5,921 research outputs found
An Adaptive Total Variation Algorithm for Computing the Balanced Cut of a Graph
We propose an adaptive version of the total variation algorithm proposed in
[3] for computing the balanced cut of a graph. The algorithm from [3] used a
sequence of inner total variation minimizations to guarantee descent of the
balanced cut energy as well as convergence of the algorithm. In practice the
total variation minimization step is never solved exactly. Instead, an accuracy
parameter is specified and the total variation minimization terminates once
this level of accuracy is reached. The choice of this parameter can vastly
impact both the computational time of the overall algorithm as well as the
accuracy of the result. Moreover, since the total variation minimization step
is not solved exactly, the algorithm is not guarantied to be monotonic. In the
present work we introduce a new adaptive stopping condition for the total
variation minimization that guarantees monotonicity. This results in an
algorithm that is actually monotonic in practice and is also significantly
faster than previous, non-adaptive algorithms
Bayesian Poisson process partition calculus with an application to Bayesian L\'evy moving averages
This article develops, and describes how to use, results concerning
disintegrations of Poisson random measures. These results are fashioned as
simple tools that can be tailor-made to address inferential questions arising
in a wide range of Bayesian nonparametric and spatial statistical models. The
Poisson disintegration method is based on the formal statement of two results
concerning a Laplace functional change of measure and a Poisson Palm/Fubini
calculus in terms of random partitions of the integers {1,...,n}. The
techniques are analogous to, but much more general than, techniques for the
Dirichlet process and weighted gamma process developed in [Ann. Statist. 12
(1984) 351-357] and [Ann. Inst. Statist. Math. 41 (1989) 227-245]. In order to
illustrate the flexibility of the approach, large classes of random probability
measures and random hazards or intensities which can be expressed as
functionals of Poisson random measures are described. We describe a unified
posterior analysis of classes of discrete random probability which identifies
and exploits features common to all these models. The analysis circumvents many
of the difficult issues involved in Bayesian nonparametric calculus, including
a combinatorial component. This allows one to focus on the unique features of
each process which are characterized via real valued functions h. The
applicability of the technique is further illustrated by obtaining explicit
posterior expressions for L\'evy-Cox moving average processes within the
general setting of multiplicative intensity models.Comment: Published at http://dx.doi.org/10.1214/009053605000000336 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The Recursive Form of Error Bounds for RFS State and Observation with Pd<1
In the target tracking and its engineering applications, recursive state
estimation of the target is of fundamental importance. This paper presents a
recursive performance bound for dynamic estimation and filtering problem, in
the framework of the finite set statistics for the first time. The number of
tracking algorithms with set-valued observations and state of targets is
increased sharply recently. Nevertheless, the bound for these algorithms has
not been fully discussed. Treating the measurement as set, this bound can be
applied when the probability of detection is less than unity. Moreover, the
state is treated as set, which is singleton or empty with certain probability
and accounts for the appearance and the disappearance of the targets. When the
existence of the target state is certain, our bound is as same as the most
accurate results of the bound with probability of detection is less than unity
in the framework of random vector statistics. When the uncertainty is taken
into account, both linear and non-linear applications are presented to confirm
the theory and reveal this bound is more general than previous bounds in the
framework of random vector statistics.In fact, the collection of such
measurements could be treated as a random finite set (RFS)
Topology of random clique complexes
In a seminal paper, Erdos and Renyi identified the threshold for connectivity
of the random graph G(n,p). In particular, they showed that if p >> log(n)/n
then G(n,p) is almost always connected, and if p << log(n)/n then G(n,p) is
almost always disconnected, as n goes to infinity.
The clique complex X(H) of a graph H is the simplicial complex with all
complete subgraphs of H as its faces. In contrast to the zeroth homology group
of X(H), which measures the number of connected components of H, the higher
dimensional homology groups of X(H) do not correspond to monotone graph
properties. There are nevertheless higher dimensional analogues of the
Erdos-Renyi Theorem.
We study here the higher homology groups of X(G(n,p)). For k > 0 we show the
following. If p = n^alpha, with alpha - 1/(2k+1), then the
kth homology group of X(G(n,p)) is almost always vanishing, and if -1/k < alpha
< -1/(k+1), then it is almost always nonvanishing.
We also give estimates for the expected rank of homology, and exhibit
explicit nontrivial classes in the nonvanishing regime. These estimates suggest
that almost all d-dimensional clique complexes have only one nonvanishing
dimension of homology, and we cannot rule out the possibility that they are
homotopy equivalent to wedges of spheres.Comment: 23 pages; final version, to appear in Discrete Mathematics. At
suggestion of anonymous referee, a section briefly summarizing the
topological prerequisites has been added to make the article accessible to a
wider audienc
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