[笔记] Convex Optimization 2015.10.14
来源:互联网 发布:maya编程语言 编辑:程序博客网 时间:2024/05/22 02:24
Existen of SVD
For any
s.t.
- Proof:
such that
where
so
so
Let
- Dual norm:
∥z∥∗=max{xTz:∥x∥≤1} - Fact:
∥⋅∥∗∗=∥⋅∥ - Proof: Sufficient to show for
x s.t.∥x∥=1 .∥x∥∗∗=max{zTx:∥z∥∗≤1}≤1
Still need to show that∥x∥∗∗≥1 - Claim: There exists a
z0 ,∥z0∥∗=1 , such that∥z0∥∗=xTz0 Proof of claim: Let
C={x} ,D={u:∥u∥<1}
ThenC andD are convex, disjoint.
So there existsa∈Rn∖{0},b∈R,s.t.aTx≥b,aTu≤b∀u∈D (⟹∥a∥∗=b )
Letz0=a∥a∥∗=ab ,
then∥z0∥∗=zT0x=∥a∥∗b=1 .Example: Let
A∈Sn++ , then∥x∥A=(xTAx)1/2 is the quadratic norm associated toA .
Have∥x∥A=∥A1/2x∥2 .
IfA=PDPT thenA1/2=PD1/2PT∈Sn++ ∥w∥A∗=max{xTw:∥x∣A≤1} =max{xTw:xTAx≤1} =max{xTw:xTPDPTx≤1} ⟹u=D1/2PTxx=PD−1/2umax{(PD−1/2u)Tw:∥u∥2≤1} =max{uTD−1/2PTw:∥u∥2≤1} =∥D−1/2PTw∥2∗ =∥D−1/2PTw∥2 =∥PTw∥D−1 =((PTw)TD−1PTw)1/2 =(wTPTD−1Pw)1/2
point-set topology
- [笔记] Convex Optimization 2015.10.14
- [笔记] Convex Optimization 2015.10.21
- [笔记] Convex Optimization 2015.10.28
- [笔记] Convex Optimization 2015.09.16
- [笔记] Convex Optimization 2015.09.23
- [笔记] Convex Optimization 2015.09.30
- [笔记] Convex Optimization 2015.11.04
- [笔记] Convex Optimization 2015.11.18
- [笔记] Convex Optimization 2015.11.25
- Convex Optimization
- 【Derivation】Convex Optimization
- convex optimization from stanford
- convex optimization --- affine and convex sets
- 凸优化(Convex Optimization)浅析
- Mathematical background for convex optimization
- 凸优化(Convex Optimization)是什么?
- 笔记:Tensor RPCA: Exact recovery of corrupted low-rank tensors via convex optimization
- 凸优化(convex optimization)第二讲:convex set
- Freemarker中将字符串转换为数字number
- windows wamp运行提示 Fatal error: Call to undefined function curl_init的解决方法
- actionbar设置背景图片和文字颜色
- 运行地址和加载地址
- 第四周 项目1 建立单链表
- [笔记] Convex Optimization 2015.10.14
- 自己写的第一本Linux教程,也小小的班门弄斧一下
- linux学习笔记--centos 安装
- poj 1651 Multiplication Puzzle(区间DP)
- BaiduMap初体验(一) 搭建百度地图的环境
- php header输出类型(备查)
- 内存中堆和栈的区别
- 使用ligerUI遇到未知参数、方法处理办法
- Android之Android Studio 快捷键整理分享