3,145 research outputs found
Minimax estimation of linear and quadratic functionals on sparsity classes
For the Gaussian sequence model, we obtain non-asymptotic minimax rates of
estimation of the linear, quadratic and the L2-norm functionals on classes of
sparse vectors and construct optimal estimators that attain these rates. The
main object of interest is the class s-sparse vectors for which we also provide
completely adaptive estimators (independent of s and of the noise variance)
having only logarithmically slower rates than the minimax ones. Furthermore, we
obtain the minimax rates on the Lq-balls where 0 < q < 2. This analysis shows
that there are, in general, three zones in the rates of convergence that we
call the sparse zone, the dense zone and the degenerate zone, while a fourth
zone appears for estimation of the quadratic functional. We show that, as
opposed to estimation of the vector, the correct logarithmic terms in the
optimal rates for the sparse zone scale as log(d/s^2) and not as log(d/s). For
the sparse class, the rates of estimation of the linear functional and of the
L2-norm have a simple elbow at s = sqrt(d) (boundary between the sparse and the
dense zones) and exhibit similar performances, whereas the estimation of the
quadratic functional reveals more complex effects and is not possible only on
the basis of sparsity described by the sparsity condition on the vector.
Finally, we apply our results on estimation of the L2-norm to the problem of
testing against sparse alternatives. In particular, we obtain a non-asymptotic
analog of the Ingster-Donoho-Jin theory revealing some effects that were not
captured by the previous asymptotic analysis.Comment: 32 page
Model selection in regression under structural constraints
The paper considers model selection in regression under the additional
structural constraints on admissible models where the number of potential
predictors might be even larger than the available sample size. We develop a
Bayesian formalism as a natural tool for generating a wide class of model
selection criteria based on penalized least squares estimation with various
complexity penalties associated with a prior on a model size. The resulting
criteria are adaptive to structural constraints. We establish the upper bound
for the quadratic risk of the resulting MAP estimator and the corresponding
lower bound for the minimax risk over a set of admissible models of a given
size. We then specify the class of priors (and, therefore, the class of
complexity penalties) where for the "nearly-orthogonal" design the MAP
estimator is asymptotically at least nearly-minimax (up to a log-factor)
simultaneously over an entire range of sparse and dense setups. Moreover, when
the numbers of admissible models are "small" (e.g., ordered variable selection)
or, on the opposite, for the case of complete variable selection, the proposed
estimator achieves the exact minimax rates.Comment: arXiv admin note: text overlap with arXiv:0912.438
Robust State Space Filtering under Incremental Model Perturbations Subject to a Relative Entropy Tolerance
This paper considers robust filtering for a nominal Gaussian state-space
model, when a relative entropy tolerance is applied to each time increment of a
dynamical model. The problem is formulated as a dynamic minimax game where the
maximizer adopts a myopic strategy. This game is shown to admit a saddle point
whose structure is characterized by applying and extending results presented
earlier in [1] for static least-squares estimation. The resulting minimax
filter takes the form of a risk-sensitive filter with a time varying risk
sensitivity parameter, which depends on the tolerance bound applied to the
model dynamics and observations at the corresponding time index. The
least-favorable model is constructed and used to evaluate the performance of
alternative filters. Simulations comparing the proposed risk-sensitive filter
to a standard Kalman filter show a significant performance advantage when
applied to the least-favorable model, and only a small performance loss for the
nominal model
Adaptive estimation of High-Dimensional Signal-to-Noise Ratios
We consider the equivalent problems of estimating the residual variance, the
proportion of explained variance and the signal strength in a
high-dimensional linear regression model with Gaussian random design. Our aim
is to understand the impact of not knowing the sparsity of the regression
parameter and not knowing the distribution of the design on minimax estimation
rates of . Depending on the sparsity of the regression parameter,
optimal estimators of either rely on estimating the regression parameter
or are based on U-type statistics, and have minimax rates depending on . In
the important situation where is unknown, we build an adaptive procedure
whose convergence rate simultaneously achieves the minimax risk over all up
to a logarithmic loss which we prove to be non avoidable. Finally, the
knowledge of the design distribution is shown to play a critical role. When the
distribution of the design is unknown, consistent estimation of explained
variance is indeed possible in much narrower regimes than for known design
distribution
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