From d0417b9e27bd0a3a1e6eb960a7297e1aff895d19 Mon Sep 17 00:00:00 2001 From: Shine wOng <1551885@tongji.edu.cn> Date: Mon, 27 Jan 2020 22:58:52 +0800 Subject: [PATCH] add a conclusion on nn, not finished yet. --- ml/logistic_regression/logistic_regression.md | 3 - ml/neural_network/images/neuron_model.jpg | Bin 0 -> 11247 bytes ml/neural_network/images/nn_model.jpg | Bin 0 -> 158098 bytes ml/neural_network/images/nn_multiclass.jpg | Bin 0 -> 13062 bytes ml/neural_network/neural_network.md | 176 ++++++++++++++++++ words.md | 7 + 6 files changed, 183 insertions(+), 3 deletions(-) create mode 100644 ml/neural_network/images/neuron_model.jpg create mode 100644 ml/neural_network/images/nn_model.jpg create mode 100644 ml/neural_network/images/nn_multiclass.jpg create mode 100644 ml/neural_network/neural_network.md diff --git a/ml/logistic_regression/logistic_regression.md b/ml/logistic_regression/logistic_regression.md index ca191d9..6003139 100644 --- a/ml/logistic_regression/logistic_regression.md +++ b/ml/logistic_regression/logistic_regression.md @@ -110,11 +110,8 @@ $$ $$ cost(h_\theta(x), y) = -<<<<<<< HEAD \left{ -======= \left\\{ ->>>>>>> 78f580ddaa701274fd48a20e5ef6e7f7b497320a \begin{aligned} -log(h_\theta(x))&, y = 1\\\\ -log(1 - h_\theta(x))&, y = 0 diff --git a/ml/neural_network/images/neuron_model.jpg b/ml/neural_network/images/neuron_model.jpg new file mode 100644 index 0000000000000000000000000000000000000000..53069d7ac82bc6e56042815d501489be9a7ac524 GIT binary patch literal 11247 zcmc(FcUV)|^LOY)igalzMNq1=Ae~j3AWaf_0O=j+y}1^;Mg#-|kxl|czz`rbl@huk zgx*AYmEHv2u>0Lz_kG^|{l5FxJ9(bmx%bZb%$zyrJZEOkJsUoo23*wExT^smA|e6= z5q^NPF~A)FB{?|-ITKsHmuDF3?<}rM)D{!^I={uj#BAz(7f4OteEn z!~h^>AR=KPI%@%N0f+!ZWCUyfB61=UQZfo+O2S*6ivS{0B4Sb!GGZ!<3#23jAtGYJ zTXF^pMji>KUksUfB_DXRM6>eUzHjRTeOJLR<@@W)k621JX&ECB*w`-lD^T{%BemS` 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z_w4ovszLHx?>iT^T0{L4wklbDJyv-fsC2{*((>Kcl?2}>!2Bm1=(p2Yoy?v2Qxw=j zxgLlvXREPoU}*QGndLCy2>Uwln4*6)G)q3)v>;FXkLwnrMl<&4k+rqJiK zeEkNeKd$SbO7xS*U`64hslWf?UU1zpHcr{bIV0Cfvg7LBds#-pp=UQ=Fys~d{?Iol z_QZOg0<73ME^~-zw*-7Bo;fDGW`0_&Ho(DN*pj;^8cPE!?Ol*8p zVX?6wJl6K5yTpe 单个神经元模型 + +每一个神经元都是一个**逻辑回归单元**(`logistic regression unit`),它接受多个输入信号,产生一个输出信号,如下图所示: + +![neuron_model](images/neuron_model.jpg) + +不妨设 + +$$ +x = \left[ +\begin{matrix} +x_0\\\\ +x_1\\\\ +x_2\\\\ +x_3 +\end{matrix} +\right],\ +\theta = \left[ +\begin{matrix} +\theta_0\\\\ +\theta_1\\\\ +\theta_2\\\\ +\theta_3 +\end{matrix} +\right] +$$ + +这里仍然有$x_0 = 1$,是手动添加的偏置单元。这样,$\hat{y}$满足逻辑回归的假设函数 + +$$ +\hat{y} = h_\theta(x) = sigmoid(\theta^T x) +$$ + +> 神经网络模型 + +神经网络是由多个层次的神经元组成的,每个层次含有数量可以是不相同的神经元,一个简单的神经网络结构如下图所示: + +![nn_model](images/nn_mode.jpg) + +可以看到,任何一个神经网络,都包含一个**输入层**,一个**输出层**,以及若干数量不确定的**隐含层**(`hidden layer`)。输入层一般用$x$或者$a^{(1)}$来表示,其中$x_j$或者$a_j^{(1)}$表示第$j$个特征的值;第一个隐含层用$a^{(2)}$来表示,$a_j^{(2)}$表示第一个隐含层的第$j$个神经元的值,后面的隐含层以此类推;用$L$表示神经网络含有的层次数量,这样输出层就可以用$y$或者$a^{(L)}$来表示了。 + +由于不同层次含有的神经元数量未必相同,通常使用$S_l$来表示第$l$层所包含的神经元数量,上图中,有$S_1 = 3, S_2 = 3, S_3 = 1$。 + +根据上面的神经元模型,相邻层次的神经元之间具有一定的信号传递关系,该传递关系实际上是对上一层次的全部神经元(加上偏置单元)的加权和做逻辑函数`sigmoid function`,各个神经元的权重可以理解为相邻两层神经元之间的练线上的数值,权重越大说明该神经元对于下一个层次的神经元的影响越大。相邻层次间神经元信号传递的权重可以构成一个矩阵,用$\Theta$表示,其中第$l$层和第$l + 1$层的权重矩阵可以表示为$\Theta^{(l)}$,它的每一个行向量都对应了第$l + 1$层的一个神经元的全部权重信息,因此该行向量的维度为$S_l + 1$,而$\Theta^{(l)}$一共具有$S_{l + 1}$行,一般地有 + +$$ +\begin{aligned} +a_1^{(2)} = sigmoid(\Theta_{10}^{(1)}x_0 + \Theta_{11}^{(1)}x_1 +\Theta_{12}^{(1)}x_2 + \Theta_{13}^{(1)}x_3)\\\\ +a_2^{(2)} = sigmoid(\Theta_{20}^{(1)}x_0 + \Theta_{21}^{(1)}x_1 +\Theta_{22}^{(1)}x_2 + \Theta_{23}^{(1)}x_3)\\\\ +a_3^{(2)} = sigmoid(\Theta_{30}^{(1)}x_0 + \Theta_{31}^{(1)}x_1 +\Theta_{32}^{(1)}x_2 + \Theta_{33}^{(1)}x_3) +\end{aligned} +$$ + +这就是相邻层次间神经元信号传递的模型。 + +> 前向传播算法 + +实际上,通过上面的公式已经可以窥见前向传播算法的大概了,将其写成矩阵形式,有 + +$$ +a^{(2)} = \left[ +\begin{matrix} +a_1^{(2)}\\\\ +a_2^{(2)}\\\\ +a_3^{(2)} +\end{matrix} +\right] = sigmod(\Theta^{(1)} a^{(1)}) +$$ + +通过给$a^{(2)}$添加偏置项$a_0^{(2)} = 1$,可以写出输出$y$的表达式 + +$$ +y = a^{(3)} = sigmoid(\Theta_{10}^{(2)}a_0^{(2)} + \Theta_{11}^{(2)}a_1^{(2)} +\Theta_{12}^{(2)}a_2^{(2)} + \Theta_{13}^{(2)}a_3^{(2)}) +$$ + +同理,也可以把它写作矩阵形式 + +$$ +y = sigmoid(\Theta^{(2)}a^{(2)}) +$$ + +实际上,对于任意复杂结构神经网络,都可以定义以下前向传播算法 + +$$ +\begin{aligned} +&a^{(1)} = x\\\\ +&z^{(2)} = \Theta^{(1)}a^{(1)}, a^{(2)} = sigmoid(z^{(2)})\\\\ +&z^{(3)} = \Theta^{(2)}a^{(2)}, a^{(3)} = sigmoid(z^{(3)})\\\\ +&\cdots +&z^{(L)} = \Theta^{(L - 1)}a^{(L - 1)}, h_\Theta(x) = a^{(L)} = sigmoid(z^{L}) +\end{aligned} +$$ + +顾名思义,为了计算神经网络的输出,需要首先计算$a^{(2)}, a^{(3)}, \cdots$,最终才能计算出$h_\Theta(x)$,因此称为前向传播算法。训练神经网络的过程,其实就是训练这里的权重矩阵$\Theta$,这将在后面具体阐述。 + +> 为什么要使用神经网络? + +神经网络是用于解决分类问题,而在此前的 [逻辑回归总结](../logistic_regression/logistic_regression.md) 中已经指出,借助**多项式回归**,逻辑回归模型理论上可以表示任意复杂的函数。既然如此,那为什么还要引入神经网络模型呢? + +设想一种具体的应用场景,现在需要对一张`20x20`像素的图片进行分类,模型采用逻辑回归模型。图片分类问题显然不是一个线性划分问题,因此需要使用多项式回归,考虑使用所有的低于四次(含)的项,容易看出其实这并不是一个相当复杂的模型,但是仅考虑四次项就已经有$C_{400}^4$,即大约有10亿项。需要指出的是,这里使用的图片还只是`20x20`像素的,对于实际中常见分辨率的图片,如`480*360`,将具有更高维的特征向量,这是现有的计算能力不能满足的。 + +而使用神经网络模型则不然,因为不需要手工指定特征向量的形式,只将初始的400个像素点作为输入即可。这是因为,神经网络的多个隐藏层,足以形成足够复杂的模型,而无需手动引入多项式。 + +实际上,神经网络的一个主要优势在于,**神经网络可以自行学习特征**,而不需要像线性回归或者逻辑回归那样手工给定特征向量。对于上面图中的简单神经网络,如果只看隐藏层以及输出层,容易发现这就是一个逻辑回归模型,而此前的层次的功能,则可以理解为对特征的学习,隐藏层的值就是学到的特征向量。由于`sigmoid`函数是一个非线性函数,再加上多个隐藏层,神经网络模型足以学习到相当复杂的特征,而无需像逻辑回归那样手工指定高次多项式。 + +> 多分类问题。 + +如逻辑回归一样,神经网络也可以被推广到解决多分类问题,两者的方法如出一辙。对于逻辑回归而言,为了解决$n(n \ge 3)$分类问题,只需要训练$n$个二类分类问题的分类器即可;而对于神经网络模型,则只需要将输出层的神经元增加到$n$个,由于前面已经指出,神经网络的最后两个层次本质上就是逻辑回归模型,因此该方法本质上也是训练了$n$个二类分类器,与逻辑回归一致。多分类的神经网络结构如下所示: + +![nn_multiclass](images/nn_multicalss.jpg) + +但是两者在实现的细节上也有一些差异。对于$n$类分类问题,逻辑回归的输出的取值往往是$\{1, 2, \cdots, n\}$,而神经网络则要沿用向量表示法,例如当$n = 3$时,神经网络的输出应该表示为 + +$$ +y \in \{\left[ +\begin{matrix} +1\\\\ +0\\\\ +0 +\end{matrix} +\right],\ +\left[ +\begin{matrix} +0\\\\ +1\\\\ +0 +\end{matrix} +\right],\ +\left[ +\begin{matrix} +0\\\\ +0\\\\ +1 +\end{matrix} +\right]\} +$$ diff --git a/words.md b/words.md index e7a0aa7..0904f74 100644 --- a/words.md +++ b/words.md @@ -2212,6 +2212,13 @@ Some Words - They were debating the proposition that "All people are created equal". - Under Proposition 71, the state will commit $3 billion to studying stem cells over the next 10 years. ++ moisture +> (n)a liquid such as water in the form of very small drops, either in the air, in a substance, or on a surface. + + - These plants need a rich soil that remains moisture. + - It was a clear day with little moisture in the air. + - Plants absorb carbon dioxide from the air and moisture from the soil. + + cosmetically + rationale + ameliorate

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