Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine L

 

 

未经做者容许，禁止转载： BooTurbo https://www.cnblogs.com/booturbo/p/14124207.html

 

Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning

翻译书名：计算机科学和机器学习中的数学——代数，拓扑，微积分及优化理论

 

目录

 

1 Introduction

      序言

2 Groups, Rings, and Fields

      群，环，域

2.1 Groups, Subgroups, Cosets

      群，子群，陪集

2.2 Cyclic Groups

      循环群

2.3 Rings and Fields

      环，域

 

Ⅰ Linear Algebra

      线性代数

3 Vector Spaces, Bases, Linear Maps

      向量空间，基，线性变换

3.1 Motivations: Linear Combinations, Linear Independence, Rank

      动机：线性组合，线性无关，秩

3.2 Vector Spaces

      向量空间

3.3 Indexed Families; the Sum Notation

      索引族，求和符号

3.4 Linear Independence, Subspaces

      线性无关，子空间

3.5 Bases of a Vector Space

      向量空间的基

3.6 Matrices

      矩阵

3.7 Linear Maps

      线性变换

3.8 Quotient Spaces

      商空间

3.9 Linear Forms and the Dual Space

      线性泛函，对偶空间

4 Matrices and Linear Maps

      矩阵与线性变换

4.1 Representation of Linear Maps by Matrices

      以矩阵形式表示线性变换

4.2 Composition of Linear Maps and Matrix Multiplication

      线性变换与矩阵乘法的组合

4.3 Change of Basis Matrix

      基变换矩阵

4.4 The Effect of a Change of Bases on Matrices

      基变换对矩阵的影响

5 Haar Bases, Haar Wavelets, Hadamard Matrices

      哈尔基，哈尔小波，阿达马矩阵

5.1 Introduction to Signal Compression Using Haar Wavelets

      使用哈尔小波进行信号压缩的相关介绍

5.2 Haar Matrices, Scaling Properties of Haar Wavelets

      哈尔矩阵，哈尔小波的尺度属性

5.3 Kronecker Product Construction of Haar Matrices

      哈尔矩阵的克罗内克积构造

5.4 Multiresolution Signal Analysis with Haar Bases

      使用哈尔基进行多分辨率信号分析

5.5 Haar Transform for Digital Images

      应用于数字图像的哈尔变换

5.6 Hadamard Matrices

      阿达马矩阵

6 Direct Sums

      直和

6.1 Sums, Direct Sums, Direct Products

      求和，直和，直积

6.2 The Rank-Nullity Theorem; Grassmann's Relation

      秩-零化度定理，格拉斯曼关系

7 Determinants

      行列式

7.1 Permutations, Signature of a Permutation

      排列，排列的符号

7.2 Alternating Multilinear Maps

      交替多重线性映射

7.3 Definition of a Determinant

      行列式的定义

7.4 Inverse Matrices and Determinants

      逆矩阵与行列式

7.5 Systems of Linear Equations and Determinants

      线性方程组与行列式

7.6 Determinant of a Linear Map

      线性映射的行列式

7.7 The Cayley-Hamilton Theorem

      凯莱-哈密顿定理

7.8 Permanents

      积和式

7.9 Summary

      总结

7.10 Further Readings

      深刻阅读

7.11 Problems

      问题

8 Gaussian Elimination, LU, Cholesky, Echelon Form

      高斯消元法，LU分解法，Cholesky分解，阶梯形矩阵

8.1 Motivating Example: Curve Interpolation

      动机示例：曲线插值

8.2 Gaussian Elimination

      高斯消元法

8.3 Elementary Matrices and Row Operations

      初等矩阵与行运算

8.4 LU-Factorization

      LU-分解因式

8.5 PA = LU Factorization

      PA等于LU分解因式

8.6 Proof of Theorem 8.5 

      定理8.5的证实

8.7 Dealing with Roundoff Errors; Pivoting Strategies

      处理舍入偏差，主元消去法

8.8 Gaussian Elimination of Tridiagonal Matrices

      三对角矩阵的高斯消元

8.9 SPD Matrices and the Cholesky Decomposition

      对称正定矩阵与Cholesky 分解

8.10 Reduced Row Echelon Form

      简化行阶梯形矩阵

8.11 RREF, Free Variables, Homogeneous Systems

      简化行阶梯形矩阵，自由变量，齐次线性方程组

8.12 Uniqueness of RREF

      简化行阶梯形矩阵的独特性

8.13 Solving Linear Systems Using RREF

      使用RREF求解线性方程组

8.14 Elementary Matrices and Columns Operations

      初等矩阵与列运算

8.15 Transvections and Dilatations

      错切与膨胀

9 Vector Norms and Matrix Norms

      向量范数和矩阵范数

9.1 Normed Vector Spaces

      赋范向量空间

9.2 Matrix Norms

     矩阵范数

9.3 Subordinate Norms

      从属范数

9.4 Inequalities Involving Subordinate Norms

      从属范数相关的不等式

9.5 Condition Numbers of Matrices

      矩阵的条件数

9.6 An Application of Norms: Inconsistent Linear Systems

      范数的应用之一：不相容线性方程组

9.7 Limits of Sequences and Series

      数列与级数的极限

9.8 The Matrix Exponential

      矩阵指数

10 Iterative Methods for Solving Linear Systems

      用于求解线性方程组的迭代法

10.1 Convergence of Sequences of Vectors and Matrices

      向量和矩阵序列的收敛

10.2 Convergence of Iterative Methods

       迭代法的收敛

10.3 Methods of Jacobi, Gauss-Seidel, and Relaxation

       雅可比法，高斯-赛德尔迭代法，松弛法

10.4 Convergence of the Methods

       这些方法的收敛

10.5 Convergence Methods for Tridiagonal Matrices

       三对角矩阵的收敛法

11 The Dual Space and Duality

       对偶空间及对偶

11.1 The Dual Space E* and Linear Forms

       对偶空间和线性泛函

11.2 Pairing and Duality Between E and E*     

       E 和 E* 之间的配对与对偶

11.3 The Duality Theorem and Some Consequences 

       对偶定理和一些结论

11.4 The Bidual and Canonical Pairings 

       双对偶和标准配对

11.5 Hyperplanes and Linear Forms

       超平面和线性泛函

11.6 Transpose of a Linear Map and of a Matrix

       线性映射的转置及矩阵的转置

11.7 Properties of the Double Transpose

       双重转置的属性

11.8 The Four Fundamental Subspaces

       四个基本子空间

12 Euclidean Spaces

       欧几里得空间

12.1 Inner Products, Euclidean Spaces

       内积，欧几里得空间

12.2 Orthogonality and Duality in Euclidean Spaces

       欧几里得空间中的正交和对偶

12.3 Adjoint of a Linear Map

       线性映射的伴随

12.4 Existence and Construction of Orthonormal Bases

       标准正交基的存在与构造

12.5 Linear Isometries (Orthogonal Transformations)

       线性等距同构（正交变换）

12.6 The Orthogonal Group, Orthogonal Matrices

       正交群，正交矩阵

12.7 The Rodrigues Formula

       罗德里格公式

12.8 QR-Decomposition for Invertible Matrices

       用于可逆矩阵的QR分解

12.9 Some Applications of Euclidean Geometry

       欧几里得几何的一些应用

13 QR-Decomposition for Arbitrary Matrices

       用于任意矩阵的QR分解

13.1 Orthogonal Reflections

       正交映射

13.2 QR-Decomposition Using Householder Matrices

       使用豪斯霍尔德矩阵进行QR分解

14 Hermitian Spaces

       埃尔米特空间

14.1 Hermitian Spaces, Pre-Hilbert Spaces

       埃尔米特空间，准希尔伯特空间

14.2 Orthogonality, Duality, Adjoint of a Linear Map

       线性映射的正交，对偶，伴随

14.3 Linear Isometries (Also Called Unitary Transformations)

       线性等距同构（又称做幺正变换）

14.4 The Unitary Group, Unitary Matrices

       酉群，酉矩阵（幺正矩阵）

14.5 Hermitian Reflections and QR-Decomposition

       埃尔米特映射和QR分解

14.6 Orthogonal Projections and Involutions

       正交投影与对合

14.7 Dual Norms

       对偶范数

15 Eigenvectors and Eigenvalues

        特征向量和特征值

15.1 Eigenvectors and Eigenvalues of a Linear Map

       线性变换的特征向量和特征值

15.2 Reduction to Upper Triangular Form

       简化成上三角形

15.3 Location of Eigenvalues

       特征值的位置

15.4 Conditioning of Eigenvalue Problems

       特征值问题的调节

15.5 Eigenvalues of the Matrix Exponential

       矩阵指数的特征值

16 Unit Quaternions and Rotations in SO(3)

       SO(3)中的单位四元数和旋转

16.1 The Group SU(2) and the Skew Field H of Quaternions

        SU(2)群 和 四元数的除环H

16.2 Representation of Rotation in SO(3) By Quaternions in SU(2)    

       以SU(2)中的四元数来表示SO(3)中的旋转

16.3 Matrix Representation of the Rotation r_q

       旋转r_q 的矩阵表示

16.4 An Algorithm to Find a Quaternion Representing a Rotation

       一种找出一个四元数来表示旋转的算法

16.5 The Exponential Map exp : su(2) → SU(2)

       指数映射exp: su(2) → SU(2)

16.6 Quaternion Interpolation 

       四元数插值

16.7 Nonexistence of a “Nice” Section from SO(3) to SU(2)       

       在SO(3)和SU(2)之间不存在优选

17 Spectral Theorems

       谱定理

17.1 Introduction

       介绍

17.2 Normal Linear Maps: Eigenvalues and Eigenvectors

       正规线性映射：特征值和特征向量

17.3 Spectral Theorem for Normal Linear Maps

       用于正规线性映射的谱定理

17.4 Self-Adjoint and Other Special Linear Maps

       自伴随和其余特殊线性映射

17.5 Normal and Other Special Matrices

       正规算子和其余特殊矩阵

17.6 Rayleigh–Ritz Theorems and Eigenvalue Interlacing

       瑞利里兹定理和特征值交错

17.7 The Courant–Fischer Theorem; Perturbation Results

       最大最小定理；摄动理论

18 Computing Eigenvalues and Eigenvectors

       计算特征值和特征向量

18.1 The Basic QR Algorithm

       基本QR算法

18.2 Hessenberg Matrices

       黑森贝格矩阵

18.3 Making the QR Method More Efficient Using Shifts

       使用移位使QR方法更高效

18.4 Krylov Subspaces; Arnoldi Iteration

       Krylov子空间；Arnoldi迭代法

18.5 GMRES

       广义最小残量方法

18.6 The Hermitian Case; Lanczos Iteration

       埃尔米特情形；兰乔斯迭代法

18.7 Power Methods

       幂迭代算法

19 Introduction to The Finite Elements Method

       介绍有限元方法

19.1 A One-Dimensional Problem: Bending of a Beam

       一维问题：梁弯曲

19.2 A Two-Dimensional Problem: An Elastic Membrane

       二维问题：弹性膜

19.3 Time-Dependent Boundary Problems

       时间依赖边界问题

20 Graphs and Graph Laplacians; Basic Facts

       图和图拉普拉斯；基本事实       

20.1 Directed Graphs, Undirected Graphs, Weighted Graphs

       有向图，无向图，加权图

20.2 Laplacian Matrices of Graphs

       图的拉普拉斯矩阵

20.3 Normalized Laplacian Matrices of Graphs

       图的归一化拉普拉斯矩阵

20.4 Graph Clustering Using Normalized Cuts

       使用归一化割进行图聚类

21 Spectral Graph Drawing

       谱图绘制

21.1 Graph Drawing and Energy Minimization

        图绘制和能量最小化

21.2 Examples of Graph Drawings

        图绘制的示例

22 Singular Value Decomposition and Polar Form

        奇异值分解和极式

22.1 Properties of f* ◦ f

        f* ◦ f 的性质

22.2 Singular Value Decomposition for Square Matrices

       用于方块矩阵的奇异值分解

22.3 Polar Form for Square Matrices

       方块矩阵的极式

22.4 Singular Value Decomposition for Rectangular Matrices

       长方阵的奇异值分解

22.5 Ky Fan Norms and Schatten Norms

       Ky Fan 范数和 Schatten范数

23 Applications of SVD and Pseudo-Inverses

       奇异值分解和伪逆的应用

23.1 Least Squares Problems and the Pseudo-Inverse

       最小二乘问题和伪逆

23.2 Properties of the Pseudo-Inverse

       伪逆的性质

23.3 Data Compression and SVD

       数据压缩和奇异值分解

23.4 Principal Components Analysis (PCA)

       主成分分析

23.5 Best Affine Approximation

       最佳仿射逼近

 

II Affine and Projective Geometry

       仿射与射影几何

24 Basics of Affine Geometry

       仿射几何基础

24.1 Affine Spaces

       仿射空间

24.2 Examples of Affine Spaces

       仿射空间示例

24.3 Chasles’s Identity

       查理特征（定理）

24.4 Affine Combinations, Barycenters

       仿射组合，质心

24.5 Affine Subspaces

       仿射子空间

24.6 Affine Independence and Affine Frames

       仿射无关性 和 仿射标架

24.7 Affine Maps

       仿射映射

24.8 Affine Groups

       仿射群

24.9 Affine Geometry: A Glimpse

       仿射几何学一览

24.10 Affine Hyperplanes

       仿射超平面

24.11 Intersection of Affine Spaces

       交叉仿射空间

25 Embedding an Affine Space in a Vector Space

       在向量空间中嵌入仿射空间

25.1 The “Hat Construction,” or Homogenizing

       帽构造 或 均质化

25.2 Affine Frames of E and Bases of Ê

       E的仿射标架和 Ê的基

25.3 Another Construction of Ê

       Ê 的另外一种构造       

25.4 Extending Affine Maps to Linear Maps

       将仿射映射拓展到线性映射中

26 Basics of Projective Geometry

       射影几何基础

26.1 Why Projective Spaces?

       为何是射影空间

26.2 Projective Spaces

       射影空间

26.3 Projective Subspaces

       射影子空间

26.4 Projective Frames

       射影框架（坐标系）

26.5 Projective Maps

       射影变换

26.6 Finding a Homography Between Two Projective Frames

       在两个射影坐标系之间找出一个单应性矩阵

26.7 Affine Patches

       仿射快

26.8 Projective Completion of an Affine Space

       仿射空间的射影闭合

26.9 Making Good Use of Hyperplanes at Infinity

       善于利用无限远超平面

26.10 The Cross-Ratio

       交比

26.11 Fixed Points of Homographies and Homologies

       单应性和透射的不动点

26.12 Duality in Projective Geometry

       射影几何中的对偶

26.13 Cross-Ratios of Hyperplanes

       超平面的交比

26.14 Complexification of a Real Projective Space

       复化实射影空间

26.15 Similarity Structures on a Projective Space

       射影空间上的类似结构

26.16 Some Applications of Projective Geometry

       射影几何的一些应用

 

III The Geometry of Bilinear Forms

       双线性型几何学

27 The Cartan–Dieudonné Theorem

       嘉当-迪厄多内定理

27.1 The Cartan–Dieudonné Theorem for Linear Isometries

       用于线性等距同构（变换）的嘉当-迪厄多内定理

27.2 Affine Isometries (Rigid Motions)

       仿射等距变换（刚体运动）

27.3 Fixed Points of Affine Maps

       仿射映射的不动点

27.4 Affine Isometries and Fixed Points

       仿射等距变换与不动点

27.5 The Cartan–Dieudonné Theorem for Affine Isometries

       用于仿射等距变换的嘉当-迪厄多内定理

28 Isometries of Hermitian Spaces

       埃尔米特空间的等距变换

28.1 The Cartan–Dieudonné Theorem, Hermitian Case

       嘉当-迪厄多内定理，埃尔米特情形

28.2 Affine Isometries (Rigid Motions)

       仿射等距变换（刚体运动）

29 The Geometry of Bilinear Forms; Witt’s Theorem

       双线性型几何；维特定理

29.1 Bilinear Forms

       双线性型

29.2 Sesquilinear Forms

       半双线性型

29.3 Orthogonality

       正交

29.4 Adjoint of a Linear Map

       伴随线性变换

29.5 Isometries Associated with Sesquilinear Forms

       有关半双线性型的等距变换

29.6 Totally Isotropic Subspaces

       全迷向子空间

29.7 Witt Decomposition

       维特分解

29.8 Symplectic Groups

       辛群

29.9 Orthogonal Groups and the Cartan–Dieudonné Theorem

       正交群与嘉当-迪厄多内定理

29.10 Witt’s Theorem

       维特定理

 

 

IV Algebra: PID’s, UFD’s, Noetherian Rings, Tensors, Modules over a PID, Normal Forms

       代数：主理想整环，惟一分解整环，诺特环，张量，主理想整环上的模，范式（标准型）

30 Polynomials, Ideals and PID’s

       多项式，环论中的（理想）和主理想整环

30.1 Multisets

       多重集

30.2 Polynomials

       多项式

30.3 Euclidean Division of Polynomials

       多项式的欧几里得除法

30.4 Ideals, PID’s, and Greatest Common Divisors

       理想，主理想整环及最大公约数

30.5 Factorization and Irreducible Factors in K[X]

       K[X] 中的因式分解和不可约因子

30.6 Roots of Polynomials

       多项式的根

30.7 Polynomial Interpolation (Lagrange, Newton, Hermite)

       多项式插值（拉格朗日，牛顿，埃尔米特）

31 Annihilating Polynomials; Primary Decomposition

       零化多项式；准素分解

31.1 Annihilating Polynomials and the Minimal Polynomial

        零化多项式和极小多项式

31.2 Minimal Polynomials of Diagonalizable Linear Maps

       可对角化线性映射的极小多项式

31.3 Commuting Families of Linear Maps

       线性映射的交换族

31.4 The Primary Decomposition Theorem

        准素分解定理

31.5 Jordan Decomposition

        若尔当分解

31.6 Nilpotent Linear Maps and Jordan Form

        幂零线性变换和若尔当形式

32 UFD’s, Noetherian Rings, Hilbert’s Basis Theorem

       惟一分解整环，诺特环，希尔伯特基定理

32.1 Unique Factorization Domains (Factorial Rings)

       惟一分解整环（析因环/惟一分解环）

32.2 The Chinese Remainder Theorem

       中国剩余定理（孙子定理）

32.3 Noetherian Rings and Hilbert’s Basis Theorem

       诺特环和希尔伯特基定理

32.4 Futher Readings

        深刻阅读

33 Tensor Algebras

       张量代数

33.1 Linear Algebra Preliminaries: Dual Spaces and Pairings

       线性代数预备知识：对偶空间和配对

33.2 Tensors Products

       张量积

33.3 Bases of Tensor Products

       张量积的基

33.4 Some Useful Isomorphisms for Tensor Products

       一些对于张量积有用的同构

33.5 Duality for Tensor Products

       用于张量积的对偶

33.6 Tensor Algebras

       张量代数

33.7 Symmetric Tensor Powers

       对称张量幂

33.8 Bases of Symmetric Powers

       对称幂的基

33.9 Some Useful Isomorphisms for Symmetric Powers

       一些对于对称幂有用的同构

33.10 Duality for Symmetric Powers

       用于对称幂的对偶

33.11 Symmetric Algebras

       对称代数

34 Exterior Tensor Powers and Exterior Algebras

       外张量幂和外代数

34.1 Exterior Tensor Powers

       外张量幂

34.2 Bases of Exterior Powers

       外幂的基

34.3 Some Useful Isomorphisms for Exterior Powers

       一些对于外幂有用的同构

34.4 Duality for Exterior Powers

       用于外幂的对偶

34.5 Exterior Algebras

       外代数

34.6 The Hodge ∗-Operator

       霍奇星算子

34.7 Left and Right Hooks

       左右弯钩

34.8 Testing Decomposability

       测试可分解性

34.9 The Grassmann-Plücker’s Equations and Grassmannians

       格拉斯曼-普吕克方程 和 格拉斯曼流形

34.10 Vector-Valued Alternating Forms

       向量值交错型

35 Introduction to Modules; Modules over a PID

       模介绍；主理想整环上的模

35.1 Modules over a Commutative Ring

       交换环上的模

35.2 Finite Presentations of Modules

       有限表现的模

35.3 Tensor Products of Modules over a Commutative Ring

       交换环上的模张量积

35.4 Torsion Modules over a PID; Primary Decomposition

       主理想整环上的挠模；准素分解

35.5 Finitely Generated Modules over a PID

       主理想整环上的有限生成模

35.6 Extension of the Ring of Scalars

       标量环的扩张

36 Normal Forms; The Rational Canonical Form

       范式；有理标准型

36.1 The Torsion Module Associated With An Endomorphism

       有关自同态的挠模

36.2 The Rational Canonical Form

       有理标准型

36.3 The Rational Canonical Form, Second Version

       有理标准型，第二种版本

36.4 The Jordan Form Revisited

       回顾若尔当标准型

36.5 The Smith Normal Form

       史密斯标准型

 

V Topology, Differential Calculus

       拓扑学，微分学

37 Topology 

       拓扑学

37.1 Metric Spaces and Normed Vector Spaces

       度量空间与赋范线性空间

37.2 Topological Spaces       

       拓扑空间

37.3 Continuous Functions, Limits

       连续函数，极限

37.4 Connected Sets

       连通集

37.5 Compact Sets and Locally Compact Spaces

       紧集和局部紧空间

37.6 Second-Countable and Separable Spaces

       第二可数和可分空间

37.7 Sequential Compactness

       序列紧性

37.8 Complete Metric Spaces and Compactness

       彻底度量空间和紧致性

37.9 Completion of a Metric Space

       度量空间的彻底化

37.10 The Contraction Mapping Theorem

       压缩映射定理（又称，Banach's Fixed Point Theorem 巴拿赫不动点定理）

37.11 Continuous Linear and Multilinear Maps

       连续线性与多重线性映射

37.12 Completion of a Normed Vector Space

       赋范向量空间的彻底化

37.13 Normed Affine Spaces

       赋范仿射空间

37.14 Futher Readings

       深刻阅读

38 A Detour On Fractals

       分形上的绕行

38.1 Iterated Function Systems and Fractals

       迭代函数系统和分形

39 Differential Calculus

       微分学

39.1 Directional Derivatives, Total Derivatives

       方向导数，全微分

39.2 Jacobian Matrices

       雅可比矩阵

39.3 The Implicit and The Inverse Function Theorems

       隐函数定理和反函数定理

39.4 Tangent Spaces and Differentials

       切空间与微分

39.5 Second-Order and Higher-Order Derivatives

       二阶导数与高阶导数

39.6 Taylor’s formula, Faà di Bruno’s formula

       泰勒公式，Faà di Bruno公式

39.7 Vector Fields, Covariant Derivatives, Lie Brackets

       向量场，协变函数，李括号

39.8 Futher Readings

       深刻阅读

 

VI Preliminaries for Optimization Theory

       优化理论所需的预备知识

40 Extrema of Real-Valued Functions

       实值函数的极值

40.1 Local Extrema and Lagrange Multipliers

       局部极值与拉格朗日乘数

40.2 Using Second Derivatives to Find Extrema

       使用二阶导数求极值

40.3 Using Convexity to Find Extrema

       使用凸性求极值

41 Newton’s Method and Its Generalizations

       牛顿法及其推广

41.1 Newton’s Method for Real Functions of a Real Argument

       牛顿法应用于实参的实函数

41.2 Generalizations of Newton’s Method

       牛顿法的推广

42 Quadratic Optimization Problems

       二次优化问题

42.1 Quadratic Optimization: The Positive Definite Case

       二次优化：正定情形

42.2 Quadratic Optimization: The General Case

       二次优化：通常情形

42.3 Maximizing a Quadratic Function on the Unit Sphere

       最大化单位球面上的二次函数

43 Schur Complements and Applications

       舒尔补及应用

43.1 Schur Complements

       舒尔补

43.2 SPD Matrices and Schur Complements

       对称正定矩阵和舒尔补

43.3 SP Semidefinite Matrices and Schur Complements

       对称半正定矩阵和舒尔补

 

VII Linear Optimization

       线性优化

44 Convex Sets, Cones, H-Polyhedra

       凸集，锥，H-多面体

44.1 What is Linear Programming?

       什么是线性规划？

44.2 Affine Subsets, Convex Sets, Hyperplanes, Half-Spaces

       仿射子集，凸集，超平面，半空间

44.3 Cones, Polyhedral Cones, and H-Polyhedra

       锥，多面锥和H-多面体

45 Linear Programs

       线性规划

45.1 Linear Programs, Feasible Solutions, Optimal Solutions

       线性规划，可行解，最优解

45.2 Basic Feasible Solutions and Vertices

       基本可行解和顶点（图论，或称节点，node）

46 The Simplex Algorithm

       单纯形法

46.1 The Idea Behind the Simplex Algorithm

       单纯形法背后的想法

46.2 The Simplex Algorithm in General

       通常的单纯形法

46.3 How to Perform a Pivoting Step Efficiently 

       如何高效地执行转换步骤

46.4 The Simplex Algorithm Using Tableaux 

       使用 Tableaux 的单纯形法

46.5 Computational Efficiency of the Simplex Method

       单纯形法的计算效率

47 Linear Programming and Duality

       线性规划与对偶

47.1 Variants of the Farkas Lemma

       法卡斯引理的变体

47.2 The Duality Theorem in Linear Programming 

       线性规划中的对偶定理

47.3 Complementary Slackness Conditions

       互补松弛条件

47.4 Duality for Linear Programs in Standard Form

       对偶用于标准型线性规划

47.5 The Dual Simplex Algorithm

       对偶单纯形法

47.6 The Primal-Dual Algorithm

       原始对偶法

 

VIII NonLinear Optimization

       非线性优化

48 Basics of Hilbert Spaces

       希尔伯特空间基础

48.1 The Projection Lemma, Duality

       射影引理，对偶

48.2 Farkas–Minkowski Lemma in Hilbert Spaces

       希尔伯特空间中的法卡斯-闵可夫斯基引理

49 General Results of Optimization Theory

       优化理论的通常结果

49.1 Optimization Problems; Basic Terminology

       优化问题；基本术语

49.2 Existence of Solutions of an Optimization Problem

       最优化问题解的存在性

49.3 Minima of Quadratic Functionals

       二次函数的极小值

49.4 Elliptic Functionals

       椭圆函数

49.5 Iterative Methods for Unconstrained Problems

       无约束优化问题的迭代法

49.6 Gradient Descent Methods for Unconstrained Problems

       无约束优化问题的梯度降低法

49.7 Convergence of Gradient Descent with Variable Stepsize

       变步长梯度降低法的收敛

49.8 Steepest Descent for an Arbitrary Norm

       任意范数的最速降低法

49.9 Newton’s Method For Finding a Minimum

       牛顿法求最小值

49.10 Conjugate Gradient Methods; Unconstrained Problems

       共轭梯度法；无约束问题

49.11 Gradient Projection for Constrained Optimization

       约束优化的梯度投影法

49.12 Penalty Methods for Constrained Optimization

       约束优化问题的惩罚算法

50 Introduction to Nonlinear Optimization

       非线性优化介绍

50.1 The Cone of Feasible Directions

       可行方向锥

50.2 Active Constraints and Qualified Constraints

       积极约束与规范约束

50.3 The Karush–Kuhn–Tucker Conditions

       卡鲁什-库恩-塔克条件

50.4 Equality Constrained Minimization

       等式约束最小化

50.5 Hard Margin Support Vector Machine; Version I

       硬间隔支持向量机，第1版

50.6 Hard Margin Support Vector Machine; Version II

       硬间隔支持向量机，第2版

50.7 Lagrangian Duality and Saddle Points

       拉格朗日对偶和鞍点

50.8 Weak and Strong Duality

       弱对偶和强对偶

50.9 Handling Equality Constraints Explicitly

       明确地处理等式约束

50.10 Dual of the Hard Margin Support Vector Machine

       硬间隔支持向量机的对偶

50.11 Conjugate Function and Legendre Dual Function

       共轭函数与勒让德对偶函数

50.12 Some Techniques to Obtain a More Useful Dual Program 

       一些获取更有用对偶规划的技巧

50.13 Uzawa’s Method

       Uzawa 算法

51 Subgradients and Subdifferentials

       次梯度和次微分

51.1 Extended Real-Valued Convex Functions

       扩充实值凸函数

51.2 Subgradients and Subdifferentials

       次梯度和次微分

51.3 Basic Properties of Subgradients and Subdifferentials

       次梯度和次微分的基本性质

51.4 Additional Properties of Subdifferentials

       次微分的其余性质

51.5 The Minimum of a Proper Convex Function

       真凸函数的最小值

51.6 Generalization of the Lagrangian Framework

       拉格朗日框架的推广

52 Dual Ascent Methods; ADMM

       对偶上升法；交替方向乘子法

52.1 Dual Ascent

       对偶上升法

52.2 Augmented Lagrangians and the Method of Multipliers

       增广拉格朗日和乘子法

52.3 ADMM: Alternating Direction Method of Multipliers

       交替方向乘子法

52.4 Convergence of ADMM

       交替方向乘子法的收敛

52.5 Stopping Criteria

       中止准则（条件）

52.6 Some Applications of ADMM

       ADMM的一些应用

52.7 Applications of ADMM to L1 -Norm Problems

        ADMM在L1范数问题上的一些应用

 

IX  Applications to Machine Learning

       机器学习中的应用

53 Ridge Regression and Lasso Regression

       岭回归和Lasso回归（最小绝对值收敛和选择算子、套索算法）

53.1 Ridge Regression

       岭回归

53.2 Lasso Regression (L1 - Regularized Regression)

       Lasso回归（L1正则回归）

54 Positive Definite Kernels

       正定核

54.1 Basic Properties of Positive Definite Kernels

       正定核的基本性质

54.2 Hilbert Space Representation of a Positive Kernel

       正定核的希尔伯特空间表示

54.3 Kernel PCA

       核主成分分析

54.4 ν-SV Regression

       v-支持向量机回归

55 Soft Margin Support Vector Machines

       软间隔支持向量机

55.1 Soft Margin Support Vector Machines; (SVM s1 )

       软间隔支持向量机(SVM s1 )

55.2 Soft Margin Support Vector Machines; (SVM s2 )

       软间隔支持向量机(SVM s2)

55.3 Soft Margin Support Vector Machines; (SVM s2‘)

       软间隔支持向量机(SVM s2‘)

55.4 Soft Margin SVM; (SVM s3 ) 

       软间隔支持向量机(SVM s3)

55.5 Soft Margin Support Vector Machines; (SVM s4 )

       软间隔支持向量机(SVM s4)

55.6 Soft Margin SVM; (SVM s5 ) 

       软间隔支持向量机(SVM s5)

55.7 Summary and Comparison of the SVM Methods

       总结及各类支持向量机法之间的比较

 

X Appendices

       附录

A Total Orthogonal Families in Hilbert Spaces

       希尔伯特空间中的彻底正交族

A.1 Total Orthogonal Families, Fourier Coefficients

       彻底正交族，傅里叶系数

A.2 The Hilbert Space L2 (K) and the Riesz-Fischer Theorem

       希尔伯特空间L2（K）和 里斯-费舍尔定理

B Zorn’s Lemma; Some Applications

        佐恩引理；一些应用

B.1 Statement of Zorn’s Lemma

        佐恩引理的描述

B.2 Proof of the Existence of a Basis in a Vector Space

        向量空间中基存在的证实

B.3 Existence of Maximal Proper Ideals

       极大真理想的存在性

Bibliography

       参考文献

 

 

 

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