Alphabeta MathConditional flow matching and transport paths
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- 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
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
Combinatorics
Differential Geometry
Functional Analysis
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Direct Matrix Factorisations: LU, Cholesky and QR
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Machine Learning Algorithms
Measure Theory
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Radon Measures and the Riesz Markov Kakutani Theorem
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
Number Theory
Probability
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Statistical Models and Experiments
Topology
- Compactness
- Compactness in Metric Spaces
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Probability paths and velocity fields
Lemma
A conditional mixture is a probability path
Lemma
Conditional continuity equations pass to the marginal path
Lemma
An affine conditional path has the stated velocity field
Lemma
A centered law and its affine dilation have an optimal affine coupling
Theorem
Conditional and marginal flow-matching objectives differ by a constant
Theorem
The Gaussian conditional affine map is quadratic-cost optimal
Algorithm
Conditional flow-matching training and exact-flow sampling
Lemma
Conditional flow-matching batch gradients and exact-flow output law
Sources
- Yaron Lipman et al., Flow Matching for Generative Modeling, §§2–3
- Peter Holderrieth and Ezra Erives, Introduction to Flow Matching and Diffusion Models, §§3.1–3.2
- Filippo Santambrogio, Optimal Transport for Applied Mathematicians, §4.1.2, Definition 4.1
- Yaron Lipman et al., Flow Matching for Generative Modeling, §3.1
- Peter Holderrieth and Ezra Erives, Introduction to Flow Matching and Diffusion Models, §3.1
- Yaron Lipman et al., Flow Matching for Generative Modeling, §3.1 and Appendix A
- Peter Holderrieth and Ezra Erives, Introduction to Flow Matching and Diffusion Models, §3.2
- Yaron Lipman et al., Flow Matching for Generative Modeling, §4 Theorem 3 and Appendix A
- Peter Holderrieth and Ezra Erives, Introduction to Flow Matching and Diffusion Models, §3.2 Example 10
- Tor Fjelde, Emile Mathieu, and Vincent Dutordoir, An Introduction to Flow Matching, Gaussian conditional path derivation
- Yaron Lipman et al., Flow Matching for Generative Modeling, §4.1
- Gabriel Peyré and Marco Cuturi, Computational Optimal Transport, §2.6 Remark 2.31
- Yaron Lipman et al., Flow Matching for Generative Modeling, §3.2 Theorem 2 and Appendix A
- Peter Holderrieth and Ezra Erives, Introduction to Flow Matching and Diffusion Models, §3.3 Theorem 12 and direct proof
- Yaron Lipman et al., Flow Matching for Generative Modeling, §4 Theorem 3 and §4.1
- Yaron Lipman et al., Flow Matching for Generative Modeling, §§3.2 and 6.2
- Peter Holderrieth and Ezra Erives, Introduction to Flow Matching and Diffusion Models, §3.3 Algorithm 3