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NONLINEAR GLOBAL FRÉCHET REGRESSION FOR RANDOM OBJECTS VIA WEAK CONDITIONAL EXPECTATION.

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  • 1Department of Statistics, University of Florida.

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|September 5, 2025
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Summary
This summary is machine-generated.

This study introduces a new nonlinear Fréchet regression model for complex object-valued data. The method extends existing techniques, offering a robust framework for analyzing diverse non-Euclidean datasets.

Keywords:
Primary 62G05, 62J02Random objectmetric spacesobject-on-object regressionreproducing kernel Hilbert spacessecondary 62G08, 62J99weak conditional expectation

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Area of Science:

  • Statistics
  • Machine Learning
  • Data Science

Background:

  • Object-valued data from metric spaces are increasingly common.
  • Existing regression models struggle with complex, non-Euclidean predictor and response variables.
  • A general framework for object-valued regression is lacking.

Purpose of the Study:

  • To develop a general nonlinear regression framework for object-valued data.
  • To introduce a weak conditional Fréchet mean using Carleman operators.
  • To extend regression analysis to complex, non-Euclidean predictor and response spaces.

Main Methods:

  • Utilizing reproducing kernel Hilbert space (RKHS) embedding for nonlinear modeling.
  • Defining a weak conditional Fréchet mean via Carleman operators.
  • Establishing relationships between conditional and weak conditional Fréchet means.

Main Results:

  • A novel global nonlinear Fréchet regression model is proposed.
  • The new model encompasses existing methods like linear kernel Fréchet regression.
  • Theoretical properties of the estimates are analyzed using the intrinsic geometry of metric spaces.

Conclusions:

  • The proposed method provides a powerful tool for analyzing complex object-valued data.
  • The framework is versatile, applicable to various non-Euclidean data types.
  • Numerical studies confirm the method's effectiveness for real-world applications.