496 research outputs found
Quantum R\'enyi and -divergences from integral representations
Smooth Csisz\'ar -divergences can be expressed as integrals over so-called
hockey stick divergences. This motivates a natural quantum generalization in
terms of quantum Hockey stick divergences, which we explore here. Using this
recipe, the Kullback-Leibler divergence generalises to the Umegaki relative
entropy, in the integral form recently found by Frenkel. We find that the
R\'enyi divergences defined via our new quantum -divergences are not
additive in general, but that their regularisations surprisingly yield the Petz
R\'enyi divergence for and the sandwiched R\'enyi divergence for
, unifying these two important families of quantum R\'enyi
divergences. Moreover, we find that the contraction coefficients for the new
quantum divergences collapse for all that are operator convex,
mimicking the classical behaviour and resolving some long-standing conjectures
by Lesniewski and Ruskai. We derive various inequalities, including new reverse
Pinsker inequalites with applications in differential privacy and also explore
various other applications of the new divergences.Comment: 44 pages. v2: improved results on reverse Pinsker inequalities +
minor clarification
-Divergence Inequalities via Functional Domination
This paper considers derivation of -divergence inequalities via the
approach of functional domination. Bounds on an -divergence based on one or
several other -divergences are introduced, dealing with pairs of probability
measures defined on arbitrary alphabets. In addition, a variety of bounds are
shown to hold under boundedness assumptions on the relative information. The
journal paper, which includes more approaches for the derivation of
f-divergence inequalities and proofs, is available on the arXiv at
https://arxiv.org/abs/1508.00335, and it has been published in the IEEE Trans.
on Information Theory, vol. 62, no. 11, pp. 5973-6006, November 2016.Comment: A conference paper, 5 pages. To be presented in the 2016 ICSEE
International Conference on the Science of Electrical Engineering, Nov.
16--18, Eilat, Israel. See https://arxiv.org/abs/1508.00335 for the full
paper version, published as a journal paper in the IEEE Trans. on Information
Theory, vol. 62, no. 11, pp. 5973-6006, November 201
On the equivalence of modes of convergence for log-concave measures
An important theme in recent work in asymptotic geometric analysis is that
many classical implications between different types of geometric or functional
inequalities can be reversed in the presence of convexity assumptions. In this
note, we explore the extent to which different notions of distance between
probability measures are comparable for log-concave distributions. Our results
imply that weak convergence of isotropic log-concave distributions is
equivalent to convergence in total variation, and is further equivalent to
convergence in relative entropy when the limit measure is Gaussian.Comment: v3: Minor tweak in exposition. To appear in GAFA seminar note
ON THE JENSEN-SHANNON DIVERGENCE AND THE VARIATION DISTANCE FOR CATEGORICAL PROBABILITY DISTRIBUTIONS
We establish a decomposition of the Jensen-Shannon divergence into a linear combination of a scaled Jeffreys' divergence and a reversed Jensen-Shannon divergence. Upper and lower bounds for the Jensen-Shannon divergence are then found in terms of the squared (total) variation distance. The derivations rely upon the Pinsker inequality and the reverse Pinsker inequality. We use these bounds to prove the asymptotic equivalence of the maximum likelihood estimate and minimum Jensen-Shannon divergence estimate as well as the asymptotic consistency of the minimum Jensen-Shannon divergence estimate. These are key properties for likelihood-free simulator-based inference.Peer reviewe
ON THE JENSEN-SHANNON DIVERGENCE AND THE VARIATION DISTANCE FOR CATEGORICAL PROBABILITY DISTRIBUTIONS
We establish a decomposition of the Jensen-Shannon divergence into a linear combination of a scaled Jeffreys' divergence and a reversed Jensen-Shannon divergence. Upper and lower bounds for the Jensen-Shannon divergence are then found in terms of the squared (total) variation distance. The derivations rely upon the Pinsker inequality and the reverse Pinsker inequality. We use these bounds to prove the asymptotic equivalence of the maximum likelihood estimate and minimum Jensen-Shannon divergence estimate as well as the asymptotic consistency of the minimum Jensen-Shannon divergence estimate. These are key properties for likelihood-free simulator-based inference.Peer reviewe
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