107,504 research outputs found
Asymptotic Bias of Stochastic Gradient Search
The asymptotic behavior of the stochastic gradient algorithm with a biased
gradient estimator is analyzed. Relying on arguments based on the dynamic
system theory (chain-recurrence) and the differential geometry (Yomdin theorem
and Lojasiewicz inequality), tight bounds on the asymptotic bias of the
iterates generated by such an algorithm are derived. The obtained results hold
under mild conditions and cover a broad class of high-dimensional nonlinear
algorithms. Using these results, the asymptotic properties of the
policy-gradient (reinforcement) learning and adaptive population Monte Carlo
sampling are studied. Relying on the same results, the asymptotic behavior of
the recursive maximum split-likelihood estimation in hidden Markov models is
analyzed, too.Comment: arXiv admin note: text overlap with arXiv:0907.102
On the computation of directional scale-discretized wavelet transforms on the sphere
We review scale-discretized wavelets on the sphere, which are directional and
allow one to probe oriented structure in data defined on the sphere.
Furthermore, scale-discretized wavelets allow in practice the exact synthesis
of a signal from its wavelet coefficients. We present exact and efficient
algorithms to compute the scale-discretized wavelet transform of band-limited
signals on the sphere. These algorithms are implemented in the publicly
available S2DW code. We release a new version of S2DW that is parallelized and
contains additional code optimizations. Note that scale-discretized wavelets
can be viewed as a directional generalization of needlets. Finally, we outline
future improvements to the algorithms presented, which can be achieved by
exploiting a new sampling theorem on the sphere developed recently by some of
the authors.Comment: 13 pages, 3 figures, Proceedings of Wavelets and Sparsity XV, SPIE
Optics and Photonics 2013, Code is publicly available at http://www.s2dw.org
A Bayesian approach to the study of white dwarf binaries in LISA data: The application of a reversible jump Markov chain Monte Carlo method
The Laser Interferometer Space Antenna (LISA) defines new demands on data
analysis efforts in its all-sky gravitational wave survey, recording
simultaneously thousands of galactic compact object binary foreground sources
and tens to hundreds of background sources like binary black hole mergers and
extreme mass ratio inspirals. We approach this problem with an adaptive and
fully automatic Reversible Jump Markov Chain Monte Carlo sampler, able to
sample from the joint posterior density function (as established by Bayes
theorem) for a given mixture of signals "out of the box'', handling the total
number of signals as an additional unknown parameter beside the unknown
parameters of each individual source and the noise floor. We show in examples
from the LISA Mock Data Challenge implementing the full response of LISA in its
TDI description that this sampler is able to extract monochromatic Double White
Dwarf signals out of colored instrumental noise and additional foreground and
background noise successfully in a global fitting approach. We introduce 2
examples with fixed number of signals (MCMC sampling), and 1 example with
unknown number of signals (RJ-MCMC), the latter further promoting the idea
behind an experimental adaptation of the model indicator proposal densities in
the main sampling stage. We note that the experienced runtimes and degeneracies
in parameter extraction limit the shown examples to the extraction of a low but
realistic number of signals.Comment: 18 pages, 9 figures, 3 tables, accepted for publication in PRD,
revised versio
On a variance related to the Ewens sampling formula
A one-parameter multivariate distribution, called the Ewens sampling formula, was introduced in 1972 to model the mutation phenomenon in genetics. The case discussed in this note goes back to Lynch’s theorem in the random binary search tree theory. We examine an additive statistics, being a sum of dependent random variables, and find an upper bound of its variance in terms of the sum of variances of summands. The asymptotically best constant in this estimate is established as the dimension increases. The approach is based on approximation of the extremal eigenvalues of appropriate integral operators and matrices
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