31,501 research outputs found
Representatives of the Thom class of a vector bundle
After a review of several methods designed to produce equivariant cohomology
classes, we apply one introduced by Berline, Getzler and Vergne, to get a
family of representatives of the universal Thom class of a vector bundle.
Surprisingly, this family does not contain the representative given by
Matha\"{\i} and Quillen. However it contains a particularly simple and
symmetric representative that we construct explicitly.Comment: 19 pages, Latex, one reference and some related comments added;
submitted to Journal of Geometry and Physic
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
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