Fast optimal transport regularized projection and application to coefficient shrinkage and filtering

Abstract

This paper explores solutions to the problem of regularized projections with respect to the optimal transport metric. Expanding recent works on optimal transport dictionary learning and non-negative matrix factorization, we derive general purpose algorithms for projecting on any set of vectors with any regularization, and we further propose fast algorithms for the special cases of projecting onto invertible or orthonormal bases. Noting that pass filters and coefficient shrinkage can be seen as regularized projections under the Euclidean metric, we show how to use our algorithms to perform optimal transport pass filters and coefficient shrinkage. We give experimental evidence that using the optimal transport distance instead of the Euclidean distance for filtering and coefficient shrinkage leads to reduced artifacts and improved denoising results.

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Code availability

A python library for our methods and all scripts necessary to reproduce our figures and results will be made available on the author’s Web site upon publication of this paper.

Notes

  1. 1.

    Since D is orthonormal, the problem is actually equivalent to simply \(\displaystyle \min \limits _{\varvec{\lambda }} {{\,\mathrm{OT}\,}}_{\gamma }(\varvec{X}, \varvec{\lambda }) + \alpha \Vert \varvec{\lambda }\Vert _2^2\).

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Funding

This work was partly supported by JSPS KAKENHI Grant Number 17H01788.

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AR and VS designed the research and wrote the paper. Experiments were performed by AR. All authors read and approved the final manuscript.

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Correspondence to Antoine Rolet.

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Rolet, A., Seguy, V. Fast optimal transport regularized projection and application to coefficient shrinkage and filtering. Vis Comput (2021). https://doi.org/10.1007/s00371-020-02029-7

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Keywords

  • Optimal transport
  • Wasserstein distance
  • Coefficient shrinkage
  • Sparse decomposition
  • Wavelet thresholding
  • Denoising