903 research outputs found
Application-tailored Linear Algebra Algorithms: A search-based Approach
In this paper, we tackle the problem of automatically generating algorithms
for linear algebra operations by taking advantage of problem-specific
knowledge. In most situations, users possess much more information about the
problem at hand than what current libraries and computing environments accept;
evidence shows that if properly exploited, such information leads to
uncommon/unexpected speedups. We introduce a knowledge-aware linear algebra
compiler that allows users to input matrix equations together with properties
about the operands and the problem itself; for instance, they can specify that
the equation is part of a sequence, and how successive instances are related to
one another. The compiler exploits all this information to guide the generation
of algorithms, to limit the size of the search space, and to avoid redundant
computations. We applied the compiler to equations arising as part of
sensitivity and genome studies; the algorithms produced exhibit, respectively,
100- and 1000-fold speedups
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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