88,002 research outputs found
Matching Kasteleyn Cities for Spin Glass Ground States
As spin glass materials have extremely slow dynamics, devious numerical
methods are needed to study low-temperature states. A simple and fast
optimization version of the classical Kasteleyn treatment of the Ising model is
described and applied to two-dimensional Ising spin glasses. The algorithm
combines the Pfaffian and matching approaches to directly strip droplet
excitations from an excited state. Extended ground states in Ising spin glasses
on a torus, which are optimized over all boundary conditions, are used to
compute precise values for ground state energy densities.Comment: 4 pages, 2 figures; minor clarification
Algorithmic and Statistical Perspectives on Large-Scale Data Analysis
In recent years, ideas from statistics and scientific computing have begun to
interact in increasingly sophisticated and fruitful ways with ideas from
computer science and the theory of algorithms to aid in the development of
improved worst-case algorithms that are useful for large-scale scientific and
Internet data analysis problems. In this chapter, I will describe two recent
examples---one having to do with selecting good columns or features from a (DNA
Single Nucleotide Polymorphism) data matrix, and the other having to do with
selecting good clusters or communities from a data graph (representing a social
or information network)---that drew on ideas from both areas and that may serve
as a model for exploiting complementary algorithmic and statistical
perspectives in order to solve applied large-scale data analysis problems.Comment: 33 pages. To appear in Uwe Naumann and Olaf Schenk, editors,
"Combinatorial Scientific Computing," Chapman and Hall/CRC Press, 201
An Exact No Free Lunch Theorem for Community Detection
A precondition for a No Free Lunch theorem is evaluation with a loss function
which does not assume a priori superiority of some outputs over others. A
previous result for community detection by Peel et al. (2017) relies on a
mismatch between the loss function and the problem domain. The loss function
computes an expectation over only a subset of the universe of possible outputs;
thus, it is only asymptotically appropriate with respect to the problem size.
By using the correct random model for the problem domain, we provide a
stronger, exact No Free Lunch theorem for community detection. The claim
generalizes to other set-partitioning tasks including core/periphery
separation, -clustering, and graph partitioning. Finally, we review the
literature of proposed evaluation functions and identify functions which
(perhaps with slight modifications) are compatible with an exact No Free Lunch
theorem
Random Costs in Combinatorial Optimization
The random cost problem is the problem of finding the minimum in an
exponentially long list of random numbers. By definition, this problem cannot
be solved faster than by exhaustive search. It is shown that a classical
NP-hard optimization problem, number partitioning, is essentially equivalent to
the random cost problem. This explains the bad performance of heuristic
approaches to the number partitioning problem and allows us to calculate the
probability distributions of the optimum and sub-optimum costs.Comment: 4 pages, Revtex, 2 figures (eps), submitted to PR
A histogram-free multicanonical Monte Carlo algorithm for the basis expansion of density of states
We report a new multicanonical Monte Carlo (MC) algorithm to obtain the
density of states (DOS) for physical systems with continuous state variables in
statistical mechanics. Our algorithm is able to obtain an analytical form for
the DOS expressed in a chosen basis set, instead of a numerical array of finite
resolution as in previous variants of this class of MC methods such as the
multicanonical (MUCA) sampling and Wang-Landau (WL) sampling. This is enabled
by storing the visited states directly in a data set and avoiding the explicit
collection of a histogram. This practice also has the advantage of avoiding
undesirable artificial errors caused by the discretization and binning of
continuous state variables. Our results show that this scheme is capable of
obtaining converged results with a much reduced number of Monte Carlo steps,
leading to a significant speedup over existing algorithms.Comment: 8 pages, 6 figures. Paper accepted in the Platform for Advanced
Scientific Computing Conference (PASC '17), June 26 to 28, 2017, Lugano,
Switzerlan
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