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Introduction to Nonparametric Estimation
  • A. Tsybakov
  • Mathematics
    Springer series in statistics
  • 22 October 2008
TLDR
The main idea is to introduce the fundamental concepts of the theory while maintaining the exposition suitable for a first approach in the field, and many important and useful results on optimal and adaptive estimation are provided.
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SIMULTANEOUS ANALYSIS OF LASSO AND DANTZIG SELECTOR
We show that, under a sparsity scenario, the Lasso estimator and the Dantzig selector exhibit similar behavior. For both methods, we derive, in parallel, oracle inequalities for the prediction risk…
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Nuclear norm penalization and optimal rates for noisy low rank matrix completion
TLDR
A new nuclear norm penalized estimator of A_0 is proposed and a general sharp oracle inequality for this estimator is established for arbitrary values of $n,m_1,m-2$ under the condition of isometry in expectation to find the best trace regression model approximating the data.
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Smooth Discrimination Analysis
Discriminant analysis for two data sets in IRd with probability densities f and g can be based on the estimation of the set G = {x : f(x) I g(x)}. We consider applications where it is appropriate to…
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Fast learning rates for plug-in classifiers
TLDR
This work constructs plug-in classifiers that can achieve not only fast, but also super-fast rates, that is, rates faster than n -1 , and establishes minimax lower bounds showing that the obtained rates cannot be improved.
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Minimax theory of image reconstruction
Image processing is an increasingly important area of research and there exists a large variety of image reconstruction methods proposed by different authors. This book is concerned with a technique…
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Exponential Screening and optimal rates of sparse estimation
TLDR
This work introduces a new estimation procedure, called Exponential Screening, that shows remarkable adaptation properties and shows that it solves optimally and simultaneously all the problems of aggregation in Gaussian regression that have been discussed in the literature.
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Oracle Inequalities and Optimal Inference under Group Sparsity
TLDR
The Group Lasso can achieve an improvement in the prediction and estimation properties as compared to the Lasso, and it is proved that the rate of convergence of the upper bounds is optimal in a minimax sense.
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Optimal Rates of Aggregation
TLDR
The notion of optimal rate of aggregation is defined in an abstract context and lower bounds valid for any method of aggregation are proved, thus establishing optimal rates of linear, convex and model selection type aggregation.
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Sparsity oracle inequalities for the Lasso
TLDR
It is shown 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, in nonparametric regression setting with random design.
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