28,000 research outputs found
Far-Field Compression for Fast Kernel Summation Methods in High Dimensions
We consider fast kernel summations in high dimensions: given a large set of
points in dimensions (with ) and a pair-potential function (the
{\em kernel} function), we compute a weighted sum of all pairwise kernel
interactions for each point in the set. Direct summation is equivalent to a
(dense) matrix-vector multiplication and scales quadratically with the number
of points. Fast kernel summation algorithms reduce this cost to log-linear or
linear complexity.
Treecodes and Fast Multipole Methods (FMMs) deliver tremendous speedups by
constructing approximate representations of interactions of points that are far
from each other. In algebraic terms, these representations correspond to
low-rank approximations of blocks of the overall interaction matrix. Existing
approaches require an excessive number of kernel evaluations with increasing
and number of points in the dataset.
To address this issue, we use a randomized algebraic approach in which we
first sample the rows of a block and then construct its approximate, low-rank
interpolative decomposition. We examine the feasibility of this approach
theoretically and experimentally. We provide a new theoretical result showing a
tighter bound on the reconstruction error from uniformly sampling rows than the
existing state-of-the-art. We demonstrate that our sampling approach is
competitive with existing (but prohibitively expensive) methods from the
literature. We also construct kernel matrices for the Laplacian, Gaussian, and
polynomial kernels -- all commonly used in physics and data analysis. We
explore the numerical properties of blocks of these matrices, and show that
they are amenable to our approach. Depending on the data set, our randomized
algorithm can successfully compute low rank approximations in high dimensions.
We report results for data sets with ambient dimensions from four to 1,000.Comment: 43 pages, 21 figure
Protocol-Dependence and State Variables in the Force-Moment Ensemble
Stress-based ensembles incorporating temperature-like variables have been
proposed as a route to an equation of state for granular materials. To test the
efficacy of this approach, we perform experiments on a two-dimensional
photoelastic granular system under three loading conditions: uniaxial
compression, biaxial compression, and simple shear. From the interparticle
forces, we find that the distributions of the normal component of the
coarse-grained force-moment tensor are exponential-tailed, while the deviatoric
component is Gaussian-distributed. This implies that the correct stress-based
statistical mechanics conserves both the force-moment tensor and the
Maxwell-Cremona force-tiling area. As such, two variables of state arise: the
tensorial angoricity () and a new temperature-like quantity
associated with the force-tile area which we name {\it keramicity} ().
Each quantity is observed to be inversely proportional to the global confining
pressure; however only exhibits the protocol-independence expected of
a state variable, while behaves as a variable of process
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