  Recently, the theory of diffusion maps was extended to a large class of local
kernels with exponential decay which were shown to represent various Riemannian
geometries on a data set sampled from a manifold embedded in Euclidean space.
Moreover, local kernels were used to represent a diffeomorphism, H, between a
data set and a feature of interest using an anisotropic kernel function,
defined by a covariance matrix based on the local derivatives, DH. In this
paper, we generalize the theory of local kernels to represent degenerate
mappings where the intrinsic dimension of the data set is higher than the
intrinsic dimension of the feature space. First, we present a rigorous method
with asymptotic error bounds for estimating DH from the training data set and
feature values. We then derive scaling laws for the singular values of the
local linear structure of the data, which allows the identification the tangent
space and improved estimation of the intrinsic dimension of the manifold and
the bandwidth parameter of the diffusion maps algorithm. Using these numerical
tools, our approach to feature identification is to iterate the diffusion map
with appropriately chosen local kernels that emphasize the features of
interest. We interpret the iterated diffusion map (IDM) as a discrete
approximation to an intrinsic geometric flow which smoothly changes the
geometry of the data space to emphasize the feature of interest. When the data
lies on a product manifold of the feature manifold with an irrelevant manifold,
we show that the IDM converges to the quotient manifold which is isometric to
the feature manifold, thereby eliminating the irrelevant dimensions. We will
also demonstrate empirically that if we apply the IDM to features that are not
a quotient of the data space, the algorithm identifies an intrinsically
lower-dimensional set embedding of the data which better represents the
features.
