Manifold reconstruction and bird vocalization problem

Abstract

It is often of interest to infer lower-dimensional structures underlying complex data. Riemannian manifolds are commonly used to model such nonlinear lower-dimensional structures. However, most nonlinear dimension reduction algorithms focus on producing lower-dimensional coordinates of the data, without explicitly estimating the underlying manifold or using the manifold structure to denoise the original observations.

In this talk, I will introduce a manifold reconstruction methodology designed to address these limitations. The approach operates under general assumptions on noisy data and allows interpolation of the estimated manifold between observed data points. The key idea is to transform a global manifold reconstruction problem into a collection of local regression problems, enabling the use of Gaussian processes for probabilistic manifold reconstruction. Applications of the method to studying bird species through their vocalizations, as well as other problems in high-dimensional data analysis, will also be discussed.