597 research outputs found
Nonlocal Multiscale Hierarchical Decomposition on Graphs
International audienceThe decomposition of images into their meaningful components is one of the major tasks in computer vision. Tadmor, Nezzar and Vese [1] have proposed a general approach for multiscale hierarchical decomposition of images. On the basis of this work, we propose a multiscale hierarchical decomposition of functions on graphs. The decomposition is based on a discrete variational framework that makes it possible to process arbitrary discrete data sets with the natural introduction of nonlocal interactions. This leads to an approach that can be used for the decomposition of images, meshes, or arbitrary data sets by taking advantage of the graph structure. To have a fully automatic decomposition, the issue of parameter selection is fully addressed. We illustrate our approach with numerous decomposition results on images, meshes, and point clouds and show the benefits
Adaptive multiresolution analysis based on synchronization
We propose an adaptive multiscale approach to data analysis based on synchronization. The approach is nonlinear, data driven in the sense that it does not rely on a priori chosen basis, and automatically determines the data scale. Numerical results for one- and two-dimensional cases illustrate that the method works effectively for the usual modulated signals such as chirps, etc., as well as for more complicated data with multiple scales. The method extends straightforwardly to functions defined on weighted graphs and grids in high dimensions. Connections with some other recent approaches to multiscale analysis are briefly discussed
Multilevel Artificial Neural Network Training for Spatially Correlated Learning
Multigrid modeling algorithms are a technique used to accelerate relaxation
models running on a hierarchy of similar graphlike structures. We introduce and
demonstrate a new method for training neural networks which uses multilevel
methods. Using an objective function derived from a graph-distance metric, we
perform orthogonally-constrained optimization to find optimal prolongation and
restriction maps between graphs. We compare and contrast several methods for
performing this numerical optimization, and additionally present some new
theoretical results on upper bounds of this type of objective function. Once
calculated, these optimal maps between graphs form the core of Multiscale
Artificial Neural Network (MsANN) training, a new procedure we present which
simultaneously trains a hierarchy of neural network models of varying spatial
resolution. Parameter information is passed between members of this hierarchy
according to standard coarsening and refinement schedules from the multiscale
modelling literature. In our machine learning experiments, these models are
able to learn faster than default training, achieving a comparable level of
error in an order of magnitude fewer training examples.Comment: Manuscript (24 pages) and Supplementary Material (4 pages). Updated
January 2019 to reflect new formulation of MsANN structure and new training
procedur
A fast direct solver for nonlocal operators in wavelet coordinates
In this article, we consider fast direct solvers for nonlocal operators. The
pivotal idea is to combine a wavelet representation of the system matrix,
yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The
latter drastically reduces the fill-in during the factorization of the system
matrix by means of a Cholesky decomposition or an LU decomposition,
respectively. This way, we end up with the exact inverse of the compressed
system matrix with only a moderate increase of the number of nonzero entries in
the matrix.
To illustrate the efficacy of the approach, we conduct numerical experiments
for different highly relevant applications of nonlocal operators: We consider
(i) the direct solution of boundary integral equations in three spatial
dimensions, issuing from the polarizable continuum model, (ii) a parabolic
problem for the fractional Laplacian in integral form and (iii) the fast
simulation of Gaussian random fields
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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