Learning a Metric Space for Neighbourhood Topology Estimation: Application to Manifold Learning

Abstract

Manifold learning algorithms rely on a neighbourhood graph to provide an estimate of the data’s local topology. Unfortunately, current methods for estimating local topology assume local Euclidean geometry and locally uniform data density, which often leads to poor data embeddings. We address these shortcomings by proposing a framework that combines local learning with parametric density estimation for local topology estimation. Given a data set \mathcal{D} ⊂\mathcal{X}, we first estimate a new metric space (\mathbb{X},d_\mathbb{X}) that characterizes the varying sample density of \mathcal{X} in \mathbb{X}, then use (\mathbb{X},d_\mathbb{X}) as a new (pilot) input space for the graph construction step of the manifold learning process. The proposed framework results in significantly improved embeddings, which we demonstrated objectively by assessing clustering accuracy.