637 research outputs found
Approximating Loops in a Shortest Homology Basis from Point Data
Inference of topological and geometric attributes of a hidden manifold from
its point data is a fundamental problem arising in many scientific studies and
engineering applications. In this paper we present an algorithm to compute a
set of loops from a point data that presumably sample a smooth manifold
. These loops approximate a {\em shortest} basis of the
one dimensional homology group over coefficients in finite field
. Previous results addressed the issue of computing the rank of
the homology groups from point data, but there is no result on approximating
the shortest basis of a manifold from its point sample. In arriving our result,
we also present a polynomial time algorithm for computing a shortest basis of
for any finite {\em simplicial complex} whose edges have
non-negative weights
On invariants and homology of spaces of knots in arbitrary manifolds
Finite-order invariants of knots in arbitrary 3-manifolds (including
non-orientable ones) are constructed and studied by methods of the topology of
discriminant sets. Obstructions to the integrability of admissible weight
systems to well-defined knot invariants are identified as 1-dimensional
cohomology classes of generalized loop spaces of the manifold. Unlike the case
of the 3-sphere, these obstructions can be non-trivial and provide invariants
of the manifold itself.
The corresponding algebraic machinery allows us to obtain on the level of the
``abstract nonsense'' some of results and problems of the theory, and to
extract from other the essential topological part
Minimum cycle and homology bases of surface embedded graphs
We study the problems of finding a minimum cycle basis (a minimum weight set
of cycles that form a basis for the cycle space) and a minimum homology basis
(a minimum weight set of cycles that generates the -dimensional
()-homology classes) of an undirected graph embedded on a
surface. The problems are closely related, because the minimum cycle basis of a
graph contains its minimum homology basis, and the minimum homology basis of
the -skeleton of any graph is exactly its minimum cycle basis.
For the minimum cycle basis problem, we give a deterministic
-time algorithm for graphs embedded on an orientable
surface of genus . The best known existing algorithms for surface embedded
graphs are those for general graphs: an time Monte Carlo
algorithm and a deterministic time algorithm. For the
minimum homology basis problem, we give a deterministic -time algorithm for graphs embedded on an orientable or non-orientable
surface of genus with boundary components, assuming shortest paths are
unique, improving on existing algorithms for many values of and . The
assumption of unique shortest paths can be avoided with high probability using
randomization or deterministically by increasing the running time of the
homology basis algorithm by a factor of .Comment: A preliminary version of this work was presented at the 32nd Annual
International Symposium on Computational Geometr
Bayesian analysis for reversible Markov chains
We introduce a natural conjugate prior for the transition matrix of a
reversible Markov chain. This allows estimation and testing. The prior arises
from random walk with reinforcement in the same way the Dirichlet prior arises
from P\'{o}lya's urn. We give closed form normalizing constants, a simple
method of simulation from the posterior and a characterization along the lines
of W. E. Johnson's characterization of the Dirichlet prior.Comment: Published at http://dx.doi.org/10.1214/009053606000000290 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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