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In the probability theory field of mathematics, Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces.[1][2] It was first proved by the French mathematician Michel Talagrand.[3] The inequality is one of the manifestations of the concentration of measure phenomenon.[2]
Roughly, the product of the probability to be in some subset of a product space (e.g. to be in one of some collection of states described by a vector) multiplied by the probability to be outside of a neighbourhood of that subspace at least a distance away, is bounded from above by the exponential factor . It becomes rapidly more unlikely to be outside of a larger neighbourhood of a region in a product space, implying a highly concentrated probability density for states described by independent variables, generically. The inequality can be used to streamline optimisation protocols by sampling a limited subset of the full distribution and being able to bound the probability to find a value far from the average of the samples.[4]
- ^ Alon, Noga; Spencer, Joel H. (2000). The Probabilistic Method (2nd ed.). John Wiley & Sons, Inc. ISBN 0-471-37046-0.
- ^ a b Ledoux, Michel (2001). The Concentration of Measure Phenomenon. American Mathematical Society. ISBN 0-8218-2864-9.
- ^ Talagrand, Michel (1995). "Concentration of measure and isoperimetric inequalities in product spaces". Publications Mathématiques de l'IHÉS. 81. Springer-Verlag: 73–205. arXiv:math/9406212. doi:10.1007/BF02699376. ISSN 0073-8301. S2CID 119668709.
- ^ Castelvecchi, Davide (21 March 2024). "Mathematician who tamed randomness wins Abel Prize". Nature. doi:10.1038/d41586-024-00839-6.