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Register a new userLog-majorization related to Renyi divergences
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Fumio Hiai
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For $alpha,z>0$ with $alpha e1$, motivated by comparison between different kinds of Renyi divergences in quantum information, we consider log-majorization between the matrix functions begin{align*} P_alpha(A,B)&:=B^{1/2}(B^{-1/2}AB^{-1/2})^alpha B^{1/2}, Q_{alpha,z}(A,B)&:=(B^{1-alphaover2z}A^{alphaover z}B^{1-alphaover2z})^z end{align*} of two positive (semi)definite matrices $A,B$. We precisely determine the parameter $alpha,z$ for which $P_alpha(A,B)prec_{log}Q_{alpha,z}(A,B)$ and $Q_{alpha,z}(A,B)prec_{log}P_alpha(A,B)$ holds, respectively.
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Denote by $P_n$ the set of $ntimes n$ positive definite matrices. Let $D = D_1oplus dots oplus D_k$, where $D_1in P_{n_1}, dots, D_k in P_{n_k}$ with $n_1+cdots + n_k=n$. Partition $Cin P_n$ according to $(n_1, dots, n_k)$ so that $Diag C = C_1oplus dots oplus C_k$. We prove the following weak log majorization result: begin{equation*} lambda (C^{-1}_1D_1oplus cdots oplus C^{-1}_kD_k)prec_{w ,log} lambda(C^{-1}D), end{equation*} where $lambda(A)$ denotes the vector of eigenvalues of $Ain Cnn$. The inequality does not hold if one replaces the vectors of eigenvalues by the vectors of singular values, i.e., begin{equation*} s(C^{-1}_1D_1oplus cdots oplus C^{-1}_kD_k)prec_{w ,log} s(C^{-1}D) end{equation*} is not true. As an application, we provide a generalization of a determinantal inequality of Matic cite[Theorem 1.1]{M}. In addition, we obtain a weak majorization result which is complementary to a determinantal inequality of Choi cite[Theorem 2]{C} and give a weak log majorization open question.
The purpose of this paper is two-fold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua-Marcus inequalities. Our results are stronger and more general than the existing ones.
Motivated by Horns log-majorization (singular value) inequality $s(AB)underset{log}{prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)underset{w}{prec} s(A)*s(B)$ for square complex matrices, we consider their Hermitian analogs $lambda(sqrt{A}Bsqrt{A}) underset{log}{prec} lambda(A)*lambda(B)$ for positive semidefinite matrices and $lambda(|Acirc B|) underset{w}{prec} lambda(|A|)*lambda(|B|)$ for general (Hermitian) matrices, where $Acirc B$ denotes the Jordan product of $A$ and $B$ and $*$ denotes the componentwise product in $R^n$. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form $lambdabig (P_{sqrt{a}}(b)big )underset{log}{prec} lambda(a)*lambda(b)$ for $a,bgeq 0$ and $lambdabig (|acirc b|big )underset{w}{prec} lambda(|a|)*lambda(|b|)$ for all $a$ and $b$, where $P_u$ and $lambda(u)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element $u$. We also describe inequalities of the form $lambda(|Abullet b|)underset{w}{prec} lambda({mathrm{diag}}(A))*lambda(|b|)$, where $A$ is a real symmetric positive semidefinite matrix and $A,bullet, b$ is the Schur product of $A$ and $b$. In the form of an application, we prove the generalized H{o}lder type inequality $||acirc b||_pleq ||a||_r,||b||_s$, where $||x||_p:=||lambda(x)||_p$ denotes the spectral $p$-norm of $x$ and $p,q,rin [1,infty]$ with $frac{1}{p}=frac{1}{r}+frac{1}{s}$. We also give precise values of the norms of the Lyapunov transformation $L_a$ and $P_a$ relative to two spectral $p$-norms.
In this paper, we develop a novel way to perturbatively calculate Renyi relative divergences $D_{gamma}(rho|| sigma) ={rm tr} rho^{gamma} sigma^{1-gamma}$ and related quantities without using replica trick as well as analytic continuation. We explicitly determine the form of the perturbative term at any order by an integral along the modular flow of the unperturbed state. By applying the prescription to a class of reduced density matrices in conformal field theory, we find that the second order term of certain linear combination of the divergences has a holographic expression in terms of bulk symplectic form, which is a one parameter generalization of the statement Fisher information = Bulk canonical energy.
We derive a new variational formula for the Renyi family of divergences, $R_alpha(Q|P)$, between probability measures $Q$ and $P$. Our result generalizes the classical Donsker-Varadhan variational formula for the Kullback-Leibler divergence. We further show that this Renyi variational formula holds over a range of function spaces; this leads to a formula for the optimizer under very weak assumptions and is also key in our development of a consistency theory for Renyi divergence estimators. By applying this theory to neural-network estimators, we show that if a neural network family satisfies one of several strengthen
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