  Recently, there has been much interest in spectral approaches to learning
manifolds---so-called kernel eigenmap methods. These methods have had some
successes, but their applicability is limited because they are not robust to
noise. To address this limitation, we look at two-manifold problems, in which
we simultaneously reconstruct two related manifolds, each representing a
different view of the same data. By solving these interconnected learning
problems together, two-manifold algorithms are able to succeed where a
non-integrated approach would fail: each view allows us to suppress noise in
the other, reducing bias. We propose a class of algorithms for two-manifold
problems, based on spectral decomposition of cross-covariance operators in
Hilbert space, and discuss when two-manifold problems are useful. Finally, we
demonstrate that solving a two-manifold problem can aid in learning a nonlinear
dynamical system from limited data.
