Alphabeta MathFinite-metric transport duality
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1 · Prerequisites
Abstract Algebra
Information Theory
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Convex and Semicontinuous Functions on Rⁿ
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Total Derivative in ℝᵐ → ℝⁿ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Couplings of finite probability vectors
Lemma
A cone generated by finitely many Euclidean vectors is closed
Definition
The finite-metric 1-Wasserstein transport cost
Lemma
Finite-metric transport and normalized dual optima are attained
Theorem
Finite-metric transport duality and complementary slackness
Sources
- Gabriel Peyré and Marco Cuturi, Computational Optimal Transport, §§2.1–2.3 and 3.1
- Filippo Santambrogio, Introduction to Optimal Transport Theory, §1.1, PDF pp. 3–4
- Wouter Kager, A Short Simple Proof of Closedness of Convex Cones and Farkas' Lemma, complete two-page paper, Lemma
- Gabriel Peyré and Marco Cuturi, Computational Optimal Transport, §§2.1–2.3 and 6.1
- Filippo Santambrogio, Introduction to Optimal Transport Theory, §§1.1 and 1.3, PDF pp. 3–6
- Filippo Santambrogio, Introduction to Optimal Transport Theory, §§1.1–1.2, PDF pp. 3–5
- Gabriel Peyré and Marco Cuturi, Computational Optimal Transport, §§2.1–2.3, 3.1, 3.3 and 6.1
- Filippo Santambrogio, Introduction to Optimal Transport Theory, §§1.1–1.3, PDF pp. 3–6
- Martin Arjovsky, Soumith Chintala and Léon Bottou, Wasserstein GAN, §§2–3, PDF pp. 3–6