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Stochastic Separation Theorems

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journal contribution
posted on 21.08.2017, 09:52 by A. N. Gorban, I. Y. Tyukin
The problem of non-iterative one-shot and non-destructive correction of unavoidable mistakes arises in all Artificial Intelligence applications in the real world. Its solution requires robust separation of samples with errors from samples where the system works properly. We demonstrate that in (moderately) high dimension this separation could be achieved with probability close to one by linear discriminants. Surprisingly, separation of a new image from a very large set of known images is almost always possible even in moderately high dimensions by linear functionals, and coefficients of these functionals can be found explicitly. Based on fundamental properties of measure concentration, we show that for $M1-\vartheta$, where $1>\vartheta>0$ is a given small constant. Exact values of $a,b>0$ depend on the probability distribution that determines how the random $M$-element sets are drawn, and on the constant $\vartheta$. These {\em stochastic separation theorems} provide a new instrument for the development, analysis, and assessment of machine learning methods and algorithms in high dimension. Theoretical statements are illustrated with numerical examples.

Funding

The work was partially supported by Innovate UK (KTP009890 and KTP010522).

History

Citation

Neural Networks, 2017, 94, pp. 255-259

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

AM (Accepted Manuscript)

Published in

Neural Networks

Publisher

Elsevier for European Neural Network Society (ENNS), International Neural Network Society (INNS), Japanese Neural Network Society (JNNS)

issn

0893-6080

eissn

1879-2782

Acceptance date

21/07/2017

Copyright date

2017

Available date

31/07/2018

Publisher version

http://www.sciencedirect.com/science/article/pii/S0893608017301776?via=ihub

Notes

The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.

Language

en