# Scientific

## A variational approach to the regularity theory for optimal transportation: Lecture 3

In this mini-course, we shall explain the variational approach to regularity

theory for optimal transportation introduced in [8]. This approach does

completely bypass the celebrated regularity theory of Caffarelli [2], which is

based on the regularity theory for the Monge-Amp ere equation as a fully

nonlinear elliptic equation with a comparison principle. Nonetheless, one

recovers the same partial regularity theory [5, 4].

The advantage of the variational approach resides in its robustness regarding

the regularity of the measures, which can be arbitrary measures [7][Theorem

1.4], and in terms of the problem formulation, e.g. by its extension to almost

minimizers [10]. The former for instance is crucial in order to tackle the

widely popular matching problem [3, 1] e.g. the optimal transportation between

(random) point clouds, as carried out in [7, 6, 9]. The latter is convenient

when treating more general than square Euclidean cost functions.

The variational approach follows de Giorgi’s philosophy for minimal surfaces.

At its core is the approximation of the displacement by the gradient of a

harmonic function. This approximation is based on the Eulerian formulation of

optimal transportation, which reveals its strict convexity and the proximity to

the $H^{-1}$-norm. In this mini-course, we shall give a pretty self-contained

derivation of this harmonic approximation result, and establish applications to

the matching problem.

#### References

- [1] L. Ambrosio, F. Stra, D. Trevisan: A PDE approach to a 2-dimensional

matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019). - [2] L.A. Caffarelli: The regularity of mappings with a convex potential.

Journal of the American Mathematical Society 5 (1992), no. 1, 99–104. - [3] S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro: Scaling hypothesis for

the Euclidean bipartite matching problem. Physical Review E, 90(1), 2014. - [4] G. De Philippis, A. Figalli: Partial regularity for optimal transport

maps. Publications Mathématiques. Institut de Hautes Études Scientifiques

121 (2015), 81–112. - [5] A. Figalli, Y.-H. Kim: Partial regularity of Brenier solutions of the

Monge-Amépre equation. Discrete and Continuous Dynamical Systems (Series A)

28 (2010), 559–565. - [6] M. Goldman, M. Huesmann: A fluctuation result for the displacement in the

optimal matching problem. arXiv e-prints, May 2021. arXiv:2105.02915. - [7] M. Goldman, M. Huesmann, F. Otto: Quantitative linearization results for

the Monge-Amp`ere equation. Communications on Pure and Applied Mathematics

(2021). - [8] M. Goldman, F. Otto: A variational proof of partial regularity for optimal

transportation maps. Annales Scientifiques de l’Ećole Normale Supérieure.

Quatriéme Série 53 (2020), no. 5, 1209–1233. - [9] M. Huesmann, F. Mattesini, F. Otto: There is no stationary cyclically

monotone Poisson matching in 2d. arXiv e-prints, September 2021.

arXiv:2109.13590. - [10] F. Otto, M. Prod’homme, T. Ried: Variational approach to regularity of

optimal transport maps: general cost functions. (English summary) Ann. PDE 7

(2021), no. 2, Paper No. 17, 74 pp.

## A variational approach to the regularity theory for optimal transportation: Lecture 2

In this mini-course, we shall explain the variational approach to regularity

theory for optimal transportation introduced in [8]. This approach does

completely bypass the celebrated regularity theory of Caffarelli [2], which is

based on the regularity theory for the Monge-Amp ere equation as a fully

nonlinear elliptic equation with a comparison principle. Nonetheless, one

recovers the same partial regularity theory [5, 4].

The advantage of the variational approach resides in its robustness regarding

the regularity of the measures, which can be arbitrary measures [7][Theorem

1.4], and in terms of the problem formulation, e.g. by its extension to almost

minimizers [10]. The former for instance is crucial in order to tackle the

widely popular matching problem [3, 1] e.g. the optimal transportation between

(random) point clouds, as carried out in [7, 6, 9]. The latter is convenient

when treating more general than square Euclidean cost functions.

The variational approach follows de Giorgi’s philosophy for minimal surfaces.

At its core is the approximation of the displacement by the gradient of a

harmonic function. This approximation is based on the Eulerian formulation of

optimal transportation, which reveals its strict convexity and the proximity to

the $H^{-1}$-norm. In this mini-course, we shall give a pretty self-contained

derivation of this harmonic approximation result, and establish applications to

the matching problem.

#### References

- [1] L. Ambrosio, F. Stra, D. Trevisan: A PDE approach to a 2-dimensional

matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019). - [2] L.A. Caffarelli: The regularity of mappings with a convex potential.

Journal of the American Mathematical Society 5 (1992), no. 1, 99–104. - [3] S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro: Scaling hypothesis for

the Euclidean bipartite matching problem. Physical Review E, 90(1), 2014. - [4] G. De Philippis, A. Figalli: Partial regularity for optimal transport

maps. Publications Mathématiques. Institut de Hautes Études Scientifiques

121 (2015), 81–112. - [5] A. Figalli, Y.-H. Kim: Partial regularity of Brenier solutions of the

Monge-Amépre equation. Discrete and Continuous Dynamical Systems (Series A)

28 (2010), 559–565. - [6] M. Goldman, M. Huesmann: A fluctuation result for the displacement in the

optimal matching problem. arXiv e-prints, May 2021. arXiv:2105.02915. - [7] M. Goldman, M. Huesmann, F. Otto: Quantitative linearization results for

the Monge-Amp`ere equation. Communications on Pure and Applied Mathematics

(2021). - [8] M. Goldman, F. Otto: A variational proof of partial regularity for optimal

transportation maps. Annales Scientifiques de l’Ećole Normale Supérieure.

Quatriéme Série 53 (2020), no. 5, 1209–1233. - [9] M. Huesmann, F. Mattesini, F. Otto: There is no stationary cyclically

monotone Poisson matching in 2d. arXiv e-prints, September 2021.

arXiv:2109.13590. - [10] F. Otto, M. Prod’homme, T. Ried: Variational approach to regularity of

optimal transport maps: general cost functions. (English summary) Ann. PDE 7

(2021), no. 2, Paper No. 17, 74 pp.

## A variational approach to the regularity theory for optimal transportation: Lecture 1

In this mini-course, we shall explain the variational approach to regularity

theory for optimal transportation introduced in [8]. This approach does

completely bypass the celebrated regularity theory of Caffarelli [2], which is

based on the regularity theory for the Monge-Amp ere equation as a fully

nonlinear elliptic equation with a comparison principle. Nonetheless, one

recovers the same partial regularity theory [5, 4].

The advantage of the variational approach resides in its robustness regarding

the regularity of the measures, which can be arbitrary measures [7][Theorem

1.4], and in terms of the problem formulation, e.g. by its extension to almost

minimizers [10]. The former for instance is crucial in order to tackle the

widely popular matching problem [3, 1] e.g. the optimal transportation between

(random) point clouds, as carried out in [7, 6, 9]. The latter is convenient

when treating more general than square Euclidean cost functions.

The variational approach follows de Giorgi’s philosophy for minimal surfaces.

At its core is the approximation of the displacement by the gradient of a

harmonic function. This approximation is based on the Eulerian formulation of

optimal transportation, which reveals its strict convexity and the proximity to

the $H^{-1}$-norm. In this mini-course, we shall give a pretty self-contained

derivation of this harmonic approximation result, and establish applications to

the matching problem.

#### References

- [1] L. Ambrosio, F. Stra, D. Trevisan: A PDE approach to a 2-dimensional

matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019). - [2] L.A. Caffarelli: The regularity of mappings with a convex potential.

Journal of the American Mathematical Society 5 (1992), no. 1, 99–104. - [3] S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro: Scaling hypothesis for

the Euclidean bipartite matching problem. Physical Review E, 90(1), 2014. - [4] G. De Philippis, A. Figalli: Partial regularity for optimal transport

maps. Publications Mathématiques. Institut de Hautes Études Scientifiques

121 (2015), 81–112. - [5] A. Figalli, Y.-H. Kim: Partial regularity of Brenier solutions of the

Monge-Amépre equation. Discrete and Continuous Dynamical Systems (Series A)

28 (2010), 559–565. - [6] M. Goldman, M. Huesmann: A fluctuation result for the displacement in the

optimal matching problem. arXiv e-prints, May 2021. arXiv:2105.02915. - [7] M. Goldman, M. Huesmann, F. Otto: Quantitative linearization results for

the Monge-Amp`ere equation. Communications on Pure and Applied Mathematics

(2021). - [8] M. Goldman, F. Otto: A variational proof of partial regularity for optimal

transportation maps. Annales Scientifiques de l’Ećole Normale Supérieure.

Quatriéme Série 53 (2020), no. 5, 1209–1233. - [9] M. Huesmann, F. Mattesini, F. Otto: There is no stationary cyclically

monotone Poisson matching in 2d. arXiv e-prints, September 2021.

arXiv:2109.13590. - [10] F. Otto, M. Prod’homme, T. Ried: Variational approach to regularity of

optimal transport maps: general cost functions. (English summary) Ann. PDE 7

(2021), no. 2, Paper No. 17, 74 pp.

## Optimal Transport for Machine Learning: Lecture 3

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this course, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book “Computational Optimal Transport”.

## Optimal Transport for Machine Learning: Lecture 2

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this course, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book “Computational Optimal Transport”.

## Optimal Transport for Machine Learning: Lecture 1

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this course, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book “Computational Optimal Transport”.

## Recent advances in dynamical optimal transport: Lecture 3

In this lecture series we present an overview of dynamical optimal transport and some of its applications to discrete probability and non-commutative analysis. Particular focus is on gradient structures and functional inequalities for dissipative quantum systems, and on homogenisation results for dynamical optimal transport.

## Recent advances in dynamical optimal transport: Lecture 2

In this lecture series we present an overview of dynamical optimal transport and some of its applications to discrete probability and non-commutative analysis. Particular focus is on gradient structures and functional inequalities for dissipative quantum systems, and on homogenisation results for dynamical optimal transport.

## Recent advances in dynamical optimal transport: Lecture 1

In this lecture series we present an overview of dynamical optimal transport and some of its applications to discrete probability and non-commutative analysis. Particular focus is on gradient structures and functional inequalities for dissipative quantum systems, and on homogenisation results for dynamical optimal transport.

## Gross substitutes, optimal transport and matching models: Lecture 3

Gross substitutes is a fundamental property in mathematics, economics and computation, almost as important as convexity. It is at the heart of optimal transport theory – although this is often underrecognized – and understanding the connection key to understanding the extension of optimal transport to other models of matching.