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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
深入阅读
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的证明
8.7 Dealing with Roundoff Errors; Pivoting Strategies
处理舍入误差，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（又称作幺正变换）
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 rq
旋转rq 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
自伴随和其他特殊线性映射
17.5 Normal and Other Special Matrices
正规算子和其他特殊矩阵
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?
为什么是射影空间
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
射影空间上的相似结构
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，唯一分解整环，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
唯一分解整环，诺特环，希尔伯特基定理
32.1 Unique Factorization Domains (Factorial Rings)
唯一分解整环（析因环/唯一分解环）
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
深入阅读
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
完全度量空间和紧致性
37.9 Completion of a Metric Space
度量空间的完全化
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
赋范向量空间的完全化
37.13 Normed Affine Spaces
赋范仿射空间
37.14 Futher Readings
深入阅读
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
深入阅读
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：一般情形
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
一般的单纯形法
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
优化理论的一般结果
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下降法
49.7 Convergence of Gradient Descent with Variable Stepsize
变步长梯度下降法的收敛
49.8 Steepest Descent for an Arbitrary Norm
任意范数的最速下降法
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
次微分的其他性质
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
停止准则（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
总结及各种支持向量机法之间的比较
X Appendices
附录
A Total Orthogonal Families in Hilbert Spaces
希尔伯特空间中的完全正交族
A.1 Total Orthogonal Families, Fourier Coefficients
完全正交族，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明
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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