Diffusion maps for changing data

Abstract 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.

[1]  Russell C. Hardie,et al.  Hyperspectral Change Detection in the Presenceof Diurnal and Seasonal Variations , 2008, IEEE Transactions on Geoscience and Remote Sensing.

[2]  C. Brislawn,et al.  Traceable integral kernels on countably generated measure spaces. , 1991 .

[3]  J. Lee,et al.  MULTISCALE ANALYSIS OF TIME SERIES OF GRAPHS , 2010 .

[4]  C. Brislawn,et al.  Kernels of trace class operators , 1988 .

[5]  J. Mercer.  Functions of Positive and Negative Type, and their Connection with the Theory of Integral Equations , 1909 .

[6]  A. Singer,et al.  Vector diffusion maps and the connection Laplacian , 2011, Communications on pure and applied mathematics.

[7]  B. Simon.  Trace ideals and their applications , 1979 .

[8]  I. Mezić,et al.  Ergodic Theory and Visualization II: Visualization of Resonances and Periodic Sets , 2008 .

[9]  Mikhail Belkin,et al.  On Learning with Integral Operators , 2010, J. Mach. Learn. Res..

[10]  Igor Mezić,et al.  Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets. , 2008, Chaos.

[11]  Mikhail Belkin,et al.  Laplacian Eigenmaps for Dimensionality Reduction and Data Representation , 2003, Neural Computation.

[12]  Ronald R. Coifman,et al.  Bi-stochastic kernels via asymmetric affinity functions , 2012, ArXiv.

[13]  Yuan Yao,et al.  Mercer's Theorem, Feature Maps, and Smoothing , 2006, COLT.

[14]  I. Mezic,et al.  Comparison of Systems using Diffusion Maps , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[15]  D. Donoho,et al.  Hessian Eigenmaps : new locally linear embedding techniques for high-dimensional data , 2003 .

[16]  D. Donoho,et al.  Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[17]  N. Aronszajn.  Theory of Reproducing Kernels. , 1950 .

[18]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[19]  Lorenzo Rosasco,et al.  Learning from Examples as an Inverse Problem , 2005, J. Mach. Learn. Res..

[20]  Amir Averbuch,et al.  Linear-projection diffusion on smooth Euclidean submanifolds , 2013 .

[21]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.

[22]  V. Burenkov.  Sobolev spaces on domains , 1998 .

[23]  Hiba Abdallah.  Processus de diffusion sur un flot de variétés riemanniennes , 2010 .

[24]  F. Mémoli,et al.  A spectral notion of Gromov–Wasserstein distance and related methods , 2011 .

[25]  Stéphane Lafon,et al.  Diffusion maps , 2006 .

[26]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[27]  Igor Mezi'c,et al.  Ergodic theory and visualization. II. Harmonic mesochronic plots visualize (quasi)periodic sets , 2008, 0808.2182.