Abstract
This paper introduces a gradient flow in infinite dimension, whose long-time dynamics is expected to be an approximation of the quantization problem for probability densities, in the sense of Graf and Luschgy (Lecture Notes in Mathematics, vol 1730. Springer, Berlin, 2000). Quantization of probability distributions is a problem which one encounters in a great variety of contexts, such as signal processing, pattern or speech recognition, economics... The present work describes a dynamical approach of the optimal quantization problem in space dimensions one and two, involving (systems of) parabolic equations. This is an account of recent work in collaboration with Caglioti et al. (Math Models Methods Appl Sci 25:1845–1885, 2015 and arXiv:1607.01198 (math.AP), to appear in Ann. Inst. H. Poincaré, Anal. Non Lin. https://doi.org/10.1016/j.anihpc.2017.12.003).
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Notes
- 1.
The Lebesgue measure on \({\mathbf {R}}^d\) is denoted by \({\mathscr {L}}^d\) throughout the present paper.
- 2.
The notation \(T\#m\) designates the push-forward of the measure m by the transformation T.
- 3.
We call “Aristotelian” a mechanical equation based on the axiom that velocity (and not acceleration, as in Newtonian mechanics) is proportional to force. See for instance the following statement: “[...] the medium causes a difference because it impedes the moving thing, most of all if it is moving in the opposite direction, but in a secondary degree even if it is at rest; [...] A, then, will move through B in time \(\Gamma \), and through \({\varDelta }\), which is thinner, in time E (if the length of B is equal to \({\varDelta }\)), in proportion to the density of the hindering body. For let B be water and \({\varDelta }\) air; then by so much as air is thinner and more incorporeal than water, A will move through \({\varDelta }\) faster than through B. [...] Then if air is twice as thin, the body will traverse B in twice the time that it does \({\varDelta }\), and the time \(\Gamma \) will be twice the time E”. (Aristotle, “Physics”, Book IV, Part 8, transl. R. P. Hardie and R. K. Gaye, Clarendon Press, Oxford, 1930).
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Golse, F. (2018). Quantization of Probability Densities: A Gradient Flow Approach. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations . PSPDE 2016. Springer Proceedings in Mathematics & Statistics, vol 258. Springer, Cham. https://doi.org/10.1007/978-3-319-99689-9_6
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