Quantization of Probability Densities: A Gradient Flow Approach
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).
KeywordsQuantization of probability densities Wasserstein distance Gradient flow Parabolic equations
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