🧕🏻 🈂️ 🧘🏽 基于格的机器学习的对象表示 👩🏽‍💻 👨🏼‍🏭 🈵

这是系列文章中的第四篇（链接到第一，第二和第三篇文章），专门研究基于格子理论的机器学习系统，称为“ VKF系统”。该程序使用基于马尔可夫链的算法，通过计算一些学习对象组之间相似性的随机子集来生成目标属性的原因。本文通过位字符串描述对象的表示形式，以便通过相应表示形式的按位乘法来计算相似性。具有离散特征的对象需要形式概念分析中的某些技术。具有连续特征的对象的情况使用逻辑回归，使用信息论和与比较区间的凸包相对应的表示将变化区域划分为子区间。

有主意！

1分立征兆

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$\langle{L,\wedge,\vee}\rangle$ $G$ () $\wedge$ - $M$ () $\vee$ - . $gIm\Leftrightarrow{g\geq{m}}$ $(G,M,I)$ $L(G,M,I)$ , $\langle{L,\wedge,\vee}\rangle$ .

$x\in{L}$ $\langle{L,\wedge,\vee}\rangle$ $\vee$ -, $x\neq\emptyset$ $y,z\in{L}$ $y<x$ $z<x$ $y\vee{z}<x$ .

$x\in{L}$ $\langle{L,\wedge,\vee}\rangle$ $\wedge$ -, $x\neq\textbf{T}$ $y,z\in{L}$ $x<y$ $x<z$ $x<y\wedge{z}$ .

$\wedge$ - , , $\vee$ - , .

不可约元素

( . $(L,L,\geq)$ )

G\M	h	i	j	k
a	1	1	1	0
b	0	1	1	1
c	1	1	0	0
d	1	0	1	0
f	0	1	0	1
g	0	0	1	1

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2.1

, . .

$E=G\cup{O}$ $G$ - $O$ . $[a,b)\subseteq\textbf{R}$ $V:G\to\textbf{R}$ $G[a,b)=\lbrace{g\in{G}: a\leq{V(g)}<b}\rbrace,$ $O[a,b)=\lbrace{g\in{O}: a\leq{V(g)}<b}\rbrace$

$E[a,b)=\lbrace{g\in{E}: a\leq{V(g)}<b}\rbrace$ .

$[a,b)\subseteq\textbf{R}$ $V:G\to\textbf{R}$

e n t [a, b) = - \frac{| G [a, b) |}{| E [a, b) |} \cdot \log_{2} (\frac{| G [a, b) |}{| E [a, b) |}) - \frac{| O [a, b) |}{| E [a, b) |} \cdot \log_{2} (\frac{| O [a, b) |}{| E [a, b) |})

${\rm{ent}}[a,b)=-\frac{\vert{G[a,b)}\vert}{\vert{E[a,b)}\vert}\cdot\log_{2}\left(\frac{\vert{G[a,b)}\vert}{\vert{E[a,b)}\vert}\right)-\frac{\vert{O[a,b)}\vert}{\vert{E[a,b)}\vert}\cdot\log_{2}\left(\frac{\vert{O[a,b)}\vert}{\vert{E[a,b)}\vert}\right)$

$a<r<b$ $[a,b)\subseteq\textbf{R}$ $V:G\to\textbf{R}$

i n f [a, r, b) = \frac{| E [a, r) |}{| E [a, b) |} \cdot e n t [a, r) + \frac{| E [r, b) |}{| E [a, b) |} \cdot e n t [r, b) .

${\rm{inf}}[a,r,b)=\frac{\vert{E[a,r)}\vert}{\vert{E[a,b)}\vert}\cdot{\rm{ent}}[a,r)+\frac{\vert{E[r,b)}\vert}{\vert{E[a,b)}\vert}\cdot{\rm{ent}}[r,b).$

— $V=r$ .

$V:G\to\textbf{R}$ $a=\min\{V\}$ $v_{0}$ , $v_{l+1}$ , $b=\max\{V\}$ . $\lbrace{v_{1}<\ldots<v_{l}}\rbrace$ .

2.2

$2l$ , $l$ — . ()

δ_{i}^{V} (g) = 1 \Leftrightarrow V (g) \geq v_{i} σ_{i}^{V} (g) = 1 \Leftrightarrow V (g) < v_{i},

$\delta_{i}^{V}(g)=1 \Leftrightarrow V(g)\geq{v_{i}} \\ \sigma_{i}^{V}(g)=1 \Leftrightarrow V(g)<v_{i},$

$1\leq{i}\leq{l}$ .

$\delta_{1}^{V}(g)\ldots\delta_{l}^{V}(g)\sigma_{1}^{V}(g)\ldots\sigma_{l}^{V}(g)$ $V$ $g\in{E}$ .

, — .

$\delta_{1}^{(1)}\ldots\delta_{l}^{(1)}\sigma_{1}^{(1)}\ldots\sigma_{l}^{(1)}$ $v_{i}\leq{V(A_{1})}<v_{j}$ $\delta_{1}^{(2)}\ldots\delta_{l}^{(2)}\sigma_{1}^{(2)}\ldots\sigma_{l}^{(2)}$ $v_{n}\leq{V(A_{2})}<v_{m}$ .

(δ_{1}^{(1)} \cdot δ_{1}^{(2)}) \dots (δ_{l}^{(1)} \cdot δ_{l}^{(2)}) (σ_{1}^{(1)} \cdot σ_{1}^{(2)}) \dots (σ_{l}^{(1)} \cdot σ_{l}^{(2)})

$(\delta_{1}^{(1)}\cdot\delta_{1}^{(2)})\ldots(\delta_{l}^{(1)}\cdot\delta_{l}^{(2)})(\sigma_{1}^{(1)}\cdot\sigma_{1}^{(2)})\ldots(\sigma_{l}^{(1)}\cdot\sigma_{l}^{(2)})$

$\min\lbrace{v_{i},v_{n}}\rbrace\leq{V((A_{1}\cup{A_{2}})'')}<\max\lbrace{v_{j},v_{m}}\rbrace$ .

, $0\ldots00\ldots0$ $\min\{V\}\leq{V((A_{1}\cup{A_{2}})'')}\leq\max\{V\}$ .

2.3

. ( 1). . , .

$p_{i_{1}}\vee\ldots\vee{p_{i_{k}}}$ $p_{i_{1}}+\ldots+{p_{i_{k}}}>\sigma$ $0<\sigma<1$ .

— $c:$ R $^{d}\to\lbrace{0,1}\rbrace$ , $\textbf{R}^{d}$ — ( $d$ ) $\lbrace{0,1}\rbrace$ — .

, $\langle{\vec{X},K}\rangle\in\text{R}^{d}\times\lbrace{0,1}\rbrace$ ,

p_{\vec{X}, K} (\vec{x}, k) = p_{\vec{X}} (\vec{x}) \cdot p_{K ∣ \vec{X}} (k ∣ \vec{x}),

$p_{\vec{X},K}(\vec{x},k)=p_{\vec{X}}(\vec{x})\cdot{p_{K\mid\vec{X}}(k\mid\vec{x})},$

$p_{\vec{X}}(\vec{x})$ — () , a $p_{K\mid\vec{X}}(k\mid\vec{x})$ — , .. $\vec{x}\in\text{R}^{d}$

p_{K ∣ \vec{X}} (k ∣ \vec{x}) = P {K = k ∣ \vec{X} = \vec{x}} .

$p_{K\mid\vec{X}}(k\mid\vec{x})=\textbf{P}\lbrace{K=k\mid\vec{X}=\vec{x}}\rbrace.$

$c:\textbf{R}^{d}\to\lbrace{0,1}\rbrace$

R (c) = P {c (\vec{X}) \neq K} .

$R(c)=\textbf{P}\left\lbrace{c(\vec{X})\neq{K}}\right\rbrace.$

$b:\textbf{R}^{d}\to\lbrace{0,1}\rbrace$ $p_{K\mid\vec{X}}(k\mid\vec{x})$

b (\vec{x}) = 1 \Leftrightarrow p_{K ∣ \vec{X}} (1 ∣ \vec{x}) > \frac{1}{2} > p_{K ∣ \vec{X}} (0 ∣ \vec{x})

$b(\vec{x})=1 \Leftrightarrow p_{K\mid\vec{X}}(1\mid\vec{x})>\frac{1}{2}>p_{K\mid\vec{X}}(0\mid\vec{x})$

$b$ :

\forall c : R^{d} \to {0, 1} [R (b) = P {b (\vec{X}) \neq K} \leq R (c)]

$\forall{c:\textbf{R}^{d}\to\lbrace{0,1}\rbrace}\left[R(b)=\textbf{P}\lbrace{b(\vec{X})\neq{K}}\rbrace\leq{R(c)}\right]$

p_{K ∣ \vec{X}} (1 ∣ \vec{x}) = \frac{p_{\vec{X} ∣ K} (\vec{x} ∣ 1) \cdot P {K = 1}}{p_{\vec{X} ∣ K} (\vec{x} ∣ 1) \cdot P {K = 1} + p_{\vec{X} ∣ K} (\vec{x} ∣ 0) \cdot P {K = 0}} = = \frac{1}{1 + \frac{p_{\vec{X} ∣ K} (\vec{x} ∣ 0) \cdot P {K = 0}}{p_{\vec{X} ∣ K} (\vec{x} ∣ 1) \cdot P {K = 1}}} = \frac{1}{1 + \exp {- a (\vec{x})}} = σ (a (\vec{x})),

$p_{K\mid\vec{X}}(1\mid\vec{x})=\frac{p_{\vec{X}\mid{K}}(\vec{x}\mid{1})\cdot\textbf{P}\lbrace{K=1}\rbrace}{p_{\vec{X}\mid{K}}(\vec{x}\mid{1})\cdot\textbf{P}\lbrace{K=1}\rbrace+p_{\vec{X}\mid{K}}(\vec{x}\mid{0})\cdot\textbf{P}\lbrace{K=0}\rbrace}= \\ =\frac{1}{1+\frac{p_{\vec{X}\mid{K}}(\vec{x}\mid{0})\cdot\textbf{P}\lbrace{K=0}\rbrace}{p_{\vec{X}\mid{K}}(\vec{x}\mid{1})\cdot\textbf{P}\lbrace{K=1}\rbrace}}=\frac{1}{1+\exp\lbrace{-a(\vec{x})}\rbrace}=\sigma(a(\vec{x})),$

$a(\vec{x})=\log\frac{p_{\vec{X}\mid{K}}(\vec{x}\mid{1})\cdot\textbf{P}\lbrace{K=1}\rbrace}{p_{\vec{X}\mid{K}}(\vec{x}\mid{0})\cdot\textbf{P}\lbrace{K=0}\rbrace}$ $\sigma(y)=\frac{1}{1+\exp\lbrace{-y}\rbrace}$ — .

2.4

$\langle\vec{x}_{1},k_{1}\rangle,\dots,\langle\vec{x}_{n},k_{n}\rangle$ $t_{j}=2k_{j}-1$ .

\log {p (t_{1}, \dots, t_{n} ∣ {\vec{x}}_{1}, \dots, {\vec{x}}_{n}, \vec{w})} = - \sum_{j = 1}^{n} \log [1 + \exp {- t_{j} \sum_{i = 1}^{m} w_{i} φ_{i} ({\vec{x}}_{j})}] .

$\log\lbrace{p(t_{1},\dots,t_{n}\mid\vec{x}_{1},\ldots,\vec{x}_{n},\vec{w})}\rbrace=-\sum_{j=1}^{n}\log\left[1+\exp\lbrace{-t_{j}\sum_{i=1}^{m}w_{i}\varphi_{i}(\vec{x}_{j})}\rbrace\right].$

L (w_{1}, \dots, w_{m}) = - \sum_{j = 1}^{n} \log [1 + \exp {- t_{j} \sum_{i = 1}^{m} w_{i} φ_{i} ({\vec{x}}_{j})}] \to max

$L(w_{1},\ldots,w_{m})=-\sum_{j=1}^{n}\log\left[1+\exp\lbrace{-t_{j}\sum_{i=1}^{m}w_{i}\varphi_{i}(\vec{x}_{j})}\rbrace\right]\to\max$

{\vec{w}}_{t + 1} = {\vec{w}}_{t} - (\nabla_{{\vec{w}}^{T}} \nabla_{\vec{w}} L ({\vec{w}}_{t}))^{- 1} \cdot \nabla_{\vec{w}} L ({\vec{w}}_{t}) .

$\vec{w}_{t+1}=\vec{w}_{t}-(\nabla_{\vec{w}^{T}}\nabla_{\vec{w}}L(\vec{w}_{t}))^{-1}\cdot\nabla_{\vec{w}}L(\vec{w}_{t}).$

$s_{j}=\frac{1}{1+\exp\lbrace{t_{j}\cdot{(w^{T}\cdot\Phi(x_{j}))}}\rbrace}$

\nabla L (\vec{w}) = - Φ^{T} d i a g (t_{1}, \dots, t_{n}) \vec{s}, \nabla \nabla L (\vec{w}) = Φ^{T} R Φ,

$\nabla{L(\vec{w})}=-\Phi^{T}{\rm{diag}}(t_{1},\ldots,t_{n})\vec{s}, \nabla\nabla{L(\vec{w})}=\Phi^{T}R\Phi,$

$R={\rm{diag}}(s_{1}(1-s_{1}), s_{2}(1-s_{2}), \ldots, s_{n}(1-s_{n}))$ —

$s_{1}(1-s_{1}), s_{2}(1-s_{2}), \ldots, s_{n}(1-s_{n})$ ${\rm{diag}}(t_{1},\ldots,t_{n})\vec{s}$ — $t_{1}s_{1}, t_{2}s_{2}, \ldots, t_{n}s_{n}$ .

{\vec{w}}_{t + 1} = {\vec{w}}_{t} + {(Φ^{T} R Φ)}^{- 1} Φ^{T} d i a g (t) \vec{s} = (Φ^{T} R Φ)^{- 1} Φ^{T} R \vec{z},

$\vec{w}_{t+1}=\vec{w}_{t}+\left(\Phi^{T}R\Phi\right)^{-1}\Phi^{T}{\rm{diag}}(t)\vec{s}= (\Phi^{T}R\Phi)^{-1}\Phi^{T}R\vec{z},$

$\vec{z}=\Phi\vec{w}_{t}+R^{-1}{\rm{diag}}(t_{1},\ldots,t_{n})\vec{s}$ — .

, - -

{\vec{w}}_{t + 1} = (Φ^{T} R Φ + λ \cdot I)^{- 1} \cdot (Φ^{T} R \vec{z}) .

$\vec{w}_{t+1}=(\Phi^{T}R\Phi+\lambda\cdot{I})^{-1}\cdot(\Phi^{T}R\vec{z}).$

"-" : 1 .

, . :

- $V_{k}$ ,

R^{2} = 1 - \exp {2 (L (w_{0}, \dots, w_{k - 1}) - L (w_{0}, \dots, w_{k - 1}, w_{k})) / n} \geq σ

$R^{2}=1-\exp\lbrace{2(L(w_{0},\ldots, w_{k-1})-L(w_{0},\ldots,w_{k-1},w_{k}))/n}\rbrace\geq\sigma$

$V_{k}$ ,

1 - \frac{L (w_{0}, \dots, w_{k - 1}, w_{k})}{L (w_{0}, \dots, w_{k - 1})} \geq σ

$1-\frac{L(w_{0},\ldots,w_{k-1},w_{k})}{L(w_{0},\ldots,w_{k-1})}\geq\sigma$

"-" Wine Quality ( . ). . ( >7), .

( 2.3) "" "". ( ) , 0 1. " " "" .

但是这对（“ pH”，“酒精”）的情况却截然不同。“酒精”重量为正，而“ pH”重量为负。但是借助明显的逻辑转换，我们得到了含义（“ pH” $\Rightarrow$ “醇”）。

作者要感谢他的同事和学生的支持和鼓励。

基于格的机器学习的对象表示

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