LearnAI

Chapter 1 引言

是时候系统学习一下人工智能了,希望什么时候可以补充一下引言吧

设备环境：Linux-Ubuntu20.04
编译环境：miniconda

Chapter 2 预备知识

2.1 数据处理

2.1.1 创建张量

1	import torch

此处为torch,~~虽然包叫pytorch~~

1 2	x = torch.arange(12) x

tensor([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11])

张量：数组，但里面的内容都是数值，维度不限

x.shape

torch.Size([12])

通过shape可以查询张量的形状;[]代表维度为1，12代表这一维度的长度

x.numel()

numel()可以访问张量中元素的总数

1 2	x = x.reshape(3, 4) x

tensor([[ 0,  1,  2,  3],
        [ 4,  5,  6,  7],
        [ 8,  9, 10, 11]])

通过reshape(c,r),可以将张量重新排列成多维形式

1 2	y = x.reshape(4, 4) y

---------------------------------------------------------------------------

RuntimeError                              Traceback (most recent call last)

Cell In[6], line 1
----> 1 y = x.reshape(4, 4)
      2 y


RuntimeError: shape '[4, 4]' is invalid for input of size 12

但重组时，col与row的乘积必须等于元素总数

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z = torch.zeros((2, 3, 4))
o = torch.ones((3, 2, 4))
print(z, '\n' ,o)

tensor([[[0., 0., 0., 0.],
         [0., 0., 0., 0.],
         [0., 0., 0., 0.]],

        [[0., 0., 0., 0.],
         [0., 0., 0., 0.],
         [0., 0., 0., 0.]]]) 
 tensor([[[1., 1., 1., 1.],
         [1., 1., 1., 1.]],

        [[1., 1., 1., 1.],
         [1., 1., 1., 1.]],

        [[1., 1., 1., 1.],
         [1., 1., 1., 1.]]])

使用zeros()构造全0矩阵，同理ones(), ~~没有除了1和0之外的数~~

1 2	x = torch.tensor([[1,2,3,4], [5,6,7,8], [9,10,11,12]]) x

tensor([[ 1,  2,  3,  4],
        [ 5,  6,  7,  8],
        [ 9, 10, 11, 12]])

使用tensor()手动构造变量

1	torch.randn(3, 4)

tensor([[-0.2559,  0.4066,  0.4201, -1.1634],
        [-2.0670,  0.6678,  0.9912,  0.2923],
        [ 0.2710,  1.9265, -1.6423, -0.9642]])

生成随机数张量，元素平均数为0，标准差为1的正态分布中随机采样

2.1.2 张量运算

基本运算

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x = torch.tensor([1.0,2, 4, 8])
y = torch.tensor([2,  3, 4, 5])
x+y,x-y,x*y,x/y,x**y,x%y

(tensor([ 3.,  5.,  8., 13.]),
 tensor([-1., -1.,  0.,  3.]),
 tensor([ 2.,  6., 16., 40.]),
 tensor([0.5000, 0.6667, 1.0000, 1.6000]),
 tensor([1.0000e+00, 8.0000e+00, 2.5600e+02, 3.2768e+04]),
 tensor([1., 2., 0., 3.]))

对于相同形状的张量，可以通过(+,-,*,/,%)作运算，具体来说：$I+J \rightarrow I_i+J_i$

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import numpy
x = torch.tensor([numpy.e,numpy.log(2),1,torch.e])
torch.exp(x)

tensor([15.1543,  2.0000,  2.7183, 15.1543], dtype=torch.float64)

exp()求解e的x次方：$y_i=e^{x_i}$

张量拼接

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x = torch.arange(12).reshape((3,4))
y = torch.tensor([[2,1,4,3],[1,2,3,4],[4,3,2,1]])
torch.cat((x,y),dim=0), torch.cat((x,y),dim=1)

(tensor([[ 0,  1,  2,  3],
         [ 4,  5,  6,  7],
         [ 8,  9, 10, 11],
         [ 2,  1,  4,  3],
         [ 1,  2,  3,  4],
         [ 4,  3,  2,  1]]),
 tensor([[ 0,  1,  2,  3,  2,  1,  4,  3],
         [ 4,  5,  6,  7,  1,  2,  3,  4],
         [ 8,  9, 10, 11,  4,  3,  2,  1]]))

通过cat()可以实现拼接，dim=0按照行，dim=1按列

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x = torch.arange(12).reshape((3,4))
y = torch.tensor([[2,1,4],[1,2,3],[4,3,2]])
torch.cat((x,y),dim=0), torch.cat((x,y),dim=1)

---------------------------------------------------------------------------

RuntimeError                              Traceback (most recent call last)

Cell In[13], line 3
      1 x = torch.arange(12).reshape((3,4))
      2 y = torch.tensor([[2,1,4],[1,2,3],[4,3,2]])
----> 3 torch.cat((x,y),dim=0), torch.cat((x,y),dim=1)


RuntimeError: Sizes of tensors must match except in dimension 0. Expected size 4 but got size 3 for tensor number 1 in the list.

若尺寸不对应则会报错

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x = torch.arange(12).reshape((3,4))
y = torch.tensor([[2,1,4,3],[1,2,3,4],[4,3,2,1]])
x == y

tensor([[False,  True, False,  True],
        [False, False, False, False],
        [False, False, False, False]])

近似于求位与运算

元素求和

1 2	x = torch.arange(12).reshape((3,4)) torch.sum(x), x.sum()

(tensor(66), tensor(66))

广播机制

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a = torch.arange(3).reshape((3, 1))
b = torch.arange(2).reshape((1, 2))
a + b

tensor([[0, 1],
        [1, 2],
        [2, 3]])

试试多维，有点废脑子

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x = torch.tensor([ [ [1],[1],[1] ] ,[[2],[3],[4] ] ])
y = torch.tensor([ [ [1,2,3]] ])
x,y,x+y

(tensor([[[1],
          [1],
          [1]],

         [[2],
          [3],
          [4]]]),
 tensor([[[1, 2, 3]]]),
 tensor([[[2, 3, 4],
          [2, 3, 4],
          [2, 3, 4]],

         [[3, 4, 5],
          [4, 5, 6],
          [5, 6, 7]]]))

张量切片

按行切片

1 2	x = torch.arange(12).reshape((3,4)) x,x[-1],x[2],x[0:2],x[0:3],x[0:-1],x[:-1],x[:],x[:0]

(tensor([[ 0,  1,  2,  3],
         [ 4,  5,  6,  7],
         [ 8,  9, 10, 11]]),
 tensor([ 8,  9, 10, 11]),
 tensor([ 8,  9, 10, 11]),
 tensor([[0, 1, 2, 3],
         [4, 5, 6, 7]]),
 tensor([[ 0,  1,  2,  3],
         [ 4,  5,  6,  7],
         [ 8,  9, 10, 11]]),
 tensor([[0, 1, 2, 3],
         [4, 5, 6, 7]]),
 tensor([[0, 1, 2, 3],
         [4, 5, 6, 7]]),
 tensor([[ 0,  1,  2,  3],
         [ 4,  5,  6,  7],
         [ 8,  9, 10, 11]]),
 tensor([], size=(0, 4), dtype=torch.int64))

自定义切片

1 2	x = torch.arange(12).reshape((3,4)) x[:,1] , x[1:3,1:] , x[0::2,1::2], x[0::2,0::2]

(tensor([1, 5, 9]),
 tensor([[ 5,  6,  7],
         [ 9, 10, 11]]),
 tensor([[ 1,  3],
         [ 9, 11]]),
 tensor([[ 0,  2],
         [ 8, 10]]))

a:b:c,在区间[a,b)中取数，step=c,a,c默认为0，b默认最后一位

暴力写入

1 2	x[1, 2] = 9 x

tensor([[ 0,  1,  2,  3],
        [ 4,  5,  9,  7],
        [ 8,  9, 10, 11]])

内存节省

y = torch.arange(4)
before = id(y)
y = y + x
id(y) == before

False

在处理变量中，一般操作会导致新建内存，如果矩阵数据特别大，容易消耗内存,可以使用原地存储

y = torch.arange(4)
before = id(y)
y += x
id(y) == before

---------------------------------------------------------------------------

RuntimeError                              Traceback (most recent call last)

Cell In[22], line 3
      1 y = torch.arange(4)
      2 before = id(y)
----> 3 y += x
      4 id(y) == before


RuntimeError: output with shape [4] doesn't match the broadcast shape [3, 4]

类型互换

y = torch.arange(4)
x = torch.arange(4,8,1)
z = torch.zeros_like(y)
print('id(z):',id(z))
z[:] = x + y
print('id(z):',id(z))

id(z): 140157950377472
id(z): 140157950377472

Pytorch与Numpy的互换

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A = x.numpy()
B = torch.tensor(A)
type(A),type(B)

(numpy.ndarray, torch.Tensor)

将大小为1的张量转化为python标量

1 2	a = torch.tensor([3.5]) a,a.item(),float(a),int(a)

(tensor([3.5000]), 3.5, 3.5, 3)

2.2数据预处理

2.2.1 生成数据集文件

import os
os.makedirs(os.path.join('..','data'), exist_ok=True)
data_file = os.path.join('..', 'data', 'house_tiny.csv')
with open(data_file, 'w') as f:
    f.write('NumRooms,Alley,Price\n')
    f.write('NA,Pave,127500\n')
    f.write('2,NA,106000\n')
    f.write('4,NA,178100\n')
    f.write('NA,NA,140000\n')

创建CSV文件：房价数据-房间数量/巷子类型/价格

import pandas as pd

data = pd.read_csv(data_file)
print(data)

   NumRooms Alley   Price
0       NaN  Pave  127500
1       2.0   NaN  106000
2       4.0   NaN  178100
3       NaN   NaN  140000

运用pandas库读取csv

2.2.2 补充丢失数据

补充丢失数据的典型方法是插值法和删除法

插值法：用一个数字替代缺失位
删除法：直接忽略缺少的数据

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inputs, outputs = data.iloc[:,0:2], data.iloc[:, 2]
inputs = inputs.fillna(inputs.mean())
print(inputs)

   NumRooms Alley
0       3.0  Pave
1       2.0   NaN
2       4.0   NaN
3       3.0   NaN

在补充房间数，使用插值法，通过插入已知数的平均值补充未知数，input.mean()读取数据平均数

1 2	inputs = pd.get_dummies(inputs,dummy_na=True) print(inputs)

   NumRooms  Alley_Pave  Alley_nan
0       3.0           1          0
1       2.0           0          1
2       4.0           0          1
3       3.0           0          1

在补充巷子时，看成是有Pave为1，无为0的矩阵形式

2.2.3 转化为张量形式

import torch

x = torch.tensor(inputs.values)
y = torch.tensor(outputs.values)

x,y

(tensor([[3., 1., 0.],
         [2., 0., 1.],
         [4., 0., 1.],
         [3., 0., 1.]], dtype=torch.float64),
 tensor([127500, 106000, 178100, 140000]))

利用tensor()构造张量

实例：删除空数据最多的一列

count = 0
count_max = 0
labels = ['NumRooms','Alley','Price']
for label in labels:
    count = data[label].isna().sum()
    if count > count_max:
        count_max = count
        flag = label
data_new = data.drop(flag,axis=1)
data_new

	NumRooms	Price
0	NaN	127500
1	2.0	106000
2	4.0	178100
3	NaN	140000

2.3 线性代数

1	import torch

2.3.1 标量

虽然直接声明常量更为常见，但是torch也支持创建标量，即长度为1的一维张量

1 2	x = torch.tensor(3.0) y = torch.tensor(2.0)

2.3.2 向量

向量的声明

向量的声明通常就是一维张量，即tensor([x,y,z,...])

1 2	x = torch.arange(4,dtype=torch.float32) x

tensor([0., 1., 2., 3.])

向量的形状

shape可以查看向量的维度，是一个元素组

1 2	y = torch.arange(12).reshape(2, 3, 2) y, y.shape

(tensor([[[ 0,  1],
          [ 2,  3],
          [ 4,  5]],

         [[ 6,  7],
          [ 8,  9],
          [10, 11]]]),
 torch.Size([2, 3, 2]))

2.3.3 矩阵

声明矩阵

一个二维张量,$R^{(M·N)}$ 对应A(m,n)

1 2	A = torch.arange(20, dtype=torch.float32).reshape(5, 4) A

tensor([[ 0.,  1.,  2.,  3.],
        [ 4.,  5.,  6.,  7.],
        [ 8.,  9., 10., 11.],
        [12., 13., 14., 15.],
        [16., 17., 18., 19.]])

矩阵转制

A.T

tensor([[ 0.,  4.,  8., 12., 16.],
        [ 1.,  5.,  9., 13., 17.],
        [ 2.,  6., 10., 14., 18.],
        [ 3.,  7., 11., 15., 19.]])

按元素相乘 Hadamard Product

1 2	B = A.clone() A * B

tensor([[  0.,   1.,   4.,   9.],
        [ 16.,  25.,  36.,  49.],
        [ 64.,  81., 100., 121.],
        [144., 169., 196., 225.],
        [256., 289., 324., 361.]])

矩阵降维

矩阵的按列降维和按行降维

A_sum_axis0 = A.sum(axis=0)
A_sum_axis1 = A.sum(axis=1)
print(A_sum_axis0, A_sum_axis0.shape)
print(A_sum_axis1, A_sum_axis1.shape)

tensor([40., 45., 50., 55.]) torch.Size([4])
tensor([ 6., 22., 38., 54., 70.]) torch.Size([5])

求平均值，按平均值降维，矩阵数据必须是dtype=torch.float32

1	A.mean(), A.mean(axis=0), A.mean(axis=1)

(tensor(9.5000),
 tensor([ 8.,  9., 10., 11.]),
 tensor([ 1.5000,  5.5000,  9.5000, 13.5000, 17.5000]))

非均值降维

1 2	sum_A = A.sum(axis=1,keepdim=True) A, sum_A, A/sum_A

(tensor([[ 0.,  1.,  2.,  3.],
         [ 4.,  5.,  6.,  7.],
         [ 8.,  9., 10., 11.],
         [12., 13., 14., 15.],
         [16., 17., 18., 19.]]),
 tensor([[ 6.],
         [22.],
         [38.],
         [54.],
         [70.]]),
 tensor([[0.0000, 0.1667, 0.3333, 0.5000],
         [0.1818, 0.2273, 0.2727, 0.3182],
         [0.2105, 0.2368, 0.2632, 0.2895],
         [0.2222, 0.2407, 0.2593, 0.2778],
         [0.2286, 0.2429, 0.2571, 0.2714]]))

矩阵处理

按列、行累加

1	A, A.cumsum(axis=0), A.cumsum(axis=1)

(tensor([[ 0.,  1.,  2.,  3.],
         [ 4.,  5.,  6.,  7.],
         [ 8.,  9., 10., 11.],
         [12., 13., 14., 15.],
         [16., 17., 18., 19.]]),
 tensor([[ 0.,  1.,  2.,  3.],
         [ 4.,  6.,  8., 10.],
         [12., 15., 18., 21.],
         [24., 28., 32., 36.],
         [40., 45., 50., 55.]]),
 tensor([[ 0.,  1.,  3.,  6.],
         [ 4.,  9., 15., 22.],
         [ 8., 17., 27., 38.],
         [12., 25., 39., 54.],
         [16., 33., 51., 70.]]))

2.3.4 点乘与乘法

向量点乘

点乘等价于每个元素相乘并求和

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y = torch.ones(4, dtype=torch.float32)
print(torch.dot(x, y))
print(torch.sum(x * y))

tensor(6.)
tensor(6.)

矩阵-向量乘法

1	torch.mv(A, x)

tensor([ 14.,  38.,  62.,  86., 110.])

矩阵-矩阵乘法

1 2	B = torch.ones(4, 3) torch.mm(A, B)

tensor([[ 6.,  6.,  6.],
        [22., 22., 22.],
        [38., 38., 38.],
        [54., 54., 54.],
        [70., 70., 70.]])

2.3.5 范数

L1范数

$||x||_1 = \sum_{i=1}^{n}{|x_i|}$

1 2	u = torch.tensor([3., 4., 12.]) torch.abs(u).sum()

tensor(19.)

L2范数

$norm = ||x||_2 = \sqrt{\sum_{i=1}^{n}{x_i^2}}$

1	torch.norm(u)

tensor(13.)