2,989 research outputs found
Transportation cost-information and concentration inequalities for bifurcating Markov chains
We investigate the transportation cost-information inequalities for
bifurcating Markov chains which are a class of processes indexed by binary
tree. These processes provide models for cell growth when each individual in
one generation gives birth to two offsprings in the next one. Transportation
cost inequalities provide useful concentra-tion inequalities. We also study
deviation inequalities for the empiri-cal means under relaxed assumptions on
the Wasserstein contraction of the Markov kernels. Applications to bifurcating
non linear autore-gressive processes are considered: deviation inequalities for
pointwise estimates of the non linear leading functions
Exact Analysis of TTL Cache Networks: The Case of Caching Policies driven by Stopping Times
TTL caching models have recently regained significant research interest,
largely due to their ability to fit popular caching policies such as LRU. This
paper advances the state-of-the-art analysis of TTL-based cache networks by
developing two exact methods with orthogonal generality and computational
complexity. The first method generalizes existing results for line networks
under renewal requests to the broad class of caching policies whereby evictions
are driven by stopping times. The obtained results are further generalized,
using the second method, to feedforward networks with Markov arrival processes
(MAP) requests. MAPs are particularly suitable for non-line networks because
they are closed not only under superposition and splitting, as known, but also
under input-output caching operations as proven herein for phase-type TTL
distributions. The crucial benefit of the two closure properties is that they
jointly enable the first exact analysis of feedforward networks of TTL caches
in great generality
Laplace deconvolution on the basis of time domain data and its application to Dynamic Contrast Enhanced imaging
In the present paper we consider the problem of Laplace deconvolution with
noisy discrete non-equally spaced observations on a finite time interval. We
propose a new method for Laplace deconvolution which is based on expansions of
the convolution kernel, the unknown function and the observed signal over
Laguerre functions basis (which acts as a surrogate eigenfunction basis of the
Laplace convolution operator) using regression setting. The expansion results
in a small system of linear equations with the matrix of the system being
triangular and Toeplitz. Due to this triangular structure, there is a common
number of terms in the function expansions to control, which is realized
via complexity penalty. The advantage of this methodology is that it leads to
very fast computations, produces no boundary effects due to extension at zero
and cut-off at and provides an estimator with the risk within a logarithmic
factor of the oracle risk. We emphasize that, in the present paper, we consider
the true observational model with possibly nonequispaced observations which are
available on a finite interval of length which appears in many different
contexts, and account for the bias associated with this model (which is not
present when ). The study is motivated by perfusion imaging
using a short injection of contrast agent, a procedure which is applied for
medical assessment of micro-circulation within tissues such as cancerous
tumors. Presence of a tuning parameter allows to choose the most
advantageous time units, so that both the kernel and the unknown right hand
side of the equation are well represented for the deconvolution. The
methodology is illustrated by an extensive simulation study and a real data
example which confirms that the proposed technique is fast, efficient,
accurate, usable from a practical point of view and very competitive.Comment: 36 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1207.223
Optimising Spatial and Tonal Data for PDE-based Inpainting
Some recent methods for lossy signal and image compression store only a few
selected pixels and fill in the missing structures by inpainting with a partial
differential equation (PDE). Suitable operators include the Laplacian, the
biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The
quality of such approaches depends substantially on the selection of the data
that is kept. Optimising this data in the domain and codomain gives rise to
challenging mathematical problems that shall be addressed in our work.
In the 1D case, we prove results that provide insights into the difficulty of
this problem, and we give evidence that a splitting into spatial and tonal
(i.e. function value) optimisation does hardly deteriorate the results. In the
2D setting, we present generic algorithms that achieve a high reconstruction
quality even if the specified data is very sparse. To optimise the spatial
data, we use a probabilistic sparsification, followed by a nonlocal pixel
exchange that avoids getting trapped in bad local optima. After this spatial
optimisation we perform a tonal optimisation that modifies the function values
in order to reduce the global reconstruction error. For homogeneous diffusion
inpainting, this comes down to a least squares problem for which we prove that
it has a unique solution. We demonstrate that it can be found efficiently with
a gradient descent approach that is accelerated with fast explicit diffusion
(FED) cycles. Our framework allows to specify the desired density of the
inpainting mask a priori. Moreover, is more generic than other data
optimisation approaches for the sparse inpainting problem, since it can also be
extended to nonlinear inpainting operators such as EED. This is exploited to
achieve reconstructions with state-of-the-art quality.
We also give an extensive literature survey on PDE-based image compression
methods
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