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Efficient Discretization of Optimal Transport.

Junqi Wang1, Pei Wang1, Patrick Shafto1,2

  • 1Department of Math & CS, Rutgers University, Newark, NJ 07102, USA.

Entropy (Basel, Switzerland)
|June 28, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces an efficient algorithm for optimal transportation (OT) problems with continuous spaces. The new method provides accurate discretizations using fewer weighted points, reducing computational cost for complex calculations.

Keywords:
discretizationentropy regularizationgradient descentoptimal transport

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Area of Science:

  • Computational Mathematics
  • Optimization Theory
  • Data Science

Background:

  • Optimal Transportation (OT) problems are computationally challenging in continuous spaces.
  • Current discretization methods using independent and identically distributed (i.i.d.) sampling require large sample sizes, leading to high computational costs.
  • Existing approaches struggle with practical implementation due to intensive computational demands.

Purpose of the Study:

  • To develop an efficient algorithm for calculating discretizations of marginal distributions in OT problems.
  • To minimize the entropy-regularized Wasserstein distance using a specified number of weighted points.
  • To provide performance bounds for the proposed discretization method.

Main Methods:

  • An algorithm is proposed to compute discretizations by minimizing the entropy-regularized Wasserstein distance.
  • The method utilizes a given number of weighted points for marginal distributions.
  • A local, parallelizable version of the discretization is developed for practical applications.

Main Results:

  • The proposed discretization plans achieve comparable performance to methods using significantly larger numbers of i.i.d. samples.
  • The algorithm demonstrates improved computational efficiency compared to existing alternatives.
  • The method's effectiveness is validated through image approximation tasks.

Conclusions:

  • The developed algorithm offers an efficient and effective solution for approximating OT problems in continuous spaces.
  • The method reduces computational burden, making OT solutions more accessible for practical applications.
  • The local, parallelizable version facilitates real-world applications, including image processing.