248 research outputs found
High-dimensional stochastic optimization with the generalized Dantzig estimator
We propose a generalized version of the Dantzig selector. We show that it
satisfies sparsity oracle inequalities in prediction and estimation. We
consider then the particular case of high-dimensional linear regression model
selection with the Huber loss function. In this case we derive the sup-norm
convergence rate and the sign concentration property of the Dantzig estimators
under a mutual coherence assumption on the dictionary
Asymptotics and Concentration Bounds for Bilinear Forms of Spectral Projectors of Sample Covariance
Let be i.i.d. Gaussian random variables with zero mean and
covariance operator taking values in a
separable Hilbert space Let be the effective rank of being the trace of and being its
operator norm. Let be
the sample (empirical) covariance operator based on The
paper deals with a problem of estimation of spectral projectors of the
covariance operator by their empirical counterparts, the spectral
projectors of (empirical spectral projectors). The focus is on
the problems where both the sample size and the effective rank are large. This framework includes and generalizes well known
high-dimensional spiked covariance models. Given a spectral projector
corresponding to an eigenvalue of covariance operator and its
empirical counterpart we derive sharp concentration bounds for
bilinear forms of empirical spectral projector in terms of sample
size and effective dimension Building upon these
concentration bounds, we prove the asymptotic normality of bilinear forms of
random operators under the assumptions that
and In a special case of eigenvalues of
multiplicity one, these results are rephrased as concentration bounds and
asymptotic normality for linear forms of empirical eigenvectors. Other results
include bounds on the bias and a method of bias
reduction as well as a discussion of possible applications to statistical
inference in high-dimensional principal component analysis
Pac-bayesian bounds for sparse regression estimation with exponential weights
We consider the sparse regression model where the number of parameters is
larger than the sample size . The difficulty when considering
high-dimensional problems is to propose estimators achieving a good compromise
between statistical and computational performances. The BIC estimator for
instance performs well from the statistical point of view \cite{BTW07} but can
only be computed for values of of at most a few tens. The Lasso estimator
is solution of a convex minimization problem, hence computable for large value
of . However stringent conditions on the design are required to establish
fast rates of convergence for this estimator. Dalalyan and Tsybakov
\cite{arnak} propose a method achieving a good compromise between the
statistical and computational aspects of the problem. Their estimator can be
computed for reasonably large and satisfies nice statistical properties
under weak assumptions on the design. However, \cite{arnak} proposes sparsity
oracle inequalities in expectation for the empirical excess risk only. In this
paper, we propose an aggregation procedure similar to that of \cite{arnak} but
with improved statistical performances. Our main theoretical result is a
sparsity oracle inequality in probability for the true excess risk for a
version of exponential weight estimator. We also propose a MCMC method to
compute our estimator for reasonably large values of .Comment: 19 page
- …