46,758 research outputs found
Fast learning rates for plug-in classifiers
It has been recently shown that, under the margin (or low noise) assumption,
there exist classifiers attaining fast rates of convergence of the excess Bayes
risk, that is, rates faster than . The work on this subject has
suggested the following two conjectures: (i) the best achievable fast rate is
of the order , and (ii) the plug-in classifiers generally converge more
slowly than the classifiers based on empirical risk minimization. We show that
both conjectures are not correct. In particular, we construct plug-in
classifiers that can achieve not only fast, but also super-fast rates, that is,
rates faster than . We establish minimax lower bounds showing that the
obtained rates cannot be improved.Comment: Published at http://dx.doi.org/10.1214/009053606000001217 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Fast learning rates for plug-in classifiers under the margin condition
It has been recently shown that, under the margin (or low noise) assumption,
there exist classifiers attaining fast rates of convergence of the excess Bayes
risk, i.e., the rates faster than . The works on this subject
suggested the following two conjectures: (i) the best achievable fast rate is
of the order , and (ii) the plug-in classifiers generally converge
slower than the classifiers based on empirical risk minimization. We show that
both conjectures are not correct. In particular, we construct plug-in
classifiers that can achieve not only the fast, but also the {\it super-fast}
rates, i.e., the rates faster than . We establish minimax lower bounds
showing that the obtained rates cannot be improved.Comment: 36 page
Relative complexity of random walks in random sceneries
Relative complexity measures the complexity of a probability preserving
transformation relative to a factor being a sequence of random variables whose
exponential growth rate is the relative entropy of the extension. We prove
distributional limit theorems for the relative complexity of certain zero
entropy extensions: RWRSs whose associated random walks satisfy the
\alpha-stable CLT (). The results give invariants for relative
isomorphism of these.Comment: Published in at http://dx.doi.org/10.1214/11-AOP688 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A low complexity algorithm for non-monotonically evolving fronts
A new algorithm is proposed to describe the propagation of fronts advected in
the normal direction with prescribed speed function F. The assumptions on F are
that it does not depend on the front itself, but can depend on space and time.
Moreover, it can vanish and change sign. To solve this problem the Level-Set
Method [Osher, Sethian; 1988] is widely used, and the Generalized Fast Marching
Method [Carlini et al.; 2008] has recently been introduced. The novelty of our
method is that its overall computational complexity is predicted to be
comparable to that of the Fast Marching Method [Sethian; 1996], [Vladimirsky;
2006] in most instances. This latter algorithm is O(N^n log N^n) if the
computational domain comprises N^n points. Our strategy is to use it in regions
where the speed is bounded away from zero -- and switch to a different
formalism when F is approximately 0. To this end, a collection of so-called
sideways partial differential equations is introduced. Their solutions locally
describe the evolving front and depend on both space and time. The
well-posedness of those equations, as well as their geometric properties are
addressed. We then propose a convergent and stable discretization of those
PDEs. Those alternative representations are used to augment the standard Fast
Marching Method. The resulting algorithm is presented together with a thorough
discussion of its features. The accuracy of the scheme is tested when F depends
on both space and time. Each example yields an O(1/N) global truncation error.
We conclude with a discussion of the advantages and limitations of our method.Comment: 30 pages, 12 figures, 1 tabl
Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators
In this paper, we are interested in the strong convergence properties of the
Ninomiya-Victoir scheme which is known to exhibit weak convergence with order
2. We prove strong convergence with order . This study is aimed at
analysing the use of this scheme either at each level or only at the finest
level of a multilevel Monte Carlo estimator: indeed, the variance of a
multilevel Monte Carlo estimator is related to the strong error between the two
schemes used on the coarse and fine grids at each level. Recently, Giles and
Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo
estimator achieving the optimal complexity for
the precision . In the same spirit, we propose a modified
Ninomiya-Victoir scheme, which may be strongly coupled with order to the
Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo estimator.
Numerical experiments show that this choice improves the efficiency, since the
order of weak convergence of the Ninomiya-Victoir scheme permits to reduce
the number of discretization levels
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