Compactness Hypothesis, Potential Functions, and Rectifying Linear Space in Machine Learning

  • Vadim MottlEmail author
  • Oleg Seredin
  • Olga Krasotkina
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11100)


Emmanuel Braverman was one of the very few thinkers who, during his extremely short life, managed to inseminate several seemingly completely different areas of science. This paper overviews one of the knowledge areas he essentially affected in the sixties years of the last century, namely, the area of Machine Learning. Later, Vladimir Vapnik proposed a more engineering-oriented name of this knowledge area – Estimation of Dependencies Based on Empirical Data. We shall consider these titles as synonyms. The aim of the paper is to briefly trace the way how three notions introduced by Braverman formed the core of the contemporary Machine Learning doctrine. These notions are: (1) compactness hypothesis, (2) potential function, and (3) the rectifying linear space, in which the former two have resulted. There will be little new in this paper. Almost all the constructions we are going to speak about had been published by numerous scientists. The novelty is, perhaps, only in that all these issues will be systematically considered together as immediate consequences of Braveman’s basic principles.


Set of real-world objects Pattern recognition Numerical regression Ordinal regression Compactness hypothesis Precedent-based learning Distance representation modalities Pseudo Euclidean liner space Regularized empirical risk minimization Potential function Distance transformation Selective fusion of distances 



We would like to acknowledge support from grants of the Russian Foundation for Basic Research 14-07-00527, 16-57-52042, 17-07-00436, 17-07-00993, 18-07-01087, 18-07-00942, and from Tula State University within the framework of the scientific project № 2017-62PUBL.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Tula State UniversityTulaRussia
  2. 2.Moscow State UniversityMoscowRussia

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