33 research outputs found

    Diffusion maps for changing data

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    Graph Laplacians and related nonlinear mappings into low dimensional spaces have been shown to be powerful tools for organizing high dimensional data. Here we consider a data set X in which the graph associated with it changes depending on some set of parameters. We analyze this type of data in terms of the diffusion distance and the corresponding diffusion map. As the data changes over the parameter space, the low dimensional embedding changes as well. We give a way to go between these embeddings, and furthermore, map them all into a common space, allowing one to track the evolution of X in its intrinsic geometry. A global diffusion distance is also defined, which gives a measure of the global behavior of the data over the parameter space. Approximation theorems in terms of randomly sampled data are presented, as are potential applications.Comment: 38 pages. 9 figures. To appear in Applied and Computational Harmonic Analysis. v2: Several minor changes beyond just typos. v3: Minor typo corrected, added DO

    Local Kernels and the Geometric Structure of Data

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    We introduce a theory of local kernels, which generalize the kernels used in the standard diffusion maps construction of nonparametric modeling. We prove that evaluating a local kernel on a data set gives a discrete representation of the generator of a continuous Markov process, which converges in the limit of large data. We explicitly connect the drift and diffusion coefficients of the process to the moments of the kernel. Moreover, when the kernel is symmetric, the generator is the Laplace-Beltrami operator with respect to a geometry which is influenced by the embedding geometry and the properties of the kernel. In particular, this allows us to generate any Riemannian geometry by an appropriate choice of local kernel. In this way, we continue a program of Belkin, Niyogi, Coifman and others to reinterpret the current diverse collection of kernel-based data analysis methods and place them in a geometric framework. We show how to use this framework to design local kernels invariant to various features of data. These data-driven local kernels can be used to construct conformally invariant embeddings and reconstruct global diffeomorphisms
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