Operator valued positive definite kernels and differentiable universality

黎曼对称空间上核的普适性

Universality of kernels on Riemannian symmetric spaces

该研究刻画了黎曼对称空间上连续正定不变核的普适性，对紧与非紧对称空间分别给出等价条件，并提供实例说明普适核的构造准则。

On directional derivatives of trace functionals of the form $A\mapsto\Tr(Pf(A))$

Given a function $f:(0,\infty)\rightarrow\RR$ and a positive semidefinite $n\times n$ matrix $P$, one may define a trace functional on positive definite $n\times n$ matrices as $A\mapsto \Tr(Pf(A))$. For differentiable functions $f$, the function $A\mapsto \Tr(Pf(A))$ is differentiable at all positive definite matrices $A$. Under certain continuity conditions on~$f$, this function may be extended to certain non-positive-definite matrices $A$, and the \emph{directional} derivatives of $\Tr(Pf(A)$ may be computed there. This note presents conditions for these directional derivatives to exist and…

Maps on positive definite matrices preserving Bregman and Jensen divergences

In this paper we determine those bijective maps of the set of all positive definite $n\times n$ complex matrices which preserve a given Bregman divergence corresponding to a differentiable convex function that satisfies certain conditions. We cover the cases of the most important Bregman divergences and present the precise structure of the mentioned transformations. Similar results concerning Jensen divergences and their preservers are also given.

岭核平均与一致逼近

Ridge Kernel Averaging and Uniform Approximation

本文针对参数化岭核生成的函数类开发框架，刻画其核为岭原子锥壳一致闭包并给出普适性准则，还分析了均值正定的随机核网络，证明其蒙特卡罗速率与紧集高概率一致界。

Norm inequalities related to operator monotone functions

Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n f(A)|||\leq\|f^{(n)}(A)\|$ and $\|f^{(n)}(\cdot)\|$ is a quasi-convex function on the set of all positive definite operators in $B(H)$. We establish some estimates of the right hand side of some Hermite-Hadamard type inequalities in which differentiable functions are involved, and norms of the maps induced by them on the set of self adjoint operators are convex, quasi-conve…

辛特征值的一阶灵敏度分析

First order sensitivity analysis of symplectic eigenvalues

本文研究正定矩阵辛特征值的一阶灵敏度，证明其方向导数存在并给出表达式，同时分析其Clarke与Michel-Penot次微分性质。