The World is not always Flat or Learning Curved Manifolds
Evgeni Begelfor, Michael Werman · 2006
Manifold learning and finding low-dimensional structure in data is an important task. Many algorithms for this purpose embed data in Euclidean space, an approach which is destined to fail on non-flat data. This paper presents a non-iterative algebraic method for embedding the data into hyperbolic and spherical spaces. We argue that these spaces are often better than Euclidean space in capturing the geometry of the data. The approach can be used to extend algorithms such as ISOMAP and SDE to the curved case. We also demonstrate the utility of these embeddings by showing how some of the standard clustering algorithms translate to these curved manifolds.