In many applications involving datasets in high dimensional spaces, it is often postulated that the data points lie on an unknown manifold of much lower dimension than the ambient space dimension. This motivates manifold reconstruction to study the geometrical and topological properties of the manifold. Given a set of points sampled from an unknown manifold, manifold reconstruction is to produce a representation with the same topology as the manifold and geometrically close to it.

We divide the reconstruction problem into three tasks : detecting the manifold dimension, estimating the tangent spaces of the manifold and constructing an implicit function whose zero-set is homeomorphic to the manifold. In this thesis, we address the second and third tasks assuming that the manifold...[ Read more ]

In many applications involving datasets in high dimensional spaces, it is often postulated that the data points lie on an unknown manifold of much lower dimension than the ambient space dimension. This motivates manifold reconstruction to study the geometrical and topological properties of the manifold. Given a set of points sampled from an unknown manifold, manifold reconstruction is to produce a representation with the same topology as the manifold and geometrically close to it.

We divide the reconstruction problem into three tasks : detecting the manifold dimension, estimating the tangent spaces of the manifold and constructing an implicit function whose zero-set is homeomorphic to the manifold. In this thesis, we address the second and third tasks assuming that the manifold dimension has been determined. We present a method to estimate the tangent space with provably small angular error. We also show how to construct an implicit function whose zero-set contains a homeomorphic approximation of the manifold and is geometrically close to the manifold. Some experimental results are presented in both tasks.