深度学习——线性代数（预备知识）
目录
1.标量:用只有一个元素的张量表示
2.向量：标量值组成的列表
3.矩阵：
4.三维矩阵
5.矩阵运算
6.标量和矩阵
7.计算元素的和
8.指定求和的轴
9.求平均值
10.求维度的平均值
11.计算总和或均值保持轴数不变
12.通过广播A除以sum_A
13计算某个轴的A元素累加和
14dot点积是相同位置按元素乘积然后相加的和
15.Ax是一个长度为m的列向量。向量*矩阵
16.两个矩阵乘法AB
17.求向量或者矩阵的长度 norm 平方和后开根号
18.向量元素绝对值的和
19 矩阵元素的平方和 开根号torch.norm(torch.ones((4, 9))) # tensor(6.)
1.标量:用只有一个元素的张量表示
x = torch.tensor([3.0]) y = torch.tensor([2.0]) x + y # tensor([5.]) x * y # tensor([6.]) x / y # tensor([1.5000]) x ** y # tensor([9.])
2.向量：标量值组成的列表
x = torch.arange(4) x # tensor([0, 1, 2, 3]) # 通过张量的索引访问元素 x[1] # tensor(1) # 通过len获得长度 len(x) # 4 # 形状：只有一个轴的张量，形状只有一个元素 x.shape # torch.Size([4])
3.矩阵：
# 通过m,n 创建一个形状m*n的矩阵
A = torch.arange(20).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]])
'''
# 矩阵转置
A.T
'''
tensor([[ 0, 4, 8, 12, 16],
[ 1, 5, 9, 13, 17],
[ 2, 6, 10, 14, 18],
[ 3, 7, 11, 15, 19]])
'''
# 对称矩阵 A = A.T
B = torch.tensor([[1, 2, 3],
[2, 0, 4],
[3, 4, 5]])
B == B.T
'''
tensor([[True, True, True],
[True, True, True],
[True, True, True]])
'''
4.三维矩阵
x = torch.arange(24).reshape(2, 3, 4)
x
'''
tensor([[[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]],
[[12, 13, 14, 15],
[16, 17, 18, 19],
[20, 21, 22, 23]]])
x[0][1][0] 为 4
'''
5.矩阵运算
# 形状相同的两个张量，按元素二元运算形状不变
A = torch.arange(20, dtype=torch.float32).reshape(5, 4)
B = A.clone() # 分配新内存,A复制给B
A
'''
tensor([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.],
[12., 13., 14., 15.],
[16., 17., 18., 19.]])
'''
A + B
'''
tensor([[ 0., 2., 4., 6.],
[ 8., 10., 12., 14.],
[16., 18., 20., 22.],
[24., 26., 28., 30.],
[32., 34., 36., 38.]])
'''
6.标量和矩阵
a = 2
X = torch.arange(24).reshape(2, 3, 4)
a + X
'''
X的每一个元素，都加上a
tensor([[[ 2, 3, 4, 5],
[ 6, 7, 8, 9],
[10, 11, 12, 13]],
[[14, 15, 16, 17],
[18, 19, 20, 21],
[22, 23, 24, 25]]])
'''
a * X
'''
X的每一个元素，都乘上a
tensor([[[ 0, 2, 4, 6],
[ 8, 10, 12, 14],
[16, 18, 20, 22]],
[[24, 26, 28, 30],
[32, 34, 36, 38],
[40, 42, 44, 46]]])
'''
7.计算元素的和
x = torch.arange(4, dtype=torch.float32) x # tensor([0., 1., 2., 3.]) x.sum() # tensor(6.)
# 表示任意形状张量的元素和 A = torch.arange((20 * 2), dtype=torch.float32).reshape(2, 5, 4) A.shape # torch.Size([2, 5, 4]) A.sum() # tensor(780)
8.指定求和的轴
A_sum_axis0 = A.sum(axis=0) A_sum_axis0
'''
A
tensor([[[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15],
[16, 17, 18, 19]],
[[20, 21, 22, 23],
[24, 25, 26, 27],
[28, 29, 30, 31],
[32, 33, 34, 35],
[36, 37, 38, 39]]])
A.sum(axis=0) 与每一个的 每一行的各个元素相加。
tensor([[20, 22, 24, 26],
[28, 30, 32, 34],
[36, 38, 40, 42],
[44, 46, 48, 50],
[52, 54, 56, 58]])
'''
A_sum_axis1 = A.sum(axis=1)
A_sum_axis1
'''
tensor([[ 40, 45, 50, 55],
[140, 145, 150, 155]])
'''
# A = torch.arange(20 * 2).reshape(2, 5, 4)
A_sum_axis01 = A.sum(axis=[0, 1])
A_sum_axis01
'''
tensor([180, 190, 200, 210])
'''
9.求平均值
A = torch.arange(20, dtype=torch.float32).reshape(5, 4)
'''
tensor([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.],
[12., 13., 14., 15.],
[16., 17., 18., 19.]])
'''
A.sum() / A.numel() A.mean() # tensor(9.5000)
10.求维度的平均值
'''
tensor([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.],
[12., 13., 14., 15.],
[16., 17., 18., 19.]])
'''
''' A.sum(axis=0) tensor([40., 45., 50., 55.]) A.shape[0] 为5 ''' A.sum(axis=0) / A.shape[0] A.mean(axis=0) # tensor([ 8., 9., 10., 11.])
11.计算总和或均值保持轴数不变
sum_A = A.sum(axis=1)
sum_A # tensor([ 6., 22., 38., 54., 70.]) 变了
sum_A = A.sum(axis=1, keepdims=True)
sum_A # 不变[5,1]
'''
tensor([[ 6.],
[22.],
[38.],
[54.],
[70.]])
'''
12.通过广播A除以sum_A
'''
A
tensor([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.],
[12., 13., 14., 15.],
[16., 17., 18., 19.]])
sum_A
tensor([[ 6.,6.,6.,6.],
[22.,22.,22.,22.],
[38.,38.,38.,38.],
[54.,54.,54.,54.],
[70.,70.,70.,70.]])
A / sum_
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]])
'''
13计算某个轴的A元素累加和
'''
A
tensor([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.],
[12., 13., 14., 15.],
[16., 17., 18., 19.]])
'''
A.cumsum(axis=1)
'''
tensor([[ 0., 1., 3., 6.],
[ 4., 9., 15., 22.],
[ 8., 17., 27., 38.],
[12., 25., 39., 54.],
[16., 33., 51., 70.]])
'''
A.cumsum(axis=0)
'''
tensor([[ 0., 1., 2., 3.],
[ 4., 6., 8., 10.],
[12., 15., 18., 21.],
[24., 28., 32., 36.],
[40., 45., 50., 55.]])
'''
14dot点积是相同位置按元素乘积然后相加的和
x = torch.arange(4, dtype=torch.float32) # tensor([0., 1., 2., 3.]) y = torch.ones(4, dtype=torch.float32) # tensor([1., 1., 1., 1.]) torch.dot(x, y) # tensor(6.) # 等价于 torch.sum(x * y)
15.Ax是一个长度为m的列向量。向量*矩阵
'''
A
tensor([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.],
[12., 13., 14., 15.],
[16., 17., 18., 19.]])
X
tensor([0., 1., 2., 3.])
'''
torch.mv(A, x) # tensor([ 14., 38., 62., 86., 110.])
16.两个矩阵乘法AB
A = torch.arange(20, dtype=torch.float32).reshape(5, 4)
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.]])
'''
17.求向量或者矩阵的长度 norm 平方和后开根号
u = torch.tensor([3.0, -4.0]) torch.norm(u) # tensor(5.)
18.向量元素绝对值的和
torch.abs(u).sum() # tensor(7.)