33 research outputs found
Diffusion maps for changing data
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
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