Learning transport operators for image manifolds

Abstract

We describe an unsupervised manifold learning algorithm that represents a surface through a compact description of operators that traverse it. The operators are based on matrix exponentials, which are the solution to a system of first-order linear differential equations. The matrix exponents are represented by a basis that is adapted to the statistics of the data so that the infinitesimal generator for a trajectory along the underlying manifold can be produced by linearly composing a few elements. The method is applied to recover topological structure from low dimensional synthetic data, and to model local structure in how natural images change over time and scale.