61 research outputs found
Honest variable selection in linear and logistic regression models via and penalization
This paper investigates correct variable selection in finite samples via
and type penalization schemes. The asymptotic
consistency of variable selection immediately follows from this analysis. We
focus on logistic and linear regression models. The following questions are
central to our paper: given a level of confidence , under which
assumptions on the design matrix, for which strength of the signal and for what
values of the tuning parameters can we identify the true model at the given
level of confidence? Formally, if is an estimate of the true
variable set , we study conditions under which
, for a given sample size , number
of parameters and confidence . We show that in identifiable
models, both methods can recover coefficients of size , up
to small multiplicative constants and logarithmic factors in and
. The advantage of the penalization over the
is minor for the variable selection problem, for the models we
consider here. Whereas the former estimates are unique, and become more stable
for highly correlated data matrices as one increases the tuning parameter of
the part, too large an increase in this parameter value may preclude
variable selection.Comment: Published in at http://dx.doi.org/10.1214/08-EJS287 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Consistent selection via the Lasso for high dimensional approximating regression models
In this article we investigate consistency of selection in regression models
via the popular Lasso method. Here we depart from the traditional linear
regression assumption and consider approximations of the regression function
with elements of a given dictionary of functions. The target for
consistency is the index set of those functions from this dictionary that
realize the most parsimonious approximation to among all linear
combinations belonging to an ball centered at and of radius
. In this framework we show that a consistent estimate of this index
set can be derived via penalized least squares, with a data dependent
penalty and with tuning sequence , where is the
sample size. Our results hold for any , for any
.Comment: Published in at http://dx.doi.org/10.1214/074921708000000101 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
Sparsity oracle inequalities for the Lasso
This paper studies oracle properties of -penalized least squares in
nonparametric regression setting with random design. We show that the penalized
least squares estimator satisfies sparsity oracle inequalities, i.e., bounds in
terms of the number of non-zero components of the oracle vector. The results
are valid even when the dimension of the model is (much) larger than the sample
size and the regression matrix is not positive definite. They can be applied to
high-dimensional linear regression, to nonparametric adaptive regression
estimation and to the problem of aggregation of arbitrary estimators.Comment: Published at http://dx.doi.org/10.1214/07-EJS008 in the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Convex Banding of the Covariance Matrix
We introduce a new sparse estimator of the covariance matrix for
high-dimensional models in which the variables have a known ordering. Our
estimator, which is the solution to a convex optimization problem, is
equivalently expressed as an estimator which tapers the sample covariance
matrix by a Toeplitz, sparsely-banded, data-adaptive matrix. As a result of
this adaptivity, the convex banding estimator enjoys theoretical optimality
properties not attained by previous banding or tapered estimators. In
particular, our convex banding estimator is minimax rate adaptive in Frobenius
and operator norms, up to log factors, over commonly-studied classes of
covariance matrices, and over more general classes. Furthermore, it correctly
recovers the bandwidth when the true covariance is exactly banded. Our convex
formulation admits a simple and efficient algorithm. Empirical studies
demonstrate its practical effectiveness and illustrate that our exactly-banded
estimator works well even when the true covariance matrix is only close to a
banded matrix, confirming our theoretical results. Our method compares
favorably with all existing methods, in terms of accuracy and speed. We
illustrate the practical merits of the convex banding estimator by showing that
it can be used to improve the performance of discriminant analysis for
classifying sound recordings
Optimal selection of reduced rank estimators of high-dimensional matrices
We introduce a new criterion, the Rank Selection Criterion (RSC), for
selecting the optimal reduced rank estimator of the coefficient matrix in
multivariate response regression models. The corresponding RSC estimator
minimizes the Frobenius norm of the fit plus a regularization term proportional
to the number of parameters in the reduced rank model. The rank of the RSC
estimator provides a consistent estimator of the rank of the coefficient
matrix; in general, the rank of our estimator is a consistent estimate of the
effective rank, which we define to be the number of singular values of the
target matrix that are appropriately large. The consistency results are valid
not only in the classic asymptotic regime, when , the number of responses,
and , the number of predictors, stay bounded, and , the number of
observations, grows, but also when either, or both, and grow, possibly
much faster than . We establish minimax optimal bounds on the mean squared
errors of our estimators. Our finite sample performance bounds for the RSC
estimator show that it achieves the optimal balance between the approximation
error and the penalty term. Furthermore, our procedure has very low
computational complexity, linear in the number of candidate models, making it
particularly appealing for large scale problems. We contrast our estimator with
the nuclear norm penalized least squares (NNP) estimator, which has an
inherently higher computational complexity than RSC, for multivariate
regression models. We show that NNP has estimation properties similar to those
of RSC, albeit under stronger conditions. However, it is not as parsimonious as
RSC. We offer a simple correction of the NNP estimator which leads to
consistent rank estimation.Comment: Published in at http://dx.doi.org/10.1214/11-AOS876 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org) (some typos corrected
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