolds. The proof techniques specific to Bayesian nonparametric statistics will be presented in Section 1.3, where we will focus on the existing methods to derive posterior contraction rates. We will then review in Section 1.4 the state of knowledge related to the questions in nonparametric estimation theory studied in Chapters 2 3 & 4.

As an integrated thesis, Chapters 2 3 & 4 contain published or unpublished but submitted research papers. In Chapters 2 & 3 we will be investigating the problem of nonparametric regression with covariates supported on unknown submanifolds, where we will obtain posterior contraction rates depending only on the intrinsic dimensionality of the data for a class of methodologies where limited results were previously known. We will then be focusing on the problem of density estimation for probability distributions supported near unknown submanifolds, by designing a new class of nonparametric Gaussian mixture models and deriving corresponding posterior contraction rates, together with the implementations of the methods.

Finally, Chapter 5 ends with a discussion.

Authors: Rosa, P

• DOI: 10.5287/ora-dmkobae16
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: curse of dimensionality; posterior contraction rates; manifold hypothesis; nonparametric estimation; Bayesian statistics
• Subjects: Mathematical statistics