Inferring manifolds from noisy data using Gaussian processes

Oregon Convention Center 

We focus on the study of a noisy data set sampled around an unknown Riemannian submanifold of a high-dimensional space. Most existing manifold learning algorithms replace the original data with lower dimensional coordinates without providing an estimate of the manifold in the observation space or using the manifold to denoise the original data. We propose a Manifold reconstruction via Gaussian processes (MrGap) algorithm for addressing these problems, allowing interpolation of the estimated manifold between fitted data points. The proposed approach is motivated by novel theoretical properties of local covariance matrices constructed from noisy samples on a manifold. Our results enable us to turn a global manifold reconstruction problem into a local regression problem, allowing the application of Gaussian processes for probabilistic manifold reconstruction. In this talk, I will review the classical manifold learning algorithms and discuss the theoretical foundation of the new method, MrGap. Simulated and real data examples will be provided to illustrate the performance. This talk is based on the joint work with David Dunson.