  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.
