783 research outputs found
Higher Dimensional Discrete Cheeger Inequalities
For graphs there exists a strong connection between spectral and
combinatorial expansion properties. This is expressed, e.g., by the discrete
Cheeger inequality, the lower bound of which states that , where is the second smallest eigenvalue of the Laplacian of
a graph and is the Cheeger constant measuring the edge expansion of
. We are interested in generalizations of expansion properties to finite
simplicial complexes of higher dimension (or uniform hypergraphs).
Whereas higher dimensional Laplacians were introduced already in 1945 by
Eckmann, the generalization of edge expansion to simplicial complexes is not
straightforward. Recently, a topologically motivated notion analogous to edge
expansion that is based on -cohomology was introduced by Gromov
and independently by Linial, Meshulam and Wallach. It is known that for this
generalization there is no higher dimensional analogue of the lower bound of
the Cheeger inequality. A different, combinatorially motivated generalization
of the Cheeger constant, denoted by , was studied by Parzanchevski,
Rosenthal and Tessler. They showed that indeed , where
is the smallest non-trivial eigenvalue of the (-dimensional
upper) Laplacian, for the case of -dimensional simplicial complexes with
complete -skeleton.
Whether this inequality also holds for -dimensional complexes with
non-complete -skeleton has been an open question. We give two proofs of
the inequality for arbitrary complexes. The proofs differ strongly in the
methods and structures employed, and each allows for a different kind of
additional strengthening of the original result.Comment: 14 pages, 2 figure
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
Cohomological invariants of a variation of flat connection
In this paper, we apply the theory of Chern-Cheeger-Simons to construct
canonical invariants associated to a -simplex whose points parametrize flat
connections on a smooth manifold . These invariants lie in degrees
-cohomology with -coefficients, for . In turn, this
corresponds to a homomorphism on the higher homology groups of the moduli space
of flat connections, and taking values in -cohomology of the underlying
smooth manifold .Comment: 15 p. Final version, to appear. arXiv admin note: text overlap with
arXiv:1310.000
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