123 research outputs found
Mixture model for designs in high dimensional regression and the LASSO
The LASSO is a recent technique for variable selection in the regression
model \bean y & = & X\beta +\epsilon, \eean where and
is a centered gaussian i.i.d. noise vector . The LASSO has been proved to perform exact support recovery
for regression vectors when the design matrix satisfies certain algebraic
conditions and is sufficiently sparse. Estimation of the vector
has also extensively been studied for the purpose of prediction under
the same algebraic conditions on and under sufficient sparsity of .
Among many other, the coherence is an index which can be used to study these
nice properties of the LASSO. More precisely, a small coherence implies that
most sparse vectors, with less nonzero components than the order ,
can be recovered with high probability if its nonzero components are larger
than the order . However, many matrices occuring in
practice do not have a small coherence and thus, most results which have
appeared in the litterature cannot be applied. The goal of this paper is to
study a model for which precise results can be obtained. In the proposed model,
the columns of the design matrix are drawn from a Gaussian mixture model and
the coherence condition is imposed on the much smaller matrix whose columns are
the mixture's centers, instead of on itself. Our main theorem states that
is as well estimated as in the case of small coherence up to a
correction parametrized by the maximal variance in the mixture model.Comment: Draft. Simulations to be included soo
Multivariate GARCH estimation via a Bregman-proximal trust-region method
The estimation of multivariate GARCH time series models is a difficult task
mainly due to the significant overparameterization exhibited by the problem and
usually referred to as the "curse of dimensionality". For example, in the case
of the VEC family, the number of parameters involved in the model grows as a
polynomial of order four on the dimensionality of the problem. Moreover, these
parameters are subjected to convoluted nonlinear constraints necessary to
ensure, for instance, the existence of stationary solutions and the positive
semidefinite character of the conditional covariance matrices used in the model
design. So far, this problem has been addressed in the literature only in low
dimensional cases with strong parsimony constraints. In this paper we propose a
general formulation of the estimation problem in any dimension and develop a
Bregman-proximal trust-region method for its solution. The Bregman-proximal
approach allows us to handle the constraints in a very efficient and natural
way by staying in the primal space and the Trust-Region mechanism stabilizes
and speeds up the scheme. Preliminary computational experiments are presented
and confirm the very good performances of the proposed approach.Comment: 35 pages, 5 figure
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