954 research outputs found
Adaptivity to Noise Parameters in Nonparametric Active Learning
This work addresses various open questions in the theory of active learning
for nonparametric classification. Our contributions are both statistical and
algorithmic: -We establish new minimax-rates for active learning under common
\textit{noise conditions}. These rates display interesting transitions -- due
to the interaction between noise \textit{smoothness and margin} -- not present
in the passive setting. Some such transitions were previously conjectured, but
remained unconfirmed. -We present a generic algorithmic strategy for adaptivity
to unknown noise smoothness and margin; our strategy achieves optimal rates in
many general situations; furthermore, unlike in previous work, we avoid the
need for \textit{adaptive confidence sets}, resulting in strictly milder
distributional requirements
Optimal Calibration for Multiple Testing against Local Inhomogeneity in Higher Dimension
Based on two independent samples X_1,...,X_m and X_{m+1},...,X_n drawn from
multivariate distributions with unknown Lebesgue densities p and q
respectively, we propose an exact multiple test in order to identify
simultaneously regions of significant deviations between p and q. The
construction is built from randomized nearest-neighbor statistics. It does not
require any preliminary information about the multivariate densities such as
compact support, strict positivity or smoothness and shape properties. The
properly adjusted multiple testing procedure is shown to be sharp-optimal for
typical arrangements of the observation values which appear with probability
close to one. The proof relies on a new coupling Bernstein type exponential
inequality, reflecting the non-subgaussian tail behavior of a combinatorial
process. For power investigation of the proposed method a reparametrized
minimax set-up is introduced, reducing the composite hypothesis "p=q" to a
simple one with the multivariate mixed density (m/n)p+(1-m/n)q as infinite
dimensional nuisance parameter. Within this framework, the test is shown to be
spatially and sharply asymptotically adaptive with respect to uniform loss on
isotropic H\"older classes. The exact minimax risk asymptotics are obtained in
terms of solutions of the optimal recovery
Bandwidth selection in kernel empirical risk minimization via the gradient
In this paper, we deal with the data-driven selection of multidimensional and
possibly anisotropic bandwidths in the general framework of kernel empirical
risk minimization. We propose a universal selection rule, which leads to
optimal adaptive results in a large variety of statistical models such as
nonparametric robust regression and statistical learning with errors in
variables. These results are stated in the context of smooth loss functions,
where the gradient of the risk appears as a good criterion to measure the
performance of our estimators. The selection rule consists of a comparison of
gradient empirical risks. It can be viewed as a nontrivial improvement of the
so-called Goldenshluger-Lepski method to nonlinear estimators. Furthermore, one
main advantage of our selection rule is the nondependency on the Hessian matrix
of the risk, usually involved in standard adaptive procedures.Comment: Published at http://dx.doi.org/10.1214/15-AOS1318 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Global and Local Two-Sample Tests via Regression
Two-sample testing is a fundamental problem in statistics. Despite its long
history, there has been renewed interest in this problem with the advent of
high-dimensional and complex data. Specifically, in the machine learning
literature, there have been recent methodological developments such as
classification accuracy tests. The goal of this work is to present a regression
approach to comparing multivariate distributions of complex data. Depending on
the chosen regression model, our framework can efficiently handle different
types of variables and various structures in the data, with competitive power
under many practical scenarios. Whereas previous work has been largely limited
to global tests which conceal much of the local information, our approach
naturally leads to a local two-sample testing framework in which we identify
local differences between multivariate distributions with statistical
confidence. We demonstrate the efficacy of our approach both theoretically and
empirically, under some well-known parametric and nonparametric regression
methods. Our proposed methods are applied to simulated data as well as a
challenging astronomy data set to assess their practical usefulness
Optimal cross-validation in density estimation with the -loss
We analyze the performance of cross-validation (CV) in the density estimation
framework with two purposes: (i) risk estimation and (ii) model selection. The
main focus is given to the so-called leave--out CV procedure (Lpo), where
denotes the cardinality of the test set. Closed-form expressions are
settled for the Lpo estimator of the risk of projection estimators. These
expressions provide a great improvement upon -fold cross-validation in terms
of variability and computational complexity. From a theoretical point of view,
closed-form expressions also enable to study the Lpo performance in terms of
risk estimation. The optimality of leave-one-out (Loo), that is Lpo with ,
is proved among CV procedures used for risk estimation. Two model selection
frameworks are also considered: estimation, as opposed to identification. For
estimation with finite sample size , optimality is achieved for large
enough [with ] to balance the overfitting resulting from the
structure of the model collection. For identification, model selection
consistency is settled for Lpo as long as is conveniently related to the
rate of convergence of the best estimator in the collection: (i) as
with a parametric rate, and (ii) with some
nonparametric estimators. These theoretical results are validated by simulation
experiments.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1240 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Locally adaptive image denoising by a statistical multiresolution criterion
We demonstrate how one can choose the smoothing parameter in image denoising
by a statistical multiresolution criterion, both globally and locally. Using
inhomogeneous diffusion and total variation regularization as examples for
localized regularization schemes, we present an efficient method for locally
adaptive image denoising. As expected, the smoothing parameter serves as an
edge detector in this framework. Numerical examples illustrate the usefulness
of our approach. We also present an application in confocal microscopy
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