5,551 research outputs found
Perturbation theory for Markov chains via Wasserstein distance
Perturbation theory for Markov chains addresses the question how small
differences in the transitions of Markov chains are reflected in differences
between their distributions. We prove powerful and flexible bounds on the
distance of the th step distributions of two Markov chains when one of them
satisfies a Wasserstein ergodicity condition. Our work is motivated by the
recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the
analysis of big data sets. By using an approach based on Lyapunov functions, we
provide estimates for geometrically ergodic Markov chains under weak
assumptions. In an autoregressive model, our bounds cannot be improved in
general. We illustrate our theory by showing quantitative estimates for
approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings
and stochastic Langevin algorithms.Comment: 31 pages, accepted at Bernoulli Journa
Tractability of the approximation of high-dimensional rank one tensors
We study the approximation of high-dimensional rank one tensors using point
evaluations and consider deterministic as well as randomized algorithms. We
prove that for certain parameters (smoothness and norm of the th derivative)
this problem is intractable while for other parameters the problem is tractable
and the complexity is only polynomial in the dimension for every fixed
. For randomized algorithms we completely characterize the set
of parameters that lead to easy or difficult problems, respectively. In the
"difficult" case we modify the class to obtain a tractable problem: The problem
gets tractable with a polynomial (in the dimension) complexity if the support
of the function is not too small.Comment: 15 pages, to appear in Constr. Appro
Confined chiral polymer nematics: ordering and spontaneous condensation
We investigate condensation of a long confined chiral nematic polymer inside
a spherical enclosure, mimicking condensation of DNA inside a viral capsid. The
Landau-de Gennes nematic free energy {\sl Ansatz} appropriate for nematic
polymers allows us to study the condensation process in detail with different
boundary conditions at the enclosing wall that simulate repulsive and
attractive polymer-surface interactions. Increasing the chirality, we observe a
transformation of the toroidal condensate into a closed surface with an
increasing genus, akin to the ordered domain formation observed in
cryo-microscopy of bacteriophages
Error bounds of MCMC for functions with unbounded stationary variance
We prove explicit error bounds for Markov chain Monte Carlo (MCMC) methods to
compute expectations of functions with unbounded stationary variance. We assume
that there is a so that the functions have finite -norm. For
uniformly ergodic Markov chains we obtain error bounds with the optimal order
of convergence and if there exists a spectral gap we almost get the
optimal order. Further, a burn-in period is taken into account and a recipe for
choosing the burn-in is provided.Comment: 13 page
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