39,876 research outputs found
Convergence Rates of Subseries
Let be a positive real sequence decreasing to such that the
series is divergent and . We show
that there exists a constant such that, for each ,
there is a subsequence for which and
.Comment: 5 pp. To appear in The American Mathematical Monthl
Convergence and Rates for Fixed-Interval Multiple-Track Smoothing Using -Means Type Optimization
We address the task of estimating multiple trajectories from unlabeled data.
This problem arises in many settings, one could think of the construction of
maps of transport networks from passive observation of travellers, or the
reconstruction of the behaviour of uncooperative vehicles from external
observations, for example. There are two coupled problems. The first is a data
association problem: how to map data points onto individual trajectories. The
second is, given a solution to the data association problem, to estimate those
trajectories. We construct estimators as a solution to a regularized
variational problem (to which approximate solutions can be obtained via the
simple, efficient and widespread -means method) and show that, as the number
of data points, , increases, these estimators exhibit stable behaviour. More
precisely, we show that they converge in an appropriate Sobolev space in
probability and with rate
Convergence Rates of Gaussian ODE Filters
A recently-introduced class of probabilistic (uncertainty-aware) solvers for
ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to
initial value problems. These methods model the true solution and its first
derivatives \emph{a priori} as a Gauss--Markov process ,
which is then iteratively conditioned on information about . This
article establishes worst-case local convergence rates of order for a
wide range of versions of this Gaussian ODE filter, as well as global
convergence rates of order in the case of and an integrated Brownian
motion prior, and analyses how inaccurate information on coming from
approximate evaluations of affects these rates. Moreover, we show that, in
the globally convergent case, the posterior credible intervals are well
calibrated in the sense that they globally contract at the same rate as the
truncation error. We illustrate these theoretical results by numerical
experiments which might indicate their generalizability to .Comment: 26 pages, 5 figure
On convergence rates equivalency and sampling strategies in functional deconvolution models
Using the asymptotical minimax framework, we examine convergence rates
equivalency between a continuous functional deconvolution model and its
real-life discrete counterpart over a wide range of Besov balls and for the
-risk. For this purpose, all possible models are divided into three
groups. For the models in the first group, which we call uniform, the
convergence rates in the discrete and the continuous models coincide no matter
what the sampling scheme is chosen, and hence the replacement of the discrete
model by its continuous counterpart is legitimate. For the models in the second
group, to which we refer as regular, one can point out the best sampling
strategy in the discrete model, but not every sampling scheme leads to the same
convergence rates; there are at least two sampling schemes which deliver
different convergence rates in the discrete model (i.e., at least one of the
discrete models leads to convergence rates that are different from the
convergence rates in the continuous model). The third group consists of models
for which, in general, it is impossible to devise the best sampling strategy;
we call these models irregular. We formulate the conditions when each of these
situations takes place. In the regular case, we not only point out the number
and the selection of sampling points which deliver the fastest convergence
rates in the discrete model but also investigate when, in the case of an
arbitrary sampling scheme, the convergence rates in the continuous model
coincide or do not coincide with the convergence rates in the discrete model.
We also study what happens if one chooses a uniform, or a more general
pseudo-uniform, sampling scheme which can be viewed as an intuitive replacement
of the continuous model.Comment: Published in at http://dx.doi.org/10.1214/09-AOS767 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
- …