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Consistency of probability measure quantization by means of power repulsion-attraction potentials

Prof. Massimo Fornasier, TUM Monaco

Sala conferenze IMATI-CNR, Pavia - Martedì 29 Maggio 2018 h.15:00


Abstract. In this talk we present the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The proof of consistency is based on the construction of a target energy functional whose unique minimizer is actually the given probability measure ω to be quantized. Then we show that the discrete functionals, defining the discrete quantizers as their minimizers, actually Γ-converge to the target energy with respect to the narrow topology on the space of probability measures. A key ingredient is the reformulation of the target functional by means of a Fourier representation, which extends the characterization of conditionally positive semi-definite functions from points in generic position to probability measures. As a byproduct of the Fourier representation, we also obtain compactness of sublevels of the target energy in terms of uniform moment bounds, which is used in the asymptotic analysis of corresponding gradient flows (see below). To model situations where the given probability is affected by noise, we additionally consider a modified energy, with the addition of a regularizing total variation term and we investigate again its point mass approximations in terms of Γ-convergence. We show that such a discrete measure representation of the total variation can be interpreted as an additional nonlinear potential, repulsive at a short range, attractive at a medium range, and at a long range not having effect, promoting a uniform distribution of the point masses. We conclude the talk with the well-posedness and asymptotic analysis of the gradient flow of the power repulsion-attraction potential in one dimension.

REFERENCES

M. Fornasier, J. Haškovec and G. Steidl. Consistency of variational continuous-domain quantization via kinetic theory, Appl. Anal., 92(6):1283-1298, 2013.
https://www-m15.ma.tum.de/foswiki/pub/M15/Allgemeines/PublicationsEN/KineticDithering_rev.pdf

M. Di Francesco, M. Fornasier, J.-C. Hütter, D. Matthes, Asymptotic behavior of gradient flows driven by nonlocal power repulsion and attraction potentials in one dimension, SIAM J. Math. Anal., 46(6):3814-3837, 2014.
http://www-m15.ma.tum.de/foswiki/pub/M15/Allgemeines/PublicationsEN/submissionE.pdf

M. Fornasier, and J.-C. Hütter, Consistency of probability measure quantization by means of power repulsion-attraction potentials, Fourier Anal. Appl., 22(3):694-749, 2016.
http://www-m15.ma.tum.de/foswiki/pub/M15/Allgemeines/PublicationsEN/varquant.pdf

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