1,106 research outputs found
Poisson-type deviation inequalities for curved continuous-time Markov chains
In this paper, we present new Poisson-type deviation inequalities for
continuous-time Markov chains whose Wasserstein curvature or -curvature
is bounded below. Although these two curvatures are equivalent for Brownian
motion on Riemannian manifolds, they are not comparable in discrete settings
and yield different deviation bounds. In the case of birth--death processes, we
provide some conditions on the transition rates of the associated generator for
such curvatures to be bounded below and we extend the deviation inequalities
established [An\'{e}, C. and Ledoux, M. On logarithmic Sobolev inequalities for
continuous time random walks on graphs. Probab. Theory Related Fields 116
(2000) 573--602] for continuous-time random walks, seen as models in null
curvature. Some applications of these tail estimates are given for
Brownian-driven Ornstein--Uhlenbeck processes and queues.Comment: Published at http://dx.doi.org/10.3150/07-BEJ6039 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature
The purpose of this paper is to extend the investigation of Poisson-type
deviation inequalities started by Joulin (Bernoulli 13 (2007) 782--798) to the
empirical mean of positively curved Markov jump processes. In particular, our
main result generalizes the tail estimates given by Lezaud (Ann. Appl. Probab.
8 (1998) 849--867, ESAIM Probab. Statist. 5 (2001) 183--201). An application to
birth--death processes completes this work.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ158 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Premixed flame shapes and polynomials
The nonlinear nonlocal Michelson-Sivashinsky equation for isolated crests of
unstable flames is studied, using pole-decompositions as starting point.
Polynomials encoding the numerically computed 2N flame-slope poles, and
auxiliary ones, are found to closely follow a Meixner Pollaczek recurrence;
accurate steady crest shapes ensue for N>=3. Squeezed crests ruled by a
discretized Burgers equation involve the same polynomials. Such explicit
approximate shape still lack for finite-N pole-decomposed periodic flames,
despite another empirical recurrence.Comment: Accepted for publication in Physica D :Nonlinear Phenomen
Measure concentration through non-Lipschitz observables and functional inequalities
Non-Gaussian concentration estimates are obtained for invariant probability
measures of reversible Markov processes. We show that the functional
inequalities approach combined with a suitable Lyapunov condition allows us to
circumvent the classical Lipschitz assumption of the observables. Our method is
general and covers diffusions as well as pure-jump Markov processes on
unbounded spaces
A Convex Relaxation for Weakly Supervised Classifiers
This paper introduces a general multi-class approach to weakly supervised
classification. Inferring the labels and learning the parameters of the model
is usually done jointly through a block-coordinate descent algorithm such as
expectation-maximization (EM), which may lead to local minima. To avoid this
problem, we propose a cost function based on a convex relaxation of the
soft-max loss. We then propose an algorithm specifically designed to
efficiently solve the corresponding semidefinite program (SDP). Empirically,
our method compares favorably to standard ones on different datasets for
multiple instance learning and semi-supervised learning as well as on
clustering tasks.Comment: Appears in Proceedings of the 29th International Conference on
Machine Learning (ICML 2012
Intertwining and commutation relations for birth-death processes
Given a birth-death process on with semigroup
and a discrete gradient depending on a positive weight , we
establish intertwining relations of the form
, where is the Feynman-Kac
semigroup with potential of another birth-death process. We provide
applications when is nonnegative and uniformly bounded from below,
including Lipschitz contraction and Wasserstein curvature, various functional
inequalities, and stochastic orderings. Our analysis is naturally connected to
the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death
processes. The proofs are remarkably simple and rely on interpolation,
commutation, and convexity.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ433 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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