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Extrinsic local regression on manifold-valued data.

Lizhen Lin1, Brian St Thomas2, Hongtu Zhu3

  • 1Department of Statistics and Data Sciences, The University of Texas at Austin, Austin, TX.

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Summary
This summary is machine-generated.

We introduce an extrinsic regression framework for manifold-valued data. This novel approach offers a computationally efficient and theoretically sound method for analyzing complex data in fields like shape analysis and medical imaging.

Keywords:
Convergence rateDifferentiable manifoldGeometryLocal regressionObject dataShape statistics

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Area of Science:

  • Statistics
  • Geometric Data Analysis

Background:

  • Regression analysis is crucial for understanding relationships between variables.
  • Modeling manifold-valued data, common in shape analysis and neuroscience, presents unique challenges.
  • Existing methods for manifold data often use intrinsic approaches, limiting their applicability in regression.

Purpose of the Study:

  • To develop a novel extrinsic regression framework for data with manifold-valued responses and Euclidean predictors.
  • To provide a computationally efficient and theoretically robust method for analyzing complex geometric data.
  • To extend the application of extrinsic methods to the regression setting for manifold-valued data.

Main Methods:

  • Embedding the response manifold into a higher-dimensional Euclidean space.
  • Obtaining a local regression estimate within this Euclidean space.
  • Projecting the estimated regression back onto the manifold.

Main Results:

  • The proposed extrinsic regression framework is general and computationally efficient.
  • Asymptotic distributions and convergence rates for the extrinsic regression estimates are derived.
  • The framework demonstrates wide applicability across various examples.

Conclusions:

  • The extrinsic regression framework offers a powerful new tool for analyzing manifold-valued data.
  • This approach overcomes limitations of existing intrinsic methods in regression contexts.
  • The derived theoretical properties and demonstrated applications highlight the framework's potential impact.