Statistical Exploration of the Manifold Hypothesis

Abstract

The manifold hypothesis is the assumption that high-dimensional data are concentrated near a
low-dimensional structure. It is a widely accepted tenet of machine learning, empirically observed
in many real-world situations, and underpins many methods and algorithms. This thesis explores
fundamental problems in understanding the statistical foundation for such manifolds and their
utility in exploring hypotheses about the underlying data-generating mechanisms. We begin by
considering how to recover the true distances and latent positions from a graph or similarity
matrix using geodesic distances. We show that this can be effectively achieved through a two-step
process of matrix factorisation followed by nonlinear dimension reduction, such as spectral
embedding and Isomap.
Building on this, we provide a statistical foundation for the manifold hypothesis. By introducing the Latent Metric Model, we demonstrate that manifold structure can emerge from
simple statistical principles such as latent variables, correlation and stationarity. This enables us
to provide a pipeline for model-based exploratory data analysis, combining well-known existing
techniques such as PCA and graph-analytic algorithms.
Finally, we explore a special case of the Latent Metric Model for hierarchical data. This
hierarchy takes the form of a tree defined through the conditional independence structure of
latent variables. This offers a novel perspective on the agglomerative clustering algorithm, in
which clusters are merged on maximum dot product. We demonstrate how our method can
recover hidden tree structure by showing how hierarchical information in this model translates
into tree geometry.
Together, these contributions aim to provide a framework for working with complex, highdimensional data, offering both theoretical insights and practical guidance.

Date of Award	17 Jun 2025
Original language	English
Awarding Institution	University of Bristol
Supervisor	Oliver T Johnson (Supervisor), Patrick Rubin-Delanchy (Supervisor) & Nick Whiteley (Supervisor)