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Register a new userNon-commutative R{e}nyi Entropic Uncertainty Principles
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Added by
Zhengwei Liu
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In this paper, we calculate the norm of the string Fourier transform on subfactor planar algebras and characterize the extremizers of the inequalities for parameters $0<p,qleq infty$. Furthermore, we establish R{e}nyi entropic uncertainty principles for subfactor planar algebras.
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In this article, we prove various smooth uncertainty principles on von Neumann bi-algebras, which unify numbers of uncertainty principles on quantum symmetries, such as subfactors, and fusion bi-algebras etc, studied in quantum Fourier analysis. We also obtain Widgerson-Wigderson type uncertainty principles for von Neumann bi-algebras. Moreover, we give a complete answer to a conjecture proposed by A. Wigderson and Y. Wigderson.
We investigate the use of parametrized families of information-theoretic measures to generalize the loss functions of generative adversarial networks (GANs) with the objective of improving performance. A new generator loss function, called least $k$th-order GAN (L$k$GAN), is first introduced, generalizing the least squares GANs (LSGANs) by using a $k$th order absolute error distortion measure with $k geq 1$ (which recovers the LSGAN loss function when $k=2$). It is shown that minimizing this generalized loss function under an (unconstrained) optimal discriminator is equivalent to minimizing the $k$th-order Pearson-Vajda divergence. Another novel GAN generator loss function is next proposed in terms of R{e}nyi cross-entropy functionals with order $alpha >0$, $alpha eq 1$. It is demonstrated that this R{e}nyi-centric generalized loss function, which provably reduces to the original GAN loss function as $alphato1$, preserves the equilibrium point satisfied by the original GAN based on the Jensen-R{e}nyi divergence, a natural extension of the Jensen-Shannon divergence. Experimental results indicate that the proposed loss functions, applied to the MNIST and CelebA datasets, under both DCGAN and StyleGAN architectures, confer performance benefits by virtue of the extra degrees of freedom provided by the parameters $k$ and $alpha$, respectively. More specifically, experiments show improvements with regard to the quality of the generated images as measured by the Frechet Inception Distance (FID) score and training stability. While it was applied to GANs in this study, the proposed approach is generic and can be used in other applications of information theory to deep learning, e.g., the issues of fairness or privacy in artificial intelligence.
The purpose of this short note was to outline the current status, then in 2011, of some research programs aiming at a categorification of parts of A.Connes non-commutative geometry and to provide an outlook on some possible subsequent developments in categorical non-commutative geometry.
In this paper we have successfully established (from first principles) that anyons do live in a 2-dimensional {it{noncommutative}} space. We have directly computed the non-trivial uncertainty relation between anyon coordinates, ${sqrt{Delta x^2Delta y^2}}=Thetasigma$, using the recently constructed anyon wave function [J. Majhi, S. Ghosh and S.K. Maiti, Phys. Rev. Lett. textbf{123}, 164801 (2019)] cite{jan}, in the framework of I. Bialynicki-Birula and Z. Bialynicka-Birula, New J. Phys. textbf{21}, 07306 (2019) cite{bel}. Furthermore we also compute the Heisenberg uncertainty relation for anyon and as a consistency check, show that the results of cite{bel} prove that, as expected, electrons live in 3-dimensional commutative space.
In this work, we establish an exact relation which connects the heat exchange between two systems initialized in their thermodynamic equilibrium states at different temperatures and the R{e}nyi divergences between the initial thermodynamic equilibrium state and the final non-equilibrium state of the total system. The relation tells us that the various moments of the heat statistics are determined by the Renyi divergences between the initial equilibrium state and the final non-equilibrium state of the global system. In particular the average heat exchange is quantified by the relative entropy between the initial equilibrium state and the final non-equilibrium state of the global system. The relation is applicable to both finite classical systems and finite quantum systems.
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