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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 D H. 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 D H 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 …