19,927 research outputs found
Tree-Independent Dual-Tree Algorithms
Dual-tree algorithms are a widely used class of branch-and-bound algorithms.
Unfortunately, developing dual-tree algorithms for use with different trees and
problems is often complex and burdensome. We introduce a four-part logical
split: the tree, the traversal, the point-to-point base case, and the pruning
rule. We provide a meta-algorithm which allows development of dual-tree
algorithms in a tree-independent manner and easy extension to entirely new
types of trees. Representations are provided for five common algorithms; for
k-nearest neighbor search, this leads to a novel, tighter pruning bound. The
meta-algorithm also allows straightforward extensions to massively parallel
settings.Comment: accepted in ICML 201
High field properties of geometrically frustrated magnets
Above the saturation field, geometrically frustrated quantum antiferromagnets
have dispersionless low-energy branches of excitations corresponding to
localized spin-flip modes. Transition into a partially magnetized state occurs
via condensation of an infinite number of degrees of freedom. The ground state
below the phase transition is a magnon crystal, which breaks only translational
symmetry and preserves spin-rotations about the field direction. We give a
detailed review of recent works on physics of such phase transitions and
present further theoretical developments. Specifically, the low-energy degrees
of freedom of a spin-1/2 kagom\'e antiferromagnet are mapped to a hard hexagon
gas on a triangular lattice. Such a mapping allows to obtain a quantitative
description of the magnetothermodynamics of a quantum kagom\'e antiferromagnet
from the exact solution for a hard hexagon gas. In particular, we find the
exact critical behavior at the transition into a magnon crystal state, the
universal value of the entropy at the saturation field, and the position of
peaks in temperature- and field-dependence of the specific heat. Analogous
mapping is presented for the sawtooth chain, which is mapped onto a model of
classical hard dimers on a chain. The finite macroscopic entropies of
geometrically frustrated magnets at the saturation field lead to a large
magnetocaloric effect.Comment: 22 pages, proceedings of YKIS2004 worksho
Spin excitations used to probe the nature of the exchange coupling in the magnetically ordered ground state of PrCaMnO
We have used time-of-flight inelastic neutron scattering to measure the spin
wave spectrum of the canonical half-doped manganite
PrCaMnO, in its magnetic and orbitally ordered phase. The
data, which cover multiple Brillouin zones and the entire energy range of the
excitations, are compared with several different models that are all consistent
with the CE-type magnetic order, but arise through different exchange coupling
schemes. The Goodenough model, i.e. an ordered state comprising strong nearest
neighbor ferromagnetic interactions along zig-zag chains with antiferromagnetic
inter-chain coupling, provides the best description of the data, provided that
further neighbor interactions along the chains are included. We are able to
rule out a coupling scheme involving formation of strongly bound ferromagnetic
dimers, i.e. Zener polarons, on the basis of gross features of the observed
spin wave spectrum. A model with weaker dimerization reproduces the observed
dispersion but can be ruled out on the basis of discrepancies between the
calculated and observed structure factors at certain positions in reciprocal
space. Adding further neighbor interactions results in almost no dimerization,
i.e. recovery of the Goodenough model. These results are consistent with
theoretical analysis of the degenerate double exchange model for half-doping,
and provide a recipe for how to interpret future measurements away from
half-doping, where degenerate double exchange models predict more complex
ground states.Comment: 14 pages, 11 figure
Sampling-based Algorithms for Optimal Motion Planning
During the last decade, sampling-based path planning algorithms, such as
Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have
been shown to work well in practice and possess theoretical guarantees such as
probabilistic completeness. However, little effort has been devoted to the
formal analysis of the quality of the solution returned by such algorithms,
e.g., as a function of the number of samples. The purpose of this paper is to
fill this gap, by rigorously analyzing the asymptotic behavior of the cost of
the solution returned by stochastic sampling-based algorithms as the number of
samples increases. A number of negative results are provided, characterizing
existing algorithms, e.g., showing that, under mild technical conditions, the
cost of the solution returned by broadly used sampling-based algorithms
converges almost surely to a non-optimal value. The main contribution of the
paper is the introduction of new algorithms, namely, PRM* and RRT*, which are
provably asymptotically optimal, i.e., such that the cost of the returned
solution converges almost surely to the optimum. Moreover, it is shown that the
computational complexity of the new algorithms is within a constant factor of
that of their probabilistically complete (but not asymptotically optimal)
counterparts. The analysis in this paper hinges on novel connections between
stochastic sampling-based path planning algorithms and the theory of random
geometric graphs.Comment: 76 pages, 26 figures, to appear in International Journal of Robotics
Researc
- …