Dual Expectation Formulation

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Table of contents

Chapter 1: Entropy-regularized Optimal Transport 
1 Introduction
2 Distances Between Probability Measures
2.1 ‘-divergences
2.2 Integral Probability Metrics and Maximum Mean discrepancy
2.3 Optimal Transport
3 Regularized Optimal Transport
3.1 Dual Formulation
3.2 The Case of Unbalanced OT
3.3 Dual Expectation Formulation
4 Entropy-Regularized Optimal Transport
4.1 Solving the Regularized Dual Problem
4.1.1 Hilbert Metric
4.1.2 Fixed Point Theorem
4.2 Sinkhorn’s Algorithm
4.3 Semi-Dual Formulation
4.3.1 Case of a Discrete Measure
4.3.2 Semi-Dual Expectation Formulation
4.3.3 Some Analytic Properties of the Semi-Dual Functional
4.4 Convergence of Entropy-Regularized OT to Standard OT
Chapter 2: Learning with Sinkhorn Divergences 
1 Introduction
2 Density Fitting
2.1 Learning with ‘-divergences
2.2 Maximum Mean Discrepancy and Optimal Transport
2.3 Regularized OT and Variants of the Regularized OT Loss
2.4 Sinkhorn Divergences : an Interpolation Between OT and MMD
3 Sinkhorn AutoDiff Algorithm
3.1 Mini-batch Sampling Loss
3.2 Sinkhorn Iterates
3.3 Learning the Cost Function Adversarially
3.4 The Optimization Procedure in Practice
4 Applications
4.1 Benchmark on Synthetic Problems
4.2 Data Clustering with Ellipses
4.3 Tuning a Generative Neural Network
4.3.1 With a Fixed Cost c
4.3.2 Learning the Cost
Chapter 3: Sample Complexity of Sinkhorn Divergences 
1 Introduction
2 Reminders on Sinkhorn Divergences
3 Approximating Optimal Transport with Sinkhorn Divergences
4 Properties of Sinkhorn Potentials
5 Approximating Sinkhorn Divergence from Samples
6 Experiments
Chapter 4: Stochastic Optimization for Large Scale OT 
1 Introduction
2 Optimal Transport: Primal, Dual and Semi-dual Formulations
2.1 Primal, Dual and Semi-dual Formulations
2.2 Stochastic Optimization Formulations
3 Discrete Optimal Transport
3.1 Discrete Optimization and Sinkhorn
3.2 Incremental Discrete Optimization with SAG when  » > 0
3.3 Numerical Illustrations on Bags of Word-Embeddings
4 Semi-Discrete Optimal Transport
4.1 Stochastic Semi-discrete Optimization with SGD
4.2 Numerical Illustrations on Synthetic Data
5 Continuous Optimal Transport Using RKHS
5.1 Kernel SGD
5.2 Speeding up Iterations with Kernel Approximation
5.2.1 Incomplete Cholesky Decomposition
5.2.2 Random Fourier Features
5.3 Comparison of the Three Algorithms on Synthetic Data
Conclusion 
Bibliography