◎ 说明： 完成作业可以使用你所熟悉的编程语言和平台，比如 C，C++、MATLAB、Python等。作业链接： 2023秋季学期人工神经网络 : http://t.csdnimg.cn/nKe3o

 

01 分类问题

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一、基本BP算法网络

  下面是分层前馈网络的示意图，分层前馈网络的随机梯度下降算法（BP算法）中，第 $n$ 层的权系数 $w_{ij}^n$ 的调整公式可以表述为：

$$\Delta w_{ij}^n=\eta \left(\sum _{k=1}^{m_n}{\delta }_k^{n+1}{w}_{jk}^{n+1}\right)y^{\prime n}\cdot y_i^{n-1}$$

▲ 图1.1.1 具有h+1层的BP网络结构示意图

1、作业基本要求

  （1） 请写出如下图结构1网络结构算法中，每个神经元权系数的更新公式 $\Delta w_{nm}$ 。神经元传递函数可以选择：sigmoid 或者 tanh （双曲正切），同学们自行选择。

$$f_1\left(x\right)=\frac{1}{1+e^{-x}};f_2\left(x\right)=\frac{1-e^{-x}}{1+e^{-x}}$$

  （2） 编写程序， 实验结构1网络算法， 求解 XOR 问题。

  注： XOR问题中， 四个样本对应的输入分别为 x ˉ 1 , 2 , 3 , 4 = { [ 1 , 1 ] , [ 1 , − 1 ] , [ − 1 , 1 ] , [ − 1 , − 1 ] } \bar x_{1,2,3,4} = \left\{ {\left[ {1,1} \right],\left[ {1, - 1} \right],\left[ { - 1,1} \right],\left[ { - 1, - 1} \right]} \right\} xˉ1,2,3,4​={[1,1],[1,−1],[−1,1],[−1,−1]} 对应的期望输出分别为： y 1 , 2 , 3 , 4 = { − 1 , 1 , 1 , − 1 } y_{1,2,3,4} = \left\{ { - 1,1,1, - 1} \right\} y1,2,3,4​={−1,1,1,−1}

▲ 图 两种结构的神经网络

2、作业选做要求

  （1）请写出 结构2 网络中每个权系数的更新公式；

  （2）编写程序实现结构2，求解 XOR 问题。

• 参考答案 : https://zhuoqing.blog.csdn.net/article/details/109734950

二、三类分类问题

1、分类问题

  设计一个前馈分层神经网络， 分别使用 BP 算法 和 RBF 算法， 对于下图中平面坐标系中三类样本进行分类。每一类样本个数都是三个，它们的坐标 $\left(x_1,x_2\right)$ 在 $\left[0,1\right]$ 之间的五个离散值分布。

▲ 图1.1.1 三类样本处在坐标系中的位置

  下面表格给出了三类样本五组随机分布的位置。可以任意选择一组分布作为训练样本训练网络。

类别	位置1	位置2	位置3
类别1	(0.75;1.0)	(0.5;0.75)	(0.25;0.0)
类别2	(0.5;0.0)	(0.0;0.0)	(1.0;0.75)
类别3	(1.0;1.0)	(0.5;0.25)	(0.75;0.5)
类别1	(1.0;0.75)	(1.0;0.5)	(0.25;0.5)
类别2	(0.0;0.0)	(0.0;0.75)	(0.75;0.0)
类别3	(1.0;1.0)	(0.75;1.0)	(0.25;0.25)
类别1	(0.25;0.5)	(0.0;0.25)	(0.75;0.0)
类别2	(0.0;0.0)	(0.75;0.5)	(1.0;0.5)
类别3	(0.75;0.75)	(1.0;0.25)	(0.0;0.75)
类别1	(0.0;0.0)	(0.25;0.5)	(0.75;0.0)
类别2	(0.75;0.5)	(0.25;0.25)	(0.5;0.25)
类别3	(1.0;0.25)	(1.0;1.0)	(0.5;0.75)
类别1	(0.75;0.5)	(0.0;0.5)	(0.5;0.75)
类别2	(0.0;0.75)	(1.0;0.5)	(0.75;1.0)
类别3	(0.0;0.0)	(0.75;0.0)	(0.75;0.75)

2、作业基本要求

（1）使用单隐层BP网络进行分类

• 绘制出网络结构，并给出算法流程描述；
• 讨论不同隐层节点个数对于分类结果影响，并给出解决 该分类问题最少隐层节点个数；
• 对每个样本增加噪声，讨论所训练网络的泛化能力。对样本增加噪声就是在两个坐标 $\left(x_1,x_2\right)$ 上增加随机数，如下所示；

$$\left(x_1+{\delta }_1,x_2+{\delta }_2\right),{\delta }_1,{\delta }_2\in rand\left(-0.25,0.25\right)$$

• 在网络训练过程中，请注意样本坐标进行归一化处理，即将坐标 $x_1,x_2$ 经过变换，使其数值分布在 $\left(-1,1\right)$ 的对称范围内，对比网络训练效果；

（2）使用RBF网络进行分类

• 使用正规化RBF网络求解，给出网络参数与仿真结果；
• 使用广义RBF网络求解，并给出隐层节点个数分别为 2,3,4时对应的分类结构；

3、作业选做要求

（1）训练数据扩充

  选择一种网络（BF，RBF，SVM），对比以下两种数据训练后的网络泛化性能：
  （1）仅仅使用没有增加噪声的九个样本进行训练；
  （2）对样本增加随机噪声，扩充到 180 个样本进行训练。扩充后的样本个数可以自己定义；

• 参考答案 : https://zhuoqing.blog.csdn.net/article/details/109745214

三、MNist分类

1、MNIST数据集合

  设计一个BP网络完成MNIST手写体识别。

▲ 图1.2.1 MNIST 数据集合示例

• MNIST数据库下载 : http://yann.lecun.com/exdb/mnist

2、作业基本要求

  （1）直接图片输入：直接构造784个节点输入，10个节点输出，自行选择中间隐层数量和节点个数，使用50000数据进行训练，10000个测试样本测试。 给出网络结构设计，训练方法以及识别结果。

3、作业选做要求

  （1） 人工提取特征：参照课件上[3-4-4]“手写体识别”中介绍的方法，利用数字的边缘方向图与灰度图的降采样作为特征，使用小的神经网络进行训练。训练样本选择5000，测试样本选择2000。

  （2） 对比和讨论人工提取特征与直接图片输入两种方法的优缺点。

三、Fashion MNist分类（选做）

1、Fashion MNIST数据集合

  设计一个BP网络完成Fashion MNIST手写体识别。 Fashion-MNIST是一组28x28灰度的衣服图像。 它比MNIST更复杂，因此可以更好地表示神经网络的实际性能。

Fashion MNIST/服饰数据集包含70000张灰度图像，其中包含60,000个示例的训练集和10,000个示例的测试集，每个示例都是一个28x28灰度图像，分为以下几类：

▲ 图1.4.1 Fashion MNIST 数据集合

• Fashion-MNIST 数据集合下载 : https://www.worldlink.com.cn/en/osdir/fashion-mnist.html

 

02 回归问题

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一、Peaks函数逼近

1、设计要求

  使用神经网络逼近 MATLAB中 的 peaks 二维函数。 下面是 peaks 函数的数学表达式。

  下面是 Peaks 函数的图像以及对应的网络样本采样点的分布。

▲ 图2.1.1 Peaks 函数的图像（左）以及训练样本采样点（右）

2、作业基本要求

  分别设计 BF，RBF 网络完成对 Peaks 函数的逼近。

  （1） 样本数据采样区域在区域内[-4,4]×[-4,4]随机采样。 样本采样的个数自行确定。

  （2） 分别使用BP网络，RBF网络实现上述函数逼近；

  （3） 在BP网络中，讨论训练样本个数与训练样本采样的分布对函数逼近效果的影响。讨论参见课件【3-4-2：函数逼近问题的讨论】。

▲ 图2.1.2 训练样本采集是否集中在函数变化剧烈的区域

  （4）在RBF网络中，讨论隐层神经元尺度参数对于函数逼近的影响。

▲ 图2.1.3 随着训练次数增加对应的函数逼近曲面

3、作业选做要求

  （1） 在BP网络中，请对比中间隐层传递函数在使用一下两种函数时对于结果的影响：

▲ 图2.1.3 两种不同的神经元传递函数

• 参考答案： 2020人工神经网络第一次作业-参考答案第五部分

 

03 数据压缩

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一、ASCII字母压缩

1、设计要求

  使用神经网络对于下面 26 个字母进行压缩。

▲ 图3.1.1 字符点阵信息与压缩网络

2、作业基本要求

  （1） 26个字母的数据文件可以从数据文件ascii8×16.txt文件中获取。其中每一行代表一个字母的数据。字母数据为8×16点阵。也可以直接在作业要求网页上，拷贝如下代码段的字符，与 ascii8×16文件中的数据是一样的。

00000000000000000000000000010000000100000001100000101000001010000010010000111100010001000100001001000010111001110000000000000000
00000000000000000000000011111000010001000100010001000100011110000100010001000010010000100100001001000100111110000000000000000000
00000000000000000000000000111110010000100100001010000000100000001000000010000000100000000100001001000100001110000000000000000000
00000000000000000000000011111000010001000100001001000010010000100100001001000010010000100100001001000100111110000000000000000000
00000000000000000000000011111100010000100100100001001000011110000100100001001000010000000100001001000010111111000000000000000000
00000000000000000000000011111100010000100100100001001000011110000100100001001000010000000100000001000000111000000000000000000000
00000000000000000000000000111100010001000100010010000000100000001000000010001110100001000100010001000100001110000000000000000000
00000000000000000000000011100111010000100100001001000010010000100111111001000010010000100100001001000010111001110000000000000000
00000000000000000000000001111100000100000001000000010000000100000001000000010000000100000001000000010000011111000000000000000000
00000000000000000000000000111110000010000000100000001000000010000000100000001000000010000000100000001000000010001000100011110000
00000000000000000000000011101110010001000100100001010000011100000101000001001000010010000100010001000100111011100000000000000000
00000000000000000000000011100000010000000100000001000000010000000100000001000000010000000100000001000010111111100000000000000000
00000000000000000000000011101110011011000110110001101100011011000101010001010100010101000101010001010100110101100000000000000000
00000000000000000000000011000111011000100110001001010010010100100100101001001010010010100100011001000110111000100000000000000000
00000000000000000000000000111000010001001000001010000010100000101000001010000010100000101000001001000100001110000000000000000000
00000000000000000000000011111100010000100100001001000010010000100111110001000000010000000100000001000000111000000000000000000000
00000000000000000000000000111000010001001000001010000010100000101000001010000010101100101100101001001100001110000000011000000000
00000000000000000000000011111100010000100100001001000010011111000100100001001000010001000100010001000010111000110000000000000000
00000000000000000000000000111110010000100100001001000000001000000001100000000100000000100100001001000010011111000000000000000000
00000000000000000000000011111110100100100001000000010000000100000001000000010000000100000001000000010000001110000000000000000000
00000000000000000000000011100111010000100100001001000010010000100100001001000010010000100100001001000010001111000000000000000000
00000000000000000000000011100111010000100100001001000100001001000010010000101000001010000001100000010000000100000000000000000000
00000000000000000000000011010110100100101001001010010010100100101010101010101010011011000100010001000100010001000000000000000000
00000000000000000000000011100111010000100010010000100100000110000001100000011000001001000010010001000010111001110000000000000000
00000000000000000000000011101110010001000100010000101000001010000001000000010000000100000001000000010000001110000000000000000000
00000000000000000000000001111110100001000000010000001000000010000001000000100000001000000100001001000010111111000000000000000000

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[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1,1,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
]

  （2） 讨论网络隐层节点个数与恢复数据误差之间的关系。

  （3）给出隐层节点在15个时，26个字母压缩恢复后的数据图像。

• 参考答案 : https://zhuoqing.blog.csdn.net/article/details/109830470

二、图像压缩（选做）

• 这个题目是选做内容。

1、设计要求

  使用神经网络对于Lena图像进行压缩。对于隐层节点数目在16～256之间选择几个数目，给出图像压缩的效果以及图像压缩MSE。

$$MSE=\frac{1}{N}\cdot \sum _{i=1}^N{\left(I_i^{org}-I_i^{re}\right)}^2$$

  其中 $I_i^{org}$ 是原始图像灰度， $I_i^{re}$ 是恢复图像的灰度值。

▲ 图3.2.1 灰白Lena测试图片

注：Lena的灰度图像可以从网络上寻找并下载。也可以选择其它的图像进行实验。

• 参考答案： 2020人工神经网络第一次作业-参考答案第八部分

  注：Lena的灰度图像可以从网络上寻找并下载。也可以选择其它的图像进行实验。下面给出了Lena下载链接：

• BMP-GRAY:Lean512 : https://www.cosy.sbg.ac.at/~pmeerw/Watermarking/lena.html
• MATLAB:Lean512.mat(262kb) : https://www.ece.rice.edu/~wakin/images/lena512.mat

 

※ 附件 ※

---

一、三类问题数据生成

  下面的 Python 程序完成三类分类问题中，三类样本随机生成的算法，同时生成不同样本集合对应位置动图。

from headm import *

pointdim = [30, 31, 32, 35, 34, 33, 36, 37, 38]
xyc = XYCoor(2, 3, 7, 8, 1, 1)
plotbox = 41

plotgif = PlotGIF()

xrange = list(arange(5)/4)*4
yrange = [xx for x in zip(*[list(arange(5)/4)]*4) for xx in x]
xyrange = [(x,y) for x,y in zip(xrange, yrange)]

printf('类别 位置1 位置2 位置3')
for i in range(5):
    random.shuffle(xyrange)

    for j in range(3):
        printff('类别%d '%(j+1),
                str(xyrange[j*3+0]).replace(' ', '').replace(',', ';') + ' ',
                str(xyrange[j*3+1]).replace(' ', '').replace(',', ';') + ' ',
                str(xyrange[j*3+2]).replace(' ', '').replace(',', ';') + ' ')

    printf('- - - -')

    for j in range(len(pointdim)):
        xd,yd = xyrange[j]

        x,y = xyc.XY2Pos(xd,yd)
        tsplocatedop(pointdim[j], x,y)

    tsprv()
    time.sleep(.05)
    plotgif.appendbox(plotbox)

plotgif.save(r'd:\temp\gif1.gif', 500)

printf('\a')

二、函数显示

  下面 Python 程序绘制 Peaks 函数3D显示。

from headm import *

from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np

fig = plt.figure()
ax = fig.gca(projection='3d')

X = np.arange(-4, 4, 0.05)
Y = np.arange(-4, 4, 0.05)
X, Y = np.meshgrid(X, Y)

def fxy(x,y):
    e1 = exp(-(x**2+(y+1)**2))
    e2 = exp(-(x**2+y**2))
    e3 = exp(-((x+1)**2+y**2))

    fvalue = 3*(1-x)**2*e1 - 10*(x/5-x**3-y**5)*e2 - 1/3*e3
    return fvalue

Z = fxy(X,Y)

surf = ax.plot_surface(X, Y, Z, cmap=cm.coolwarm,
                       linewidth=0, antialiased=False)

ax.set_zlim(-8, 8)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))

fig.colorbar(surf, shrink=0.5, aspect=5)

plt.show()

---

■ 相关文献链接:

• 2023秋季学期人工神经网络
• 2020人工神经网络第一次作业-解答第一部分
• 2020人工神经网络第一次作业-参考答案第二部分
• MNIST数据库下载
• 2020人工神经网络第一次作业-参考答案第五部分-CSDN博客
• 2020人工神经网络第一次作业-参考答案第七部分
• 2020人工神经网络第一次作业-参考答案第八部分-CSDN博客
• BMP-GRAY:Lean512
• MATLAB:Lean512.mat(262kb)

● 相关图表链接:

• 图1.1.1 具有h+1层的BP网络结构示意图
• 图1.1.2 两种结构的神经网络
• 图1.1.1 三类样本处在坐标系中的位置
• 表1-1 五组三类分类坐标位置
• 图1.2.1 MNIST 数据集合示例
• 图2.1.1 Peaks 函数的图像（左）以及训练样本采样点（右）
• 图2.1.2 训练样本采集是否集中在函数变化剧烈的区域
• 图2.1.3 随着训练次数增加对应的函数逼近曲面
• 图2.1.3 两种不同的神经元传递函数
• 图3.1.1 字符点阵信息与压缩网络
• 图3.2.1 灰白Lena测试图片