Alphabeta MathVariational autoencoders and pathwise training
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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
Bayesian Computational Statistics
Combinatorics
Linear Algebra
Mathematical Statistics
Measure Theory
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue Integral and the Convergence Theorems
- The Radon Nikodym Theorem and Lebesgue Decomposition
Numerical Optimization Inverse Problems and Uncertainty Quantification
Probability
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- 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
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Integral Logarithm and the Equivalence of Its Characterisations
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Variational autoencoder model, encoder, and ELBO
Corollary
The VAE evidence equals the ELBO plus reverse KL
Theorem
Dominated pathwise differentiation for VAE training
Algorithm
Reparameterized variational training
Lemma
The fixed-horizon reparameterized variational-training procedure terminates
Sources
- Diederik P. Kingma and Max Welling, Auto-Encoding Variational Bayes, arXiv:1312.6114v1
- David M. Blei, Alp Kucukelbir, and Jon D. McAuliffe, Variational Inference: A Review for Statisticians
- Ronghuo Zheng, Zhiting Hu, and Yuntian Deng, 10-708 Probabilistic Graphical Models, Lecture 13: Variational Inference II
- Kingma and Welling, Auto-Encoding Variational Bayes, §2.4 Algorithm 1 (procedure background)