arXiv subjects

Nurzhan Nurushev

Publications and source records attributed to Nurzhan Nurushev.

3 recordsLinked to original sources

arXiv2021

Uncertainty quantification for robust variable selection and multiple testing

We study the problem of identifying the set of \emph{active} variables, termed in the literature as \emph{variable selection} or \emph{multiple hypothesis testing}, depending on the pursued criteria. For a general \emph{robust setting} of non-normal, possibly dependent observations and a generalized notion of \emph{active set}, we propose a procedure that is used simultaneously for the both tasks, variable selection and multiple testing. The procedure is based on the \emph{risk hull minimization} method, but can also be obtained as a result of an empirical Bayes approach or a penalization strategy. We address its quality via various criteria: the Hamming risk, FDR, FPR, FWER, NDR, FNR,and various \emph{multiple testing risks}, e.g., MTR=FDR+NDR; and discuss a weak optimality of our results. Finally, we introduce and study, for the first time, the \emph{uncertainty quantification problem} in the variable selection and multiple testing context in our robust setting.

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arXiv2019

General framework for projection structures

In the first part, we develop a general framework for projection structures and study several inference problems within this framework. We propose procedures based on data dependent measures (DDM) and make connections with empirical Bayes and penalization methods. The main inference problem is the uncertainty quantification (UQ), but on the way we solve the estimation, DDM-contraction problems, and a weak version of the structure recovery problem. The approach is local in that the quality of the inference procedures is measured by the local quantity, the oracle rate, which is the best trade-off between the approximation error by a projection structure and the complexity of that approximating projection structure. Like in statistical learning settings, we develop distribution-free theory as no particular model is imposed, we only assume certain mild condition on the stochastic part of the projection predictor. We introduce the excessive bias restriction (EBR) under which we establish the local confidence optimality of the constructed confidence ball. The proposed general framework unifies a very broad class of high-dimensional models and structures, interesting and important on their own right. In the second part, we apply the developed theory and demonstrate how the general results deliver a whole avenue of local and global minimax results (many new ones, some known results from the literature are improved) for particular models and structures as consequences, including white noise model and density estimation with smoothness structure, linear regression and dictionary learning with sparsity structures, biclustering and stochastic block models with clustering structure, covariance matrix estimation with banding and sparsity structures, and many others. Various adaptive minimax results over various scales follow also from our local results.

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arXiv2018

Needles and straw in a haystack: robust confidence for possibly sparse sequences

In the general signal+noise model we construct an empirical Bayes posterior which we then use for uncertainty quantification for the unknown, possibly sparse, signal. We introduce a novel excessive bias restriction (EBR) condition, which gives rise to a new slicing of the entire space that is suitable for uncertainty quantification. Under EBR and some mild conditions on the noise, we establish the local (oracle) optimality of the proposed confidence ball. In passing, we also get the local optimal (oracle) results for estimation and posterior contraction problems. Adaptive minimax results (also for the estimation and posterior contraction problems) over various sparsity classes follow from our local results.

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