Fundamental theorems

To make the results of scientific research well-founded, introduce the techniques and practices of theorem formulations and proofs in machine learning.

Why does one need to convey an important message, a scientific result, as a theorem?

1. Theorems are the most important messages in the field of research.
2. Theorems present results in the language of mathematics by generality and rigor.
3. Theorems are at the heart of mathematics and play a central role in its aesthetics.

Theorems present the message immediately and leave reasoning for later. The direct narration puts reason first and the results later.

• How does direct narration transform into fast narration?
• How to find, state, and prove theorems in our work?

Both narration styles refer to progressions

1. Textbook: Definition \(\to\) (Axiom set) \(\to\) Theorem \(\to\) Proof \(\to\) Corollaries \(\to\) Examples \(\to\) Impact to applications
2. Scientific discovery: Application problems \(\to\) Problem generalisations \(\to\) Useful algebraic platform \(\to\) Definitions \(\to\) Axiom set

In practice, we mimic the first part of the progression, then learn to discover patterns and formulate theorems.

Theorems of Machine Learning

1. Fundamental theorem of linear algebra S
2. Singular values decomposition and spectral theorem W
3. Gauss–Markov-(Aitken) theorem W
4. Principal component analysis W
5. Karhunen–Loève theorem W
6. Kolmogorov–Arnold representation theorem W
7. Universal approximation theorem by Cybenko W
8. Deep neural network theorem Mark
9. Inverse function theorem and Jacobian W
10. No free lunch theorem by Wolpert W
11. RKHS by Aronszajn and Mercer's theorem W
12. Representer theorem by Schölkopf, Herbrich, and Smola W
13. Convolution theorem (FT, convolution, correlation with CNN examples) W
14. Fourier inversion theorem W
15. Wiener–Khinchin theorem about autocorrelation and spectral decomposition W
16. Parseval's theorem (and uniform, non-uniform convergence) W
17. Probably approximately correct learning with the theorem about compression means learnability
18. Bernstein–von Mises theorem W
19. Holland's schema theorem W
20. Variational approximation
21. Convergence of random variables and Kloek's theorem W
22. Exponential family of distributions and Nelder's theorem
23. Multi-armed bandit theorem
24. Copulas and Sklar's theorem W
25. Boosting theorem Freud, Shapire, 1996, 1995
26. Bootstrap theorem (statistical estimations): Ergodic theorem
27. Miscellaneous BershteinFonMises-1, BershteinFonMises-2, PАС_learning (compression induces learning), PAC_learning_compress

Theorem types

• Uniqueness, existence
• Universality
• ConvergenceYouTube
• Complexity
• Properties of estimations
• Bounds

A paper with theorems includes

1. Introduction: the main message briefly
2. If necessary (it could be introduced during the talk)
   1. Axiom sets
   2. Definitions
   3. Algebraic structures
   4. Notations
3. Theorem formulation and exact proof
   1. The author's variant of the proof could be ameliorated
4. Corollaries
5. Theorem significance and applications

References

Principles

1. Mathematical statistics by A.A. Borovkov, 1998
2. Learning Theory from First Principles by Francis Bach, 2021
3. Theoretical foundations of potential function method in pattern recognition by M. A. Aizerman, E. M. Braverman, and L. I. Rozonoer // Avtomatica i Telemekhanica, 1964. Vol. 25, pp. 917-936.

Proof techniques

1. Proofs and Mathematical Reasoning by Agata Stefanowicz, 2014
2. The nuts and bolts of proofs by Antonella Cupillari, 2013
3. Theorems, Corollaries, Lemmas, and Methods of Proof by Richard J. Rossi, 1956
4. Problem Books in Mathematics by P.R. Halmos (editor), 1990
5. Les contre-exemples en mathématique par Bertrand Hauchecorne, 2007
6. Kolmogorov and Mathematical Logic by Vladimir A. Uspensky // The Journal of Symbolic Logic, Vol. 57, No. 2 (Jun., 1992), 385-412.
7. Что такое аксиоматический метод? В.А. Успенский, 2001
8. Аксиоматический метод. Е.Е. Золин, 2015

Methodology

1. Introduction to Metamathematics by Stephen Cole Kleene, 1950
2. Science and Method by Henri Poincaré, 1908
3. A Summary of Scientific Method by Peter Kosso, 2011
4. Being a Researcher: An Informatics Perspective by Carlo Ghezzi, 2020
5. The definitive glossary of higher mathematical jargon by Math Vault, 2015
6. The definitive guide to learning higher mathematics: 10 principles to mathematical transcendence by Math Vault, 2020
7. List of mathematical jargon on Wikipedia
8. Пикабу. Типичные методы доказательства, 2018 (если вы чувствуете, что несет не туда)

Supplementary material

1. Three works by Greg Yang arXiv:1910.12478, arXiv:2006.14548, arXiv:2009.10685 Youtube Rus
2. Theorems on flows by Johann Brehmera and Kyle Cranmera arXiv:2003.13913v2

• GitHub project to upload your text Intelligent-Systems-Phystech/FundamentalTheoremsML