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

Research output: Contribution to journal › Article › peer-review

36 Scopus citations

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

Bibliographical 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.

Access to Document

10.1093/amrx/abs007

Cite this

@article{5e6b5d04950747039a3931b31423b566,

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",

month = mar,

day = "11",

doi = "10.1093/amrx/abs007",

language = "English (US)",

volume = "2012",

pages = "209--253",

journal = "Applied Mathematics Research eXpress",

issn = "1687-1200",

publisher = "Oxford University Press",

number = "2",

}

TY - JOUR

T1 - Regularized Regression and Density Estimation based on Optimal Transport

AU - Burger, M.

AU - Franek, M.

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.

PY - 2012/3/11

Y1 - 2012/3/11

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).

UR - http://hdl.handle.net/10754/599489

UR - https://academic.oup.com/amrx/article-lookup/doi/10.1093/amrx/abs007

UR - http://www.scopus.com/inward/record.url?scp=84866348492&partnerID=8YFLogxK

U2 - 10.1093/amrx/abs007

DO - 10.1093/amrx/abs007

M3 - Article

SN - 1687-1200

VL - 2012

SP - 209

EP - 253

JO - Applied Mathematics Research eXpress

JF - Applied Mathematics Research eXpress

IS - 2

ER -

Regularized Regression and Density Estimation based on Optimal Transport

Abstract

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this