Optimal Transport for Machine Learning Applications J Saketha

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Optimal Transport for Machine Learning Applications J Saketha Nath, IIT Hyderabad

Optimal Transport for Machine Learning Applications J Saketha Nath, IIT Hyderabad

Special Case: Assignment Problem

Special Case: Assignment Problem

Special Case: Assignment Problem

Special Case: Assignment Problem

Special Case: Assignment Problem

Special Case: Assignment Problem

Special Case: Assignment Problem Costs are known

Special Case: Assignment Problem Costs are known

Special Case: Assignment Problem Costs are known

Special Case: Assignment Problem Costs are known

Special Case: Assignment Problem Costs are known • Linear Program • Hungarian Method

Special Case: Assignment Problem Costs are known • Linear Program • Hungarian Method

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem Bijection?

Optimal Transport Problem Bijection?

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem

Optimal Transport Problem

OT: Monge’s Formulation

OT: Monge’s Formulation

OT: Monge’s Formulation Monge, 1781; Tolstoi, 1930;

OT: Monge’s Formulation Monge, 1781; Tolstoi, 1930;

OT: Monge’s Formulation Optimal Transport Map

OT: Monge’s Formulation Optimal Transport Map

OT: Monge’s Formulation Optimal Transport Map Discrete OT is Linear Program

OT: Monge’s Formulation Optimal Transport Map Discrete OT is Linear Program

Applications of Optimal Map

Applications of Optimal Map

Application: Color Transfer https: //optimaltransport. github. io/slides-peyre/Applications. pdf

Application: Color Transfer https: //optimaltransport. github. io/slides-peyre/Applications. pdf

Application: Color Transfer https: //optimaltransport. github. io/slides-peyre/Applications. pdf

Application: Color Transfer https: //optimaltransport. github. io/slides-peyre/Applications. pdf

Application: Color Transfer https: //optimaltransport. github. io/slides-peyre/Applications. pdf

Application: Color Transfer https: //optimaltransport. github. io/slides-peyre/Applications. pdf

ML Application: Domain Adaptation [Courty, 2015]

ML Application: Domain Adaptation [Courty, 2015]

ML Application: Domain Adaptation Optimal Transport over empirical distributions [Courty, 2015]

ML Application: Domain Adaptation Optimal Transport over empirical distributions [Courty, 2015]

ML Application: Domain Adaptation Optimal Transport over empirical distributions W/O distr constraint OT =

ML Application: Domain Adaptation Optimal Transport over empirical distributions W/O distr constraint OT = NN classifier [Courty, 2015]

ML Application: Domain Adaptation Optimal Transport over empirical distributions [Courty, 2015]

ML Application: Domain Adaptation Optimal Transport over empirical distributions [Courty, 2015]

ML Application: Domain Adaptation Optimal Transport over empirical distributions [Courty, 2015]

ML Application: Domain Adaptation Optimal Transport over empirical distributions [Courty, 2015]

• ML applications

• ML applications

Kantorovich’s Relaxation

Kantorovich’s Relaxation

Kantorovich’s Relaxation

Kantorovich’s Relaxation

Kantorovich’s Relaxation [Kantorovich, 1942]

Kantorovich’s Relaxation [Kantorovich, 1942]

Kantorovich’s Relaxation Optimal Transport Plan

Kantorovich’s Relaxation Optimal Transport Plan

Kantorovich’s Relaxation Entropy regularization fast Sinkhorn Solver [Cuturi, 2013]

Kantorovich’s Relaxation Entropy regularization fast Sinkhorn Solver [Cuturi, 2013]

ML Application: Domain Adaptation Plan -> Map? Optimal Transport [Courty, 2015]

ML Application: Domain Adaptation Plan -> Map? Optimal Transport [Courty, 2015]

Barycenter Problem

Barycenter Problem

Barycenter Problem

Barycenter Problem

ML Application: Domain Adaptation Optimal Transport [Courty, 2015]

ML Application: Domain Adaptation Optimal Transport [Courty, 2015]

• OT lifts ground metric between points to distributions!

• OT lifts ground metric between points to distributions!

• OT lifts ground metric between points to distributions!

• OT lifts ground metric between points to distributions!

• OT lifts ground metric between points to distributions!

• OT lifts ground metric between points to distributions!

• OT lifts ground metric between points to distributions! Entropic regularization (Sinkhorn distance)

• OT lifts ground metric between points to distributions! Entropic regularization (Sinkhorn distance) • Fast computation • Sample Complexity

Wasserstein provides nice geometry

Wasserstein provides nice geometry

Application: Shape interpolation Solomon et. al. , 2015

Application: Shape interpolation Solomon et. al. , 2015

Applications of Wasserstein distance

Applications of Wasserstein distance

Words to documents [Kusner et. al. , 2015]

Words to documents [Kusner et. al. , 2015]

Unsupervised Learning

Unsupervised Learning

Generative nets

Generative nets

Wasserstein as unsupervised loss

Wasserstein as unsupervised loss

Wasserstein loss: Generative Nets [Genevay et. al. , 2018]

Wasserstein loss: Generative Nets [Genevay et. al. , 2018]

Wasserstein loss: Generative Nets [Genevay et. al. , 2018]

Wasserstein loss: Generative Nets [Genevay et. al. , 2018]

Topic models [Rolet et. al. , 2016]

Topic models [Rolet et. al. , 2016]

Wasserstein as Supervised loss

Wasserstein as Supervised loss

Wasserstein as Supervised loss

Wasserstein as Supervised loss

Wasserstein as Supervised loss MLE

Wasserstein as Supervised loss MLE

Wasserstein as Supervised loss

Wasserstein as Supervised loss

Wasserstein as Supervised loss [Frogner et. al. , 2015]

Wasserstein as Supervised loss [Frogner et. al. , 2015]

Wasserstein as Supervised loss Use any metric on labels! [Frogner et. al. , 2015]

Wasserstein as Supervised loss Use any metric on labels! [Frogner et. al. , 2015]

Wasserstein as Supervised loss Use any metric on labels! [Frogner et. al. , 2015]

Wasserstein as Supervised loss Use any metric on labels! [Frogner et. al. , 2015]

Adversarial Attacks

Adversarial Attacks

Distributional Robustness

Distributional Robustness

Distributional Robustness

Distributional Robustness

Distributional Robustness [Sinha et. al. , 2018]

Distributional Robustness [Sinha et. al. , 2018]

Applications of Wasserstein distance Dual of Wasserstein Applications: GANs

Applications of Wasserstein distance Dual of Wasserstein Applications: GANs

Dual of OT

Dual of OT

Dual of OT

Dual of OT

Dual of OT

Dual of OT

Dual of OT

Dual of OT

Wasserstein loss: Generative Nets [Genevay et. al. , 2018]

Wasserstein loss: Generative Nets [Genevay et. al. , 2018]

Wasserstein loss: Generative Adversarial Nets [Genevay et. al. , 2018]

Wasserstein loss: Generative Adversarial Nets [Genevay et. al. , 2018]

Wasserstein loss: Generative Adversarial Nets [Genevay et. al. , 2018]

Wasserstein loss: Generative Adversarial Nets [Genevay et. al. , 2018]

Applications of Wasserstein distance Generalizations Gromov Wasserstein

Applications of Wasserstein distance Generalizations Gromov Wasserstein

Registered spaces t im t p O ns a Tr l a r po

Registered spaces t im t p O ns a Tr l a r po

Unregistered spaces! ? ?

Unregistered spaces! ? ?

Unregistered spaces!

Unregistered spaces!

Gromov Wasserstein

Gromov Wasserstein

Gromov Wasserstein

Gromov Wasserstein

Gromov Wasserstein

Gromov Wasserstein

Summary • Finding optimal map/plan – Domain adaption • Wasserstein distance preserves natural geometry

Summary • Finding optimal map/plan – Domain adaption • Wasserstein distance preserves natural geometry • Used as loss function in probabilistic modelling • Dual problem provides insights and explains GANs • Generalizations • Gromov Wasserstein • Multi-marginal OT • Unbalanced OT

References • Gabriel Peyré and Marco Cuturi (2018). Computational Optimal Transport. https: //optimaltransport. github.

References • Gabriel Peyré and Marco Cuturi (2018). Computational Optimal Transport. https: //optimaltransport. github. io/ • Santambrogio (2015). Optimal Transport for Applied Mathematicians • Villani (2008). Optimal Transport, Old and New • Marco Cuturi (2013). Sinkhorn distances: lightspeed computation of optimal transport. • Jean Feydy et. al. (2019). Interpolating between Optimal Transport and MMD using Sinkhorn Divergences.

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