We consider problems of dimensionality reduction and learning data representations for continuous spaces with two or more independent degrees of freedom. Such problems occur, for example, when observing shapes with several components that move independently. Mathematically, if the parameter space of each continuous independent motion is a manifold, then their combination is known as a product manifold. In this paper, we present a new paradigm for non-linear independent component analysis called manifold factorization. Our factorization algorithm is based on spectral graph methods for manifold learning and the separability of the Laplacian operator on product spaces. Recovering the factors of a manifold yields meaningful lower-dimensional representations and provides a new way to focus on particular aspects of the data space while ignoring others. We demonstrate the potential use of our method for an important and challenging problem in structural biology: mapping the motions of proteins and other large molecules using cryo-electron microscopy datasets.
Product Manifold Learning