Regularized Regression and Density Estimation based on Optimal Transport

M. Burger; M. Franek; C.-B. Schonlieb

doi:10.1093/amrx/abs007

Regularized Regression and Density Estimation based on Optimal Transport

M. Burger, M. Franek, C.-B. Schonlieb

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Abstract

The aim of this paper is to investigate a novel nonparametric approach for estimating and smoothing density functions as well as probability densities from discrete samples based on a variational regularization method with the Wasserstein metric as a data fidelity. The approach allows a unified treatment of discrete and continuous probability measures and is hence attractive for various tasks. In particular, the variational model for special regularization functionals yields a natural method for estimating densities and for preserving edges in the case of total variation regularization. In order to compute solutions of the variational problems, a regularized optimal transport problem needs to be solved, for which we discuss several formulations and provide a detailed analysis. Moreover, we compute special self-similar solutions for standard regularization functionals and we discuss several computational approaches and results. © 2012 The Author(s).

Original language	English (US)
Pages (from-to)	209-253
Number of pages	45
Journal	Applied Mathematics Research eXpress
Volume	2012
Issue number	2
DOIs	https://doi.org/10.1093/amrx/abs007
State	Published - Mar 11 2012
Externally published	Yes

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title = "Regularized Regression and Density Estimation based on Optimal Transport",

abstract = "The aim of this paper is to investigate a novel nonparametric approach for estimating and smoothing density functions as well as probability densities from discrete samples based on a variational regularization method with the Wasserstein metric as a data fidelity. The approach allows a unified treatment of discrete and continuous probability measures and is hence attractive for various tasks. In particular, the variational model for special regularization functionals yields a natural method for estimating densities and for preserving edges in the case of total variation regularization. In order to compute solutions of the variational problems, a regularized optimal transport problem needs to be solved, for which we discuss several formulations and provide a detailed analysis. Moreover, we compute special self-similar solutions for standard regularization functionals and we discuss several computational approaches and results. {\textcopyright} 2012 The Author(s).",

author = "M. Burger and M. Franek and C.-B. Schonlieb",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-I1-007-43 Acknowledgements: The work of M.B. has been supported by the German Science Foundation (DFG) through project Regularization with Singular Energies. C.B.S acknowledges the financial support provided by the Cambridge Centre for Analysis (CCA), the DFG Graduiertenkolleg 1023 Identification in Mathematical Models: Synergy of Stochastic and Numerical Methods and the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts. Further, this publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

year = "2012",

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doi = "10.1093/amrx/abs007",

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AU - Schonlieb, C.-B.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-I1-007-43 Acknowledgements: The work of M.B. has been supported by the German Science Foundation (DFG) through project Regularization with Singular Energies. C.B.S acknowledges the financial support provided by the Cambridge Centre for Analysis (CCA), the DFG Graduiertenkolleg 1023 Identification in Mathematical Models: Synergy of Stochastic and Numerical Methods and the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts. Further, this publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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N2 - The aim of this paper is to investigate a novel nonparametric approach for estimating and smoothing density functions as well as probability densities from discrete samples based on a variational regularization method with the Wasserstein metric as a data fidelity. The approach allows a unified treatment of discrete and continuous probability measures and is hence attractive for various tasks. In particular, the variational model for special regularization functionals yields a natural method for estimating densities and for preserving edges in the case of total variation regularization. In order to compute solutions of the variational problems, a regularized optimal transport problem needs to be solved, for which we discuss several formulations and provide a detailed analysis. Moreover, we compute special self-similar solutions for standard regularization functionals and we discuss several computational approaches and results. © 2012 The Author(s).

AB - The aim of this paper is to investigate a novel nonparametric approach for estimating and smoothing density functions as well as probability densities from discrete samples based on a variational regularization method with the Wasserstein metric as a data fidelity. The approach allows a unified treatment of discrete and continuous probability measures and is hence attractive for various tasks. In particular, the variational model for special regularization functionals yields a natural method for estimating densities and for preserving edges in the case of total variation regularization. In order to compute solutions of the variational problems, a regularized optimal transport problem needs to be solved, for which we discuss several formulations and provide a detailed analysis. Moreover, we compute special self-similar solutions for standard regularization functionals and we discuss several computational approaches and results. © 2012 The Author(s).

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Regularized Regression and Density Estimation based on Optimal Transport

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