Metric Learning for Hermitian Manifolds

Abstract

In recent years, manifold learning has emerged as one of the most promising approaches for performing nonparametric dimension reduction. While numerous manifold learning algorithms of varying degrees of complexity have been proposed, these algorithms rarely preserve the intrinsic geometry of the underlying data manifold. In 2013, Perrault-Joncas and Meila proposed a new approach to addressing this problem: instead of trying to design a manifold embedding algorithm that preserves geometry in the widest possible set of circumstances, their approach was to augment the output of any reasonable embedding algorithm with a computation of the pushforward metric, thus allowing us to recover any geometric quantity of interest of the underlying manifold. In this thesis, we apply this idea to the case of Hermitian manifolds, which are complex manifolds equipped with a \compatible" Riemannian metric. Most manifold embedding algorithms do not preserve the complex geometric structure of the input data set. In this thesis, we propose a complex version of the popular Local Tangent Space Alignment (LTSA) algorithm, which by design outputs a holomorphic embedding of the input data set into a low dimensional space. We then propose two algorithms for computing the pushforward Hermitian metric with respect to this embedding, one is based on the transformation properties of a metric under change of coordinates, while the other is based on the Diusionmaps Graph Laplacian as in the work of Perrault-Joncas and Meila