Density Preserving Maps
Yunqian Ma, Yun Fu · 2011
Much of the recent work in manifold learning and nonlinear dimensionality reduction focuses on distance-based methods, i.e., methods that aim to preserve the local or global (geodesic) distances between data points on a submanifold of Euclidean space. While this is a promising approach when the data manifold is known to have no intrinsic curvature (which is the case for common examples such as the “Swiss roll”), classical results in Riemannian geometry show that it is impossible to map a d-dimensional data manifold with intrinsic curvature into Rd in a manner that preserves distances. Consequently, distance-based methods of dimensionality reduction distort intrinsically curved data spaces, and they often do so in unpredictable ways. In this chapter, we discuss an alternative paradigm of manifold learning. We show that it is possible to perform nonlinear dimensionality reduction by preserving the underlying density of the data, for a much larger class of data manifolds than intrinsically flat ones, and demonstrate a proof-of-concept algorithm demonstrating the promise of this approach.