Alphabeta MathKantorovich–Rubinstein duality on compact metric spaces
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Information Theory
Linear Algebra
Machine Learning Algorithms
Measure Theory
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
- limsup, liminf, and Subsequential Limits
- 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
- Topology of ℝ
Set Theory
Statistical Models and Experiments
2 · Logical flowchart
3 · Items
Definition
The coupling definition of the 1-Wasserstein cost on a compact metric space
Lemma
A compact metric space has a finite Borel nearest-center quantization at every positive scale
Lemma
The McShane formula extends a 1-Lipschitz function from a finite subset
Lemma
A compact metric space is second-countable under countable choice
Lemma
A finite coupling lifts to a Borel coupling with a controlled metric cost
Theorem
Kantorovich–Rubinstein duality on compact second-countable metric spaces
Theorem
Kantorovich–Rubinstein duality on compact metric spaces under countable choice
Lemma
Borel product probabilities on compact metric spaces by sections
Theorem
Kantorovich–Rubinstein duality on compact metric spaces
Sources
- Dmitriy Panchenko, Lecture Notes on Probability Theory, §4.4, pp. 121–124
- Filippo Santambrogio, Introduction to Optimal Transport Theory, §§1.1–1.3, PDF pp. 1–7
- Martin Arjovsky, Soumith Chintala and Léon Bottou, Wasserstein GAN, §§2–3
- Dmitriy Panchenko, Lecture Notes on Probability Theory, §4.4
- Filippo Santambrogio, Introduction to Optimal Transport Theory, §1.1, PDF pp. 3–4
- Filippo Santambrogio, Introduction to Optimal Transport Theory, §1.3, PDF pp. 5–6
- Filippo Santambrogio, Introduction to Optimal Transport Theory, §§1–2
- Dmitriy Panchenko, Lecture Notes on Probability Theory, §4.4, Theorem 4.14 and proof, pp. 121–124
- Martin Arjovsky, Soumith Chintala and Léon Bottou, Wasserstein GAN, §§2–3, PDF pp. 3–6
- John K. Hunter, Measure Theory, product-measure construction by measurable sections (adapted here to product-topology Borel sets)