ai的数学基础

所需要的数学理论
1. 优化理论，AI中的部分应用包括：SVM（Support Vector Machine支持向量机）、深度学习BP (back propagation反向传播)、决策树等等
2. 线性代数与矩阵分析，AI中的部分应用包括：PCA(Principal Component Analysis主成分分析)、ICA（Independent Component Analysis独立成分分析）、推荐系统等等
3. 概率与统计，AI中的部分应用包括：朴素贝叶斯等等
4. AI中的各种算法基本上是优化理论、线性代数与矩阵分析和概率与统计的高度融合。

ICA
ICA是一种用来从多变量（多维）统计数据里找到隐含的因素或成分的方法，被认为是主成分分（Principal Component Analysis, PCA）和因子分析（Factor Analysis）的一种扩展。
线性代数的范数
1-范数：

‖ A ‖_{1} = m a x {\sum | a i_{1} |, \sum | a i_{2} |, . . ., \sum | a i_{n} |}

$\|A\|_1 = max\{\sum |ai_1|,\sum |ai_2|,...,\sum |ai_n|\}$
列范数，A每一列元素绝对值之和的最大值。
2-范数:

‖ A ‖_{2} = (m a x {λ i (A^{'} A)})^{0.5}

$\|A\|_2 = (max\{\lambda i(A'A)\})^{0.5}$
谱范数,即A’A特征值λi中最大者λm的平方根,其中A’为A的转置矩阵

\infty

$\infty$ -范数：

‖ A ‖_{\infty} = m a x {\sum | a 1_{j} |, \sum | a 2_{j} |, . . ., \sum | a n_{j} |}

$\|A\|_\infty = max\{\sum |a1_j|,\sum |a2_j|,...,\sum |an_j|\}$

行范数,A每一行元素绝对值之和的最大值)(其中为∑|a1j| 第一行元素绝对值的和，其余类似）。

附件：常用的mathjax函数

具体的参考文档请查看网站： http://www.onemathematicalcat.org/MathJaxDocumentation/TeXSyntax.htm#alphaList

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$

\frac{x + 7}{y + z + 3}

$\def\specialFrac#1#2{\frac{x + #1}{y + #2}} \specialFrac{7}{z+3}$

(x + 1)^{2} = (x + 1) (x + 1) = x^{2} + 2 x + 1

$% Note: (x+1)^2 is NOT x^2 + 1 (x+1)^2 % original expression = (x+1)(x+1) % definition of exponent = x^2 + 2x + 1 % FOIL, combine like terms$
下面列出了MathJax所支持的Tex宏指令。如果宏在扩展中定义，则扩展名的名称将遵循宏名称。如果扩展名在括号中，则在首次使用宏或环境时将自动加载扩展名

#
%
&
^
_
{
}
~
'

\   (backslash-space)
\!
\#
\$
\%
\&
\,
\:
\;
\>
\\
\_
\{
\|
\}

A

\AA                     mediawiki-texvc
\above
\abovewithdelims
\acute
\alef                   mediawiki-texvc
\aleph
\alpha
\Alpha                  mediawiki-texvc
\amalg
\And
\and                    mediawiki-texvc
\ang                    mediawiki-texvc
\angle
\approx
\approxeq               AMSsymbols
\arccos
\arcsin
\arctan
\arg
\array
\Arrowvert
\arrowvert
\ast
\asymp
\atop
\atopwithdelims

B

\backepsilon            AMSsymbols
\backprime              AMSsymbols
\backsim                AMSsymbols
\backsimeq              AMSsymbols
\backslash
\bar
\barwedge               AMSsymbols
\Bbb
\Bbb                    mediawiki-texvc
\Bbbk                   AMSsymbols
\bbox                  [bbox]
\bcancel                cancel
\because                AMSsymbols
\begin
\begingroup             begingroup      non-standard
\beta
\Beta                   mediawiki-texvc
\beth                   AMSsymbols
\between                AMSsymbols
\bf
\Big
\big
\bigcap
\bigcirc
\bigcup
\Bigg
\bigg
\Biggl
\biggl
\Biggm
\biggm
\Biggr
\biggr
\Bigl
\bigl
\Bigm
\bigm
\bigodot
\bigoplus
\bigotimes
\Bigr
\bigr
\bigsqcup
\bigstar                AMSsymbols
\bigtriangledown
\bigtriangleup
\biguplus
\bigvee
\bigwedge
\binom                  AMSmath
\blacklozenge           AMSsymbols
\blacksquare            AMSsymbols
\blacktriangle          AMSsymbols
\blacktriangledown      AMSsymbols
\blacktriangleleft      AMSsymbols
\blacktriangleright     AMSsymbols
\bmod
\bold                   mediawiki-texvc
\boldsymbol            [boldsymbol]
\bot
\bowtie
\Box                    AMSsymbols
\boxdot                 AMSsymbols
\boxed                  AMSmath
\boxminus               AMSsymbols
\boxplus                AMSsymbols
\boxtimes               AMSsymbols
\brace
\bracevert
\brack
\breve
\buildrel
\bull                   mediawiki-texvc
\bullet
\Bumpeq                 AMSsymbols
\bumpeq                 AMSsymbols

C

\C                      mediawiki-texvc
\cal
\cancel                 cancel
\cancelto               cancel
\cap
\Cap                    AMSsymbols
\cases
\cdot
\cdotp
\cdots
\ce                     mhchem
\cee                    mhchem
\centerdot              AMSsymbols
\cf                     mhchem
\cfrac                  AMSmath
\check
\checkmark              AMSsymbols
\chi
\Chi                    mediawiki-texvc
\choose
\circ
\circeq                 AMSsymbols
\circlearrowleft        AMSsymbols
\circlearrowright       AMSsymbols
\circledast             AMSsymbols
\circledcirc            AMSsymbols
\circleddash            AMSsymbols
\circledR               AMSsymbols
\circledS               AMSsymbols
\class                 [HTML]           non-standard
\clubs                  mediawiki-texvc
\clubsuit
\cnums                  mediawiki-texvc
\colon
\color                  color
\colorbox               color
\complement             AMSsymbols
\Complex                mediawiki-texvc
\cong
\coppa                  mediawiki-texvc
\Coppa                  mediawiki-texvc
\coprod
\cos
\cosh
\cot
\coth
\cr
\csc
\cssId                 [HTML]           non-standard
\cup
\Cup                    AMSsymbols
\curlyeqprec            AMSsymbols
\curlyeqsucc            AMSsymbols
\curlyvee               AMSsymbols
\curlywedge             AMSsymbols
\curvearrowleft         AMSsymbols
\curvearrowright        AMSsymbols

D

\dagger
\Dagger                 mediawiki-texvc
\daleth                 AMSsymbols
\Darr                   mediawiki-texvc
\dashleftarrow          AMSsymbols
\dashrightarrow         AMSsymbols
\dashv
\dbinom                 AMSmath
\ddagger
\ddddot                 AMSmath
\dddot                  AMSmath
\ddot
\ddots
\DeclareMathOperator    AMSmath
\definecolor            color
\def                   [newcommand]
\deg
\Delta
\delta
\det
\dfrac                  AMSmath
\diagdown               AMSsymbols
\diagup                 AMSsymbols
\diamond
\Diamond                AMSsymbols
\diamonds               mediawiki-texvc
\diamondsuit
\digamma                AMSsymbols
\Digamma                mediawiki-texvc
\dim
\displaylines
\displaystyle
\div
\divideontimes          AMSsymbols
\dot
\doteq
\Doteq                  AMSsymbols
\doteqdot               AMSsymbols
\dotplus                AMSsymbols
\dots
\dotsb
\dotsc
\dotsi
\dotsm
\dotso
\doublebarwedge         AMSsymbols
\doublecap              AMSsymbols
\doublecup              AMSsymbols
\Downarrow
\downarrow
\downdownarrows         AMSsymbols
\downharpoonleft        AMSsymbols
\downharpoonright       AMSsymbols

E

\ell
\empty                  mediawiki-texvc
\emptyset
\enclose                enclose         non-standard
\end
\endgroup               begingroup      non-standard
\enspace
\epsilon
\Epsilon                mediawiki-texvc
\eqalign
\eqalignno
\eqcirc                 AMSsymbols
\eqref                 [AMSmath]
\eqsim                  AMSsymbols
\eqslantgtr             AMSsymbols
\eqslantless            AMSsymbols
\equiv
\eta
\Eta                    mediawiki-texvc
\eth                    AMSsymbols
\euro                   mediawiki-texvc
\exist                  mediawiki-texvc
\exists
\exp

F

\fallingdotseq          AMSsymbols
\fbox
\fcolorbox              color
\Finv                   AMSsymbols
\flat
\forall
\frac
\frac                   AMSmath
\frak
\frown

G

\Game                   AMSsymbols
\Gamma
\gamma
\gcd
\gdef                   begingroup
\ge
\geneuro                mediawiki-texvc
\geneuronarrow          mediawiki-texvc
\geneurowide            mediawiki-texvc
\genfrac                AMSmath
\geq
\geqq                   AMSsymbols
\geqslant               AMSsymbols
\gets
\gg
\ggg                    AMSsymbols
\gggtr                  AMSsymbols
\gimel                  AMSsymbols
\global                 begingroup
\gnapprox               AMSsymbols
\gneq                   AMSsymbols
\gneqq                  AMSsymbols
\gnsim                  AMSsymbols
\grave
\gt
\gtrapprox              AMSsymbols
\gtrdot                 AMSsymbols
\gtreqless              AMSsymbols
\gtreqqless             AMSsymbols
\gtrless                AMSsymbols
\gtrsim                 AMSsymbols
\gvertneqq              AMSsymbols

H

\H                      mediawiki-texvc
\hAar                   mediawiki-texvc
\harr                   mediawiki-texvc
\Harr                   mediawiki-texvc
\hat
\hbar
\hbox
\hdashline
\hearts                 mediawiki-texvc
\heartsuit
\hline
\hom
\hookleftarrow
\hookrightarrow
\hphantom
\href                  [HTML]
\hskip
\hslash                 AMSsymbols
\hspace
\Huge
\huge
\idotsint               AMSmath

I

\iff
\iiiint                 AMSmath
\iiint
\iint
\Im
\image                  mediawiki-texvc
\imath
\impliedby              AMSsymbols
\implies                AMSsymbols
\in
\inf
\infin                  mediawiki-texvc
\infty
\injlim                 AMSmath
\int
\intercal               AMSsymbols
\intop
\isin                   mediawiki-texvc
\iota
\Iota                   mediawiki-texvc
\it

J

\jmath
\Join                   AMSsymbols

K

\kappa
\Kappa                  mediawiki-texvc
\ker
\kern
\koppa                  mediawiki-texvc
\Koppa                  mediawiki-texvc

L

\label                 [AMSmath]
\Lambda
\lambda
\land
\lang                   mediawiki-texvc
\langle
\LARGE
\Large
\large
\larr                   mediawiki-texvc
\Larr                   mediawiki-texvc
\lArr                   mediawiki-texvc
\LaTeX
\lbrace
\lbrack
\lceil
\ldotp
\ldots
\le
\leadsto                AMSsymbols
\left
\Leftarrow
\leftarrow
\leftarrowtail          AMSsymbols
\leftharpoondown
\leftharpoonup
\leftleftarrows         AMSsymbols
\Leftrightarrow
\leftrightarrow
\leftrightarrows        AMSsymbols
\leftrightharpoons      AMSsymbols
\leftrightsquigarrow    AMSsymbols
\leftroot
\leftthreetimes         AMSsymbols
\leq
\leqalignno
\leqq                   AMSsymbols
\leqslant               AMSsymbols
\lessapprox             AMSsymbols
\lessdot                AMSsymbols
\lesseqgtr              AMSsymbols
\lesseqqgtr             AMSsymbols
\lessgtr                AMSsymbols
\lesssim                AMSsymbols
\let                   [newcommand]
\lfloor
\lg
\lgroup
\lhd                    AMSsymbols
\lim
\liminf
\limits
\limsup
\ll
\llap
\llcorner               AMSsymbols
\Lleftarrow             AMSsymbols
\lll                    AMSsymbols
\llless                 AMSsymbols
\lmoustache
\ln
\lnapprox               AMSsymbols
\lneq                   AMSsymbols
\lneqq                  AMSsymbols
\lnot
\lnsim                  AMSsymbols
\log
\Longleftarrow
\longleftarrow
\Longleftrightarrow
\longleftrightarrow
\longmapsto
\Longrightarrow
\longrightarrow
\looparrowleft          AMSsymbols
\looparrowright         AMSsymbols
\lor
\lower
\lozenge                AMSsymbols
\lrarr                  mediawiki-texvc
\Lrarr                  mediawiki-texvc
\lrArr                  mediawiki-texvc
\lrcorner               AMSsymbols
\Lsh                    AMSsymbols
\lt
\ltimes                 AMSsymbols
\lVert                  AMSmath
\lvert                  AMSmath
\lvertneqq              AMSsymbols

M

\maltese                AMSsymbols
\mapsto
\mathbb
\mathbf
\mathbin
\mathcal
\mathchoice            [mathchoice]
\mathclose
\mathfrak
\mathinner
\mathit
\mathop
\mathopen
\mathord
\mathpunct
\mathrel
\mathring               AMSmath
\mathrm
\mathscr
\mathsf
\mathstrut
\mathtip                action          non-standard
\mathtt
\matrix
\max
\mbox
\measuredangle          AMSsymbols
\mho                    AMSsymbols
\mid
\middle
\min
\mit
\mkern
\mmlToken                               non-standard
\mod
\models
\moveleft
\moveright
\mp
\mskip
\mspace
\mu
\Mu                     mediawiki-texvc
\multimap               AMSsymbols

N

\N                      mediawiki-texvc
\nabla
\natural
\natnums                mediawiki-texvc
\ncong                  AMSsymbols
\ne
\nearrow
\neg
\negmedspace            AMSmath
\negthickspace          AMSmath
\negthinspace
\neq
\newcommand            [newcommand]
\newenvironment        [newcommand]
\Newextarrow            extpfeil
\newline
\nexists                AMSsymbols
\ngeq                   AMSsymbols
\ngeqq                  AMSsymbols
\ngeqslant              AMSsymbols
\ngtr                   AMSsymbols
\ni
\nLeftarrow             AMSsymbols
\nleftarrow             AMSsymbols
\nLeftrightarrow        AMSsymbols
\nleftrightarrow        AMSsymbols
\nleq                   AMSsymbols
\nleqq                  AMSsymbols
\nleqslant              AMSsymbols
\nless                  AMSsymbols
\nmid                   AMSsymbols
\nobreakspace           AMSmath
\nolimits
\normalsize
\not
\notag                 [AMSmath]
\notin
\nparallel              AMSsymbols
\nprec                  AMSsymbols
\npreceq                AMSsymbols
\nRightarrow            AMSsymbols
\nrightarrow            AMSsymbols
\nshortmid              AMSsymbols
\nshortparallel         AMSsymbols
\nsim                   AMSsymbols
\nsubseteq              AMSsymbols
\nsubseteqq             AMSsymbols
\nsucc                  AMSsymbols
\nsucceq                AMSsymbols
\nsupseteq              AMSsymbols
\nsupseteqq             AMSsymbols
\ntriangleleft          AMSsymbols
\ntrianglelefteq        AMSsymbols
\ntriangleright         AMSsymbols
\ntrianglerighteq       AMSsymbols
\nu
\Nu                     mediawiki-texvc
\nVDash                 AMSsymbols
\nVdash                 AMSsymbols
\nvDash                 AMSsymbols
\nvdash                 AMSsymbols
\nwarrow

O

\O                      mediawiki-texvc
\odot
\officialeuro           mediawiki-texvc
\oint
\oldstyle
\Omega
\omega
\omicron
\Omicron                mediawiki-texvc
\ominus
\operatorname           AMSmath
\oplus
\or                     mediawiki-texvc
\oslash
\otimes
\over
\overbrace
\overleftarrow
\overleftrightarrow
\overline
\overrightarrow
\overset
\overwithdelims
\owns

P

\P                      mediawiki-texvc
\pagecolor              mediawiki-texvc
\parallel
\part                   mediawiki-texvc
\partial
\perp
\phantom
\Phi
\phi
\Pi
\pi
\pitchfork              AMSsymbols
\plusmn                 mediawiki-texvc
\pm
\pmatrix
\pmb
\pmod
\pod
\Pr
\prec
\precapprox             AMSsymbols
\preccurlyeq            AMSsymbols
\preceq
\precnapprox            AMSsymbols
\precneqq               AMSsymbols
\precnsim               AMSsymbols
\precsim                AMSsymbols
\prime
\prod
\projlim                AMSmath
\propto
\Psi
\psi

Q

\Q                      mediawiki-texvc
\qquad
\quad

R

\R                      mediawiki-texvc
\raise
\rang                   mediawiki-texvc
\rangle
\rarr                   mediawiki-texvc
\Rarr                   mediawiki-texvc
\rArr                   mediawiki-texvc
\rbrace
\rbrack
\rceil
\Re
\real                   mediawiki-texvc
\reals                  mediawiki-texvc
\Reals                  mediawiki-texvc
\ref                   [AMSmath]
\renewcommand          [newcommand]
\renewenvironment      [newcommand]
\require                               non-standard
\restriction            AMSsymbols
\rfloor
\rgroup
\rhd                    AMSsymbols
\rho
\Rho                    mediawiki-texvc
\right
\Rightarrow
\rightarrow
\rightarrowtail         AMSsymbols
\rightharpoondown
\rightharpoonup
\rightleftarrows        AMSsymbols
\rightleftharpoons
\rightleftharpoons      AMSsymbols
\rightrightarrows       AMSsymbols
\rightsquigarrow        AMSsymbols
\rightthreetimes        AMSsymbols
\risingdotseq           AMSsymbols
\rlap
\rm
\rmoustache
\root
\Rrightarrow            AMSsymbols
\Rsh                    AMSsymbols
\rtimes                 AMSsymbols
\Rule                                  non-standard
\rVert                  AMSmath
\rvert                  AMSmath

S

\S
\sampi                  mediawiki-texvc
\Sampi                  mediawiki-texvc
\scr
\scriptscriptstyle
\scriptsize
\scriptstyle
\sdot                   mediawiki-texvc
\searrow
\sect                   mediawiki-texvc
\sec
\setminus
\sf
\sharp
\shortmid               AMSsymbols
\shortparallel          AMSsymbols
\shoveleft              AMSmath
\shoveright             AMSmath
\sideset                AMSmath
\Sigma
\sigma
\sim
\simeq
\sin
\sinh
\skew
\small
\smallfrown             AMSsymbols
\smallint
\smallsetminus          AMSsymbols
\smallsmile             AMSsymbols
\smash
\smile
\spades                 mediawiki-texvc
\Space                                 non-standard
\space
\spadesuit
\sphericalangle         AMSsymbols
\sqcap
\sqcup
\sqrt
\sqsubset               AMSsymbols
\sqsubseteq
\sqsupset               AMSsymbols
\sqsupseteq
\square                 AMSsymbols
\stackrel
\star
\stigma                 mediawiki-texvc
\Stigma                 mediawiki-texvc
\strut
\style                 [HTML]          non-stanard
\sub                    mediawiki-texvc
\sube                   mediawiki-texvc
\subset
\Subset                 AMSsymbols
\subseteq
\subseteqq              AMSsymbols
\subsetneq              AMSsymbols
\subsetneqq             AMSsymbols
\substack               AMSmath
\succ
\succapprox             AMSsymbols
\succcurlyeq            AMSsymbols
\succeq
\succnapprox            AMSsymbols
\succneqq               AMSsymbols
\succnsim               AMSsymbols
\succsim                AMSsymbols
\sum
\sup
\supe                   mediawiki-texvc
\supset
\Supset                 AMSsymbols
\supseteq
\supseteqq              AMSsymbols
\supsetneq              AMSsymbols
\supsetneqq             AMSsymbols
\surd
\swarrow

T

\tag                   [AMSmath]
\tan
\tanh
\tau
\Tau                    mediawiki-texvc
\tbinom                 AMSmath
\TeX
\text
\textbf
\textit
\textrm
\textsf
\textstyle
\texttt
\texttip                action         non-standard
\textvisiblespace       mediawiki-texvc
\tfrac                  AMSmath
\therefore              AMSsymbols
\Theta
\theta
\thetasym               mediawiki-texvc
\thickapprox            AMSsymbols
\thicksim               AMSsymbols
\thinspace
\tilde
\times
\tiny
\Tiny                                  non-standard
\to
\toggle                 action         non-standard
\top
\triangle
\triangledown           AMSsymbols
\triangleleft
\trianglelefteq         AMSsymbols
\triangleq              AMSsymbols
\triangleright
\trianglerighteq        AMSsymbols
\tt
\twoheadleftarrow       AMSsymbols
\twoheadrightarrow      AMSsymbols

U

\uarr                   mediawiki-texvc
\uArr                   mediawiki-texvc
\Uarr                   mediawiki-texvc
\ulcorner               AMSsymbols
\underbrace
\underleftarrow
\underleftrightarrow
\underline
\underrightarrow
\underset
\unicode               [unicode]       non-standard
\unlhd                  AMSsymbols
\unrhd                  AMSsymbols
\Uparrow
\uparrow
\Updownarrow
\updownarrow
\upharpoonleft          AMSsymbols
\upharpoonright         AMSsymbols
\uplus
\uproot
\Upsilon
\upsilon
\upuparrows             AMSsymbols
\urcorner               AMSsymbols

V

\varcoppa               mediawiki-texvc
\varDelta               AMSsymbols
\varepsilon
\varGamma               AMSsymbols
\varinjlim              AMSmath
\varkappa               AMSsymbols
\varLambda              AMSsymbols
\varliminf              AMSmath
\varlimsup              AMSmath
\varnothing             AMSsymbols
\varOmega               AMSsymbols
\varphi
\varPhi                 AMSsymbols
\varpi
\varPi                  AMSsymbols
\varprojlim             AMSmath
\varpropto              AMSsymbols
\varPsi                 AMSsymbols
\varrho
\varsigma
\varSigma               AMSsymbols
\varstigma              mediawiki-texvc
\varsubsetneq           AMSsymbols
\varsubsetneqq          AMSsymbols
\varsupsetneq           AMSsymbols
\varsupsetneqq          AMSsymbols
\vartheta
\varTheta               AMSsymbols
\vartriangle            AMSsymbols
\vartriangleleft        AMSsymbols
\vartriangleright       AMSsymbols
\varUpsilon             AMSsymbols
\varXi                  AMSsymbols
\vcenter
\vdash
\Vdash                  AMSsymbols
\vDash                  AMSsymbols
\vdots
\vec
\vee
\veebar                 AMSsymbols
\verb                  [verb]
\Vert
\vert
\vline                  mediawiki-texvc
\vphantom
\Vvdash                 AMSsymbols

W

\wedge
\weierp                 mediawiki-texvc
\widehat
\widetilde
\wp
\wr

X

\Xi
\xi
\xcancel                cancel
\xleftarrow             AMSmath
\xlongequal             extpfeil
\xmapsto                extpfeil
\xrightarrow            AMSmath
\xtofrom                extpfeil
\xtwoheadleftarrow      extpfeil
\xtwoheadrightarrow     extpfeil

Y

\yen                    AMSsymbols

Z

\Z                      mediawiki-texvc
\zeta
\Zeta

环境：当

\begin{XXX} ... \end{XXX}

$\begin{XXX} ... \end{XXX}$ 中的XXX是下面列出的某一个的时候，就会提供一个LaTeX环境。

align                  [AMSmath]
align*                 [AMSmath]
alignat                [AMSmath]
alignat*               [AMSmath]
aligned                [AMSmath]
alignedat              [AMSmath]
array

Bmatrix
bmatrix

cases
CD                      AMSmath

eqnarray
eqnarray*
equation
equation*

gather                 [AMSmath]
gather*                [AMSmath]
gathered               [AMSmath]

matrix
multline               [AMSmath]
multline*              [AMSmath]

pmatrix

smallmatrix             AMSmath
split                  [AMSmath]
subarray                AMSmath

Vmatrix
vmatrix

附件：常用的mathjax函数

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