Topology Preservation

Abstract

Overview. This chapter reviews methods that reduce the dimen-sionality by preserving the topology of data rather than their pairwise distances. Topology preservation appears more powerful than but also more complex to implement than distance preservation. The described methods are separated into two classes according to the kind of topol-ogy they use. The simplest methods rely on a predefined topology whereas more recent methods prefer a topology built according to the data set to be re-embedded. 5.1 State of the art As demonstrated in the previous chapter, nonlinear dimensionality reduction can be achieved by distance preservation. Numerous methods use distances, which are intuitively simple to understand and very easy to compute. Unfortunately , the principle of distance preservation also has a major drawback. Indeed, the appealing quantitative nature of a distance function also makes it very constraining. Characterizing a manifold with distances turns out to support and bolt it with rigid steel beams. In many cases, though, the embedding of a manifold requires some flexibility: some subregions must be locally stretched or shrunk in order to embed them in a lower-dimensional space. As stated in Section 1.4, the important point about a manifold is its topol-ogy, i.e., the neighborhood relationships between subregions of the manifold. More precisely, a manifold can be entirely characterized by giving relative or comparative proximities: a first region is close to a second one but far from a third one. To some extent, distances give too much information: the exact measure of a distance depends not exclusively on the manifold itself but also, for much too large a part, on a given embedding of the manifold. Actually, comparative information between distances, like inequalities or ranks, suffices to characterize a manifold, for any embedding.