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Adaptive exact recovery in sparse nonparametric models.

Natalia Stepanova1, Marie Turcicova2

  • 1School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, K1S 5B6 Ottawa, ON Canada.

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Summary
This summary is machine-generated.

This study identifies nonzero components of unknown functions in high-dimensional models. A novel selection procedure achieves exact variable selection, adapting to model sparsity.

Keywords:
Exact selectionFunctional ANOVA modelGaussian white noiseHamming riskSharp selection boundarySparsity

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Area of Science:

  • Statistics
  • Machine Learning
  • High-Dimensional Data Analysis

Background:

  • Observing an unknown function f(t) of d variables in a Gaussian white noise model.
  • Assuming f(t) is a sum of k-variate functions (1 <= k <= s), with only a few being nonzero.
  • Addressing the challenge in high-dimensional settings where d -> infinity and s can also grow.

Purpose of the Study:

  • To identify the nonzero components of the unknown function f(t).
  • To develop a variable selection procedure for high-dimensional models with growing complexity.
  • To determine conditions for successful and impossible exact variable selection.

Main Methods:

  • Utilizing a Gaussian white noise model with intensity epsilon > 0.
  • Developing a variable selection procedure adaptive to model sparsity (parameter beta).
  • Deriving theoretical conditions for exact variable selection.

Main Results:

  • Established conditions under which exact variable selection is possible.
  • Proposed an adaptive selection procedure that achieves exact variable selection.
  • Identified conditions that preclude exact variable selection.

Conclusions:

  • The developed procedure enables exact variable selection in high-dimensional, sparse settings.
  • The findings provide a theoretical framework for understanding variable selection limitations.
  • This work advances the field of statistical inference for complex models.