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Learning on manifolds without manifold learning.

H N Mhaskar1, Ryan O'Dowd1

  • 1Institute of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711, United States of America.

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|October 20, 2024
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Summary
This summary is machine-generated.

This study introduces a novel one-shot function approximation method for machine learning. It bypasses manifold estimation, offering optimal approximation rates for rough functions on hyperspheres.

Keywords:
Approximation on unknown manifoldsDirect approximation with error boundsLocalized polynomial kernels on the sphere

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Area of Science:

  • Machine Learning
  • Data Science
  • Applied Mathematics

Background:

  • Function approximation from random data is a core machine learning challenge.
  • The manifold hypothesis posits data lies on a lower-dimensional submanifold.
  • Existing methods often involve multi-step processes with inherent approximation errors.

Purpose of the Study:

  • To develop a one-shot function approximation technique.
  • To avoid the errors associated with estimating manifold properties.
  • To approximate functions on data residing on a hypersphere.

Main Methods:

  • Projecting the unknown data manifold onto an ambient hypersphere.
  • Utilizing a sequence of localized spherical polynomial kernels.
  • Developing a one-shot approximation strategy without manifold preprocessing.

Main Results:

  • Achieved optimal approximation rates for relatively "rough" functions.
  • Demonstrated a method that does not require manifold structure estimation beyond its dimension.
  • Introduced a more direct approach to function approximation on manifolds.

Conclusions:

  • The proposed method offers an efficient alternative to traditional two-step approaches.
  • This technique reduces approximation errors by eliminating intermediate estimation steps.
  • The localized spherical polynomial kernels provide a powerful tool for manifold learning and function approximation.