Diffeomorphic Dimensionality Reduction
C. Walder and B. Schölkopf
In: The Twenty-Second Annual Conference on Neural Information Processing Systems (NIPS 2008), 8-11 Dec 2008, Vancouver, Canada.
This paper introduces a new approach to constructing meaningful lower dimensional representations of sets of data points. We argue that constraining the mapping between the high and low dimensional spaces to be a diffeomorphism is a natural way of ensuring that pairwise distances are approximately preserved. Accordingly we develop an algorithm which diffeomorphically maps the data near to a lower dimensional subspace and then projects onto that subspace. The problem of solving for the mapping is transformed into one of solving for an Eulerian flow field which we compute using ideas from kernel methods. We demonstrate the efficacy of our approach on various real world data sets.