Alphabeta MathSelf-supervised and contrastive representations
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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
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
Lemma
Convexity, Lipschitz bounds, and oscillation of the source contrastive losses
Lemma
A finite-dimensional vector contraction bound for one-sided Rademacher averages
Definition
Latent-class contrastive representation learning
Lemma
Jensen comparison with the average mean-classifier loss
Lemma
Class-collision loss is bounded by within-class covariance deviation
Lemma
Class-conditional sub-Gaussian concentration yields a contrastive margin bound
Lemma
Uniform generalization bound for the one-negative contrastive loss
Lemma
Population contrastive excess of an approximate empirical minimizer
Theorem
Contrastive transfer bound with class collisions
Corollary
Contrastive transfer for an approximate empirical minimizer
Theorem
Contrastive transfer with a within-class covariance penalty
Corollary
Sub-Gaussian contrastive transfer with a mean-classifier margin loss
Sources
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Section 2.2
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025, Lectures 15–16
- Andreas Maurer, A vector-contraction inequality for Rademacher complexities, Theorem 3 and Corollary 4
- Yunwen Lei, Tianbao Yang, Yiming Ying, and Ding-Xuan Zhou, Generalization Analysis for Contrastive Representation Learning, Lemma A.1
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Sections 2–5
- Yunwen Lei, Tianbao Yang, Yiming Ying, and Ding-Xuan Zhou, Generalization Analysis for Contrastive Representation Learning, Sections 2–4
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Lemma 4.3 and proof
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Equation (8), Equation (9), and Lemma 4.4
- Nikunj Saunshi et al., Contrastive Unsupervised Representation Learning, Supplementary Lemma A.1 and proof
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Lemma 5.1 and Supplementary Appendix A.2
- Nikunj Saunshi et al., Supplementary Appendix A.2, complete proof of Lemma 5.1
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Supplementary Appendix A.3
- Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar, Foundations of Machine Learning, 2nd ed., Theorem 3.3 and complete proof
- Andreas Maurer, A vector-contraction inequality for Rademacher complexities, Theorem 3, Corollary 4, and complete proof
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Theorem 4.1 and proof route
- Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar, Foundations of Machine Learning, 2nd ed., empirical-to-population risk argument in Chapter 3
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Theorem 4.1, Lemmas 4.3–4.4 and proof route
- Nikunj Saunshi et al., Supplementary Appendices A.1 and A.3, complete collision and generalization arguments
- Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar, Foundations of Machine Learning, 2nd ed., Chapter 3 empirical-risk comparison
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Theorem 4.5, Lemma 4.4 and proof route
- Nikunj Saunshi et al., Supplementary Appendices A.1 and A.3
- Nikunj Saunshi et al., A Theoretical Analysis of Contrastive Unsupervised Representation Learning, Corollary 5.1.1 and Lemma 5.1