Alphabeta MathSelf-supervised and contrastive representations — Examples
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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
Combinatorics
Linear Algebra
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- 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
Mathematical Statistics
Measure Theory
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- 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 Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Radon Nikodym Theorem and Lebesgue Decomposition
Numerical Linear Algebra and Approximation
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 Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
Statistical Learning and Classical Models
2 · Logical flowchart
3 · Items
Example
A negative sample collides with the positive class
Example
High within-class variance raises contrastive loss despite perfect mean separation
Example
An augmentation that erases the label-relevant class coordinate
Example
SimCLR reports an empirical projection-head effect
Example
The hinge margin comparison does not extend to logistic contrastive loss
Sources
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Section 4.2, Equation (8)
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Section 5.1 and Figure 1(b)
- Ting Chen et al., A Simple Framework for Contrastive Learning of Visual Representations, Section 3
- Ting Chen et al., A Simple Framework for Contrastive Learning of Visual Representations, default setting in Section 4 and Section 4.2, Figure 8
- Ting Chen et al., A Simple Framework for Contrastive Learning of Visual Representations, Supplementary Appendix B.4
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Lemma 5.1 and Corollary 5.1.1