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An algorithm for computing Schubert varieties of best fit with applications.

Karim Karimov1, Michael Kirby1, Chris Peterson1

  • 1Department of Mathematics, College of Natural Sciences, Colorado State University, Fort Collins, CO, United States.

Frontiers in Artificial Intelligence
|December 11, 2023
PubMed
Summary

This study introduces the Schubert variety as a geometric tool for representing multiple subspaces. It develops a method to find a representative matrix that best fits these subspaces, applicable to machine learning.

Keywords:
GPU parallel computingSchubert variety of best fitabstract nodegeometry of learningmanifold approximationneural networksubspace classification

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Area of Science:

  • Algebraic Geometry
  • Linear Algebra
  • Machine Learning

Background:

  • Representing collections of subspaces is crucial in various mathematical and computational fields.
  • Existing methods may lack a unified geometric framework for subspace representation.
  • The Grassmannian manifold provides a space for studying subspaces, but direct representation can be challenging.

Purpose of the Study:

  • To introduce the geometric framework of the Schubert variety for representing collections of subspaces.
  • To develop a method for finding a representative matrix (K) that approximates a given set of subspaces.
  • To integrate this subspace representation into artificial neural network architectures.

Main Methods:

  • Formulation of a non-convex optimization problem to find a representative matrix K.
  • Utilizing linear combinations of column spaces to define relationships between subspaces.
  • Integration into artificial neural network architectures as a learnable computational unit (abstract node).

Main Results:

  • A method to find a representative matrix K such that subspaces V_i closely intersect its column space.
  • The proposed method can learn K in situ or sequentially within a learning problem.
  • Demonstrated applicability to classification problems on sets of data.

Conclusions:

  • The Schubert variety provides a powerful geometric framework for subspace representation and analysis.
  • The developed optimization approach enables finding a 'best fit' Schubert variety for data.
  • The integration into neural networks offers a novel approach for data processing and feature learning.