Department seminar by Nan Wu

Please Note: This seminar will be given online.

Graph Laplacian based Gaussian processes on restricted domains

In nonparametric regression, it is common for the inputs to fall in a restricted subset of Euclidean space. Typical kernel-based methods that do not take into account the intrinsic geometry of the domain across which observations are collected may produce sub-optimal results. In this talk, we focus on solving this problem in the context of Gaussian process models, proposing a new class of graph Laplacian based Gaussian processes, which learn a covariance that respects the geometry of the input domain. The graph Laplacian is constructed from a kernel depending only on the Euclidean coordinates of the inputs.  The covariance is defined through finitely many eigenpairs of the graph Laplacian. When the subset is a submanifold of Euclidean space, we show that the covariance incorporates the intrinsic geometry of the manifold by relating it to the heat kernel. The graph Laplacian based Gaussian process is computationally efficient, as we can benefit from the full knowledge about the kernel to extend the covariance structure to newly arriving samples by a Nystrom type extension. We show the performance of our method in various applications.