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

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 already found applications in the
asymptotic analysis of corresponding gradient ﬂows. 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.
http://cvgmt.sns.it/seminar/345/
            

When	Wed Feb 19, 2014 4pm – 5pm Coordinated Universal Time
Where	Aula Riunioni - Department of Mathematics, University of Pisa (map)