Nonlinear barycentric dimensionality reduction

Many high-dimensional datasets can be mapped onto lower-dimensional linear simplexes, parametrized by barycentric coordinates. We present an unsupervised algorithm that is able to find the barycentric coordinates and corresponding vertices of such a high-dimensional dataset, by combining manifold learning with a distance geometry based algorithm for finding a maximal volume inscribed simplex. The performance of the algorithm is demonstrated on a Swiss-roll dataset that is restricted to a simplex, and on the spectral unmixing of hyperspectral imagery.

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

[2]  N. Sukumar,et al.  Conforming polygonal finite elements , 2004 .

[3]  Mario Winter,et al.  N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data , 1999, Optics & Photonics.

[4]  Stefan Vandewalle,et al.  Barycentric interpolation and exact integration formulas for the finite volume element method , 2008 .

[5]  Thierry Dana-Picard.  Some applications of barycentric computations , 2000 .

[6]  Tony R. Martinez,et al.  Iterative Non-linear Dimensionality Reduction with Manifold Sculpting , 2007, NIPS.

[7]  Joe D. Warren,et al.  Barycentric coordinates for convex polytopes , 1996, Adv. Comput. Math..

[8]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[9]  R. Singer,et al.  Mars - Large scale mixing of bright and dark surface materials and implications for analysis of spectral reflectance , 1979 .

[10]  Thomas L. Ainsworth,et al.  Exploiting manifold geometry in hyperspectral imagery , 2005, IEEE Transactions on Geoscience and Remote Sensing.

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

[12]  J. Boardman,et al.  Geometric mixture analysis of imaging spectrometry data , 1994, Proceedings of IGARSS '94 - 1994 IEEE International Geoscience and Remote Sensing Symposium.

[13]  Mark Meyer,et al.  Generalized Barycentric Coordinates on Irregular Polygons , 2002, J. Graphics, GPU, & Game Tools.

[14]  Mikhail Belkin,et al.  Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering , 2001, NIPS.