785 research outputs found

### Kullback-Leibler aggregation and misspecified generalized linear models

In a regression setup with deterministic design, we study the pure
aggregation problem and introduce a natural extension from the Gaussian
distribution to distributions in the exponential family. While this extension
bears strong connections with generalized linear models, it does not require
identifiability of the parameter or even that the model on the systematic
component is true. It is shown that this problem can be solved by constrained
and/or penalized likelihood maximization and we derive sharp oracle
inequalities that hold both in expectation and with high probability. Finally
all the bounds are proved to be optimal in a minimax sense.Comment: Published in at http://dx.doi.org/10.1214/11-AOS961 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Linear and convex aggregation of density estimators

We study the problem of linear and convex aggregation of $M$ estimators of a
density with respect to the mean squared risk. We provide procedures for linear
and convex aggregation and we prove oracle inequalities for their risks. We
also obtain lower bounds showing that these procedures are rate optimal in a
minimax sense. As an example, we apply general results to aggregation of
multivariate kernel density estimators with different bandwidths. We show that
linear and convex aggregates mimic the kernel oracles in asymptotically exact
sense for a large class of kernels including Gaussian, Silverman's and
Pinsker's ones. We prove that, for Pinsker's kernel, the proposed aggregates
are sharp asymptotically minimax simultaneously over a large scale of Sobolev
classes of densities. Finally, we provide simulations demonstrating performance
of the convex aggregation procedure.Comment: 22 page

### Entropic optimal transport is maximum-likelihood deconvolution

We give a statistical interpretation of entropic optimal transport by showing
that performing maximum-likelihood estimation for Gaussian deconvolution
corresponds to calculating a projection with respect to the entropic optimal
transport distance. This structural result gives theoretical support for the
wide adoption of these tools in the machine learning community

### Optimal learning with $Q$-aggregation

We consider a general supervised learning problem with strongly convex and
Lipschitz loss and study the problem of model selection aggregation. In
particular, given a finite dictionary functions (learners) together with the
prior, we generalize the results obtained by Dai, Rigollet and Zhang [Ann.
Statist. 40 (2012) 1878-1905] for Gaussian regression with squared loss and
fixed design to this learning setup. Specifically, we prove that the
$Q$-aggregation procedure outputs an estimator that satisfies optimal oracle
inequalities both in expectation and with high probability. Our proof
techniques somewhat depart from traditional proofs by making most of the
standard arguments on the Laplace transform of the empirical process to be
controlled.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1190 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Optimal rates for plug-in estimators of density level sets

In the context of density level set estimation, we study the convergence of
general plug-in methods under two main assumptions on the density for a given
level $\lambda$. More precisely, it is assumed that the density (i) is smooth
in a neighborhood of $\lambda$ and (ii) has $\gamma$-exponent at level
$\lambda$. Condition (i) ensures that the density can be estimated at a
standard nonparametric rate and condition (ii) is similar to Tsybakov's margin
assumption which is stated for the classification framework. Under these
assumptions, we derive optimal rates of convergence for plug-in estimators.
Explicit convergence rates are given for plug-in estimators based on kernel
density estimators when the underlying measure is the Lebesgue measure. Lower
bounds proving optimality of the rates in a minimax sense when the density is
H\"older smooth are also provided.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ184 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### Optimal detection of sparse principal components in high dimension

We perform a finite sample analysis of the detection levels for sparse
principal components of a high-dimensional covariance matrix. Our minimax
optimal test is based on a sparse eigenvalue statistic. Alas, computing this
test is known to be NP-complete in general, and we describe a computationally
efficient alternative test using convex relaxations. Our relaxation is also
proved to detect sparse principal components at near optimal detection levels,
and it performs well on simulated datasets. Moreover, using polynomial time
reductions from theoretical computer science, we bring significant evidence
that our results cannot be improved, thus revealing an inherent trade off
between statistical and computational performance.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1127 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Uncoupled isotonic regression via minimum Wasserstein deconvolution

Isotonic regression is a standard problem in shape-constrained estimation
where the goal is to estimate an unknown nondecreasing regression function $f$
from independent pairs $(x_i, y_i)$ where $\mathbb{E}[y_i]=f(x_i), i=1, \ldots
n$. While this problem is well understood both statistically and
computationally, much less is known about its uncoupled counterpart where one
is given only the unordered sets $\{x_1, \ldots, x_n\}$ and $\{y_1, \ldots,
y_n\}$. In this work, we leverage tools from optimal transport theory to derive
minimax rates under weak moments conditions on $y_i$ and to give an efficient
algorithm achieving optimal rates. Both upper and lower bounds employ
moment-matching arguments that are also pertinent to learning mixtures of
distributions and deconvolution.Comment: To appear in Information and Inference: a Journal of the IM

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