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In signal processing, bandpass sampling is an effective technique for sampling signals that have most of their energy concentrated within a narrow frequency band. This type of signal is known as a bandpass signal. The key principle of bandpass sampling involves sampling the signal at a rate that is greater than twice the signal's bandwidth to prevent aliasing.
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ANISOTROPIC FUNCTION ESTIMATION USING MULTI-BANDWIDTH GAUSSIAN PROCESSES.

Anirban Bhattacharya1, Debdeep Pati2, David Dunson1

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Summary
This summary is machine-generated.

This study introduces a Bayesian method for identifying important predictors in complex regression. The approach achieves optimal estimation rates, adapting to unknown dimensions and surface smoothness.

Keywords:
AdaptiveAnisotropicBayesian nonparametricsFunction estimationGaussian processRate of convergence

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Area of Science:

  • Statistics
  • Machine Learning
  • Bayesian Inference

Background:

  • Nonparametric regression with multiple predictors often involves identifying relevant variables.
  • Estimating anisotropic multivariate regression surfaces requires adapting to varying predictor importance and smoothness.

Purpose of the Study:

  • To develop a Bayesian procedure for estimating anisotropic multivariate regression surfaces.
  • To achieve minimax optimal rates of posterior contraction, adapting to unknown dimensions and anisotropic smoothness.
  • To compare the proposed method with a homogeneous Gaussian process approach.

Main Methods:

  • Utilizing a Gaussian process prior with dimension-specific scalings.
  • Assigning carefully-chosen hyperpriors to these scalings.
  • Analyzing the theoretical properties of the Bayesian procedure for posterior contraction rates.

Main Results:

  • The proposed Bayesian approach achieves minimax optimal rates of posterior contraction (up to a log factor).
  • The method successfully adapts to the unknown effective dimension of the regression surface.
  • A homogeneous Gaussian process with a single bandwidth demonstrates sub-optimal rates in anisotropic settings.

Conclusions:

  • The developed Bayesian procedure offers an adaptive and efficient method for anisotropic nonparametric regression.
  • Dimension-specific scalings in Gaussian process priors are crucial for optimal performance in anisotropic scenarios.
  • The findings highlight the limitations of homogeneous Gaussian processes for complex, anisotropic regression problems.