5,691 research outputs found
Confirmation Sampling for Exact Nearest Neighbor Search
Locality-sensitive hashing (LSH), introduced by Indyk and Motwani in STOC â98, has been an extremely influential framework for nearest neighbor search in high-dimensional data sets. While theoretical work has focused on the approximate nearest neighbor problem, in practice LSH data structures with suitably chosen parameters are used to solve the exact nearest neighbor problem (with some error probability). Sublinear query time is often possible in practice even for exact nearest neighbor search, intuitively because the nearest neighbor tends to be significantly closer than other data points. However, theory offers little advice on how to choose LSH parameters outside of pre-specified worst-case settings.
We introduce the technique of confirmation sampling for solving the exact nearest neighbor problem using LSH. First, we give a general reduction that transforms a sequence of data structures that each find the nearest neighbor with a small, unknown probability, into a data structure that returns the nearest neighbor with probability 1âÎŽ
, using as few queries as possible. Second, we present a new query algorithm for the LSH Forest data structure with L trees that is able to return the exact nearest neighbor of a query point within the same time bound as an LSH Forest of Ω(L) trees with internal parameters specifically tuned to the query and data
Next nearest neighbour Ising models on random graphs
This paper develops results for the next nearest neighbour Ising model on
random graphs. Besides being an essential ingredient in classic models for
frustrated systems, second neighbour interactions interactions arise naturally
in several applications such as the colour diversity problem and graphical
games. We demonstrate ensembles of random graphs, including regular
connectivity graphs, that have a periodic variation of free energy, with either
the ratio of nearest to next nearest couplings, or the mean number of nearest
neighbours. When the coupling ratio is integer paramagnetic phases can be found
at zero temperature. This is shown to be related to the locked or unlocked
nature of the interactions. For anti-ferromagnetic couplings, spin glass phases
are demonstrated at low temperature. The interaction structure is formulated as
a factor graph, the solution on a tree is developed. The replica symmetric and
energetic one-step replica symmetry breaking solution is developed using the
cavity method. We calculate within these frameworks the phase diagram and
demonstrate the existence of dynamical transitions at zero temperature for
cases of anti-ferromagnetic coupling on regular and inhomogeneous random
graphs.Comment: 55 pages, 15 figures, version 2 with minor revisions, to be published
J. Stat. Mec
Vibrations of closed-shell Lennard-Jones icosahedral and cuboctahedral clusters and their effect on the cluster ground state energy
Vibrational spectra of closed shell Lennard-Jones icosahedral and
cuboctahedral clusters are calculated for shell numbers between 2 and 9.
Evolution of the vibrational density of states with the cluster shell number is
examined and differences between icosahedral and cuboctahedral clusters
described. This enabled a quantum calculation of quantum ground state energies
of the clusters in the quasiharmonic approximation and a comparison of the
differences between the two types of clusters. It is demonstrated that in the
quantum treatment, the closed shell icosahedral clusters binding energies
differ from those of cuboctahedral clusters more than is the case in classical
treatment
Quantum Monte Carlo Study of Strongly Correlated Electrons: Cellular Dynamical Mean-Field Theory
We study the Hubbard model using the Cellular Dynamical Mean-Field Theory
(CDMFT) with quantum Monte Carlo (QMC) simulations. We present the algorithmic
details of CDMFT with the Hirsch-Fye QMC method for the solution of the
self-consistently embedded quantum cluster problem. We use the one- and
two-dimensional half-filled Hubbard model to gauge the performance of CDMFT+QMC
particularly for small clusters by comparing with the exact results and also
with other quantum cluster methods. We calculate single-particle Green's
functions and self-energies on small clusters to study their size dependence in
one- and two-dimensions.Comment: 14 pages, 18 figure
Resampling methods for parameter-free and robust feature selection with mutual information
Combining the mutual information criterion with a forward feature selection
strategy offers a good trade-off between optimality of the selected feature
subset and computation time. However, it requires to set the parameter(s) of
the mutual information estimator and to determine when to halt the forward
procedure. These two choices are difficult to make because, as the
dimensionality of the subset increases, the estimation of the mutual
information becomes less and less reliable. This paper proposes to use
resampling methods, a K-fold cross-validation and the permutation test, to
address both issues. The resampling methods bring information about the
variance of the estimator, information which can then be used to automatically
set the parameter and to calculate a threshold to stop the forward procedure.
The procedure is illustrated on a synthetic dataset as well as on real-world
examples
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