Diffusion Harmonics and Dual Geometry on Carnot Manifold

The "curse of dimensionality" motivates the importance of techniques for computing low-dimensional approximations of high-dimensional data. It is often necessary to use nonlinear techniques to recover a low-dimensional manifold embedded via a nonlinear map in a high-dimensional space; this family of techniques is referred to as "manifold learning." The accuracy of manifold-learning-based approximations is founded on asymptotic results that assume the data is drawn from a low-dimensional Riemannian manifold. However, in natural datasets, this assumption is often overly restrictive. In the first part of this thesis we examine a more general class of manifolds known as Carnot manifolds, a type of sub-Riemannian manifold that governs natural phenomena such as chemical kinetics and configuration spaces of jointed objects. We find that diffusion maps can be generalized to Carnot manifolds and that the projection onto diffusion harmonics gives an almost isometric embedding; as a side effect, the diffusion distance is a computationally fast estimate for the shortest distance between two points on a Carnot manifold. We apply this theory to biochemical network data and observe that the chemical kinetics of the EGFR network are governed by a Carnot, but not Riemannian, manifold.;In the second part of this thesis we examine the Heisenberg group, a classical example of a Carnot manifold. We obtain a representation-theoretic proof that the eigenfunctions of the sub-Laplacian on SU(2) approach the eigenfunctions of the sub-Laplacian on the Heisenberg group, in the limit as the radius of the sphere becomes large, in analogy with the limiting relationship between the Fourier series on the circle and the Fourier transform on the line. This result also illustrates how projecting onto the sub-Laplacian eigenfunctions of a non-compact Carnot manifold can be locally approximated by projecting onto the sub-Laplacian eigenfunctions of a tangent compact Carnot manifold.;In the third part, we examine the question of "dual geometries" on certain Carnot manifolds, in particular the group SU(2). The spectrum of the (sub-)Laplacian does not give local, geometric information about a manifold; two eigenfunctions with nearby eigenvalues may be predominantly supported on different parts of the manifold. We examine an alternative structure for (sub)-Laplacian eigenfunctions that does incorporate such local information. One can define an "affinity" that measures the similarity between functions by summing over their inner products restricted to elements of a partition of the manifold. Given all pairwise affinities, in certain cases, it is possible to reconstruct the entire basis of eigenfunctions. However, if the manifold has symmetries, as in the case of the 3-sphere, the eigenspaces will have degree greater than one and the choice of basis of eigenfunctions must be paired with the choice of partition in order to reconstruct the eigenfunctions. The "dual" relationship between eigenfunctions and partitions gives some insights into how one might characterize the geometric information in a manifold by means of its (sub)-Laplacian eigenfunctions.