线性代数
标量由只有一个元素的张量表示
import torch
x = torch.tensor([3.0])
y = torch.tensor([2.0])
x+y,x*y,x/y,x**y
(tensor([5.]), tensor([6.]), tensor([1.5000]), tensor([9.]))
将向量视为标量值组成的列表
x = torch.arange(4)
x
tensor([0, 1, 2, 3])
通过张量的索引来访问任一元素
x[3]
tensor(3)
访问张量长度
len(x)
4
只有一个轴的张量，形状只有一个元素
x.shape
torch.Size([4])
通过指定两个分量m和n来创建一个形状为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]])
矩阵的转置
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 等于其转置: A = A^T
B = torch.tensor([[1,2,3],[2,0,4],[3,4,5]])
B
tensor([[1, 2, 3],
[2, 0, 4],
[3, 4, 5]])
B == B.T
tensor([[True, True, True],
[True, True, True],
[True, True, True]])
可以构建具有更多轴的数据结构
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]]])
给定具有相同形状的任何两个张量，任何按元素二元运算的结果都将是相同形状的张量
A = torch.arange(20,dtype = torch.float32).reshape(5,4)
B = A.clone() # 通过分配新内存，将A的一个副本分配给B
A,A+B
(tensor([[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.],
[12., 13., 14., 15.],
[16., 17., 18., 19.]]),
tensor([[ 0., 2., 4., 6.],
[ 8., 10., 12., 14.],
[16., 18., 20., 22.],
[24., 26., 28., 30.],
[32., 34., 36., 38.]]))
两矩阵的按元素乘法,哈达玛积
A * B
tensor([[ 0., 1., 4., 9.],
[ 16., 25., 36., 49.],
[ 64., 81., 100., 121.],
[144., 169., 196., 225.],
[256., 289., 324., 361.]])
a = 2
X = torch.arange(24).reshape(2,3,4)
a+X,(a*X).shape
(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]]]),
torch.Size([2, 3, 4]))
计算元素和
x = torch.arange(4,dtype = torch.float32)
x,x.sum()
(tensor([0., 1., 2., 3.]), tensor(6.))
表示任意形状张量的元素和
A = torch.arange(20*2).reshape(2,5,4)
A.shape,A.sum()
(torch.Size([2, 5, 4]), tensor(780))
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]]])
指定求和汇总张量的轴
A_sum_axis0 = A.sum(axis = 0)
A_sum_axis0,A_sum_axis0.shape
(tensor([[20, 22, 24, 26],
[28, 30, 32, 34],
[36, 38, 40, 42],
[44, 46, 48, 50],
[52, 54, 56, 58]]),
torch.Size([5, 4]))
A_sum_axis1 = A.sum(axis = 1)
A_sum_axis1,A_sum_axis1.shape
(tensor([[ 40, 45, 50, 55],
[140, 145, 150, 155]]),
torch.Size([2, 4]))
A.sum(axis = [0,1]),A.sum(axis = [0,1]).shape
(tensor([180, 190, 200, 210]), torch.Size([4]))
平均值
A = A.float()
A.mean(),A.sum()/A.numel()
(tensor(19.5000), tensor(19.5000))
A.mean(axis = 0),A.sum(axis = 0)/A.shape[0]
(tensor([[10., 11., 12., 13.],
[14., 15., 16., 17.],
[18., 19., 20., 21.],
[22., 23., 24., 25.],
[26., 27., 28., 29.]]),
tensor([[10., 11., 12., 13.],
[14., 15., 16., 17.],
[18., 19., 20., 21.],
[22., 23., 24., 25.],
[26., 27., 28., 29.]]))
计算总和或均值时保持轴数不变
sum_A = A.sum(axis = 1,keepdims =True)
sum_A
tensor([[[ 40., 45., 50., 55.]],
[[140., 145., 150., 155.]]])
通过广播将A 除以 sum_A
A/sum_A
tensor([[[0.0000, 0.0222, 0.0400, 0.0545],
[0.1000, 0.1111, 0.1200, 0.1273],
[0.2000, 0.2000, 0.2000, 0.2000],
[0.3000, 0.2889, 0.2800, 0.2727],
[0.4000, 0.3778, 0.3600, 0.3455]],
[[0.1429, 0.1448, 0.1467, 0.1484],
[0.1714, 0.1724, 0.1733, 0.1742],
[0.2000, 0.2000, 0.2000, 0.2000],
[0.2286, 0.2276, 0.2267, 0.2258],
[0.2571, 0.2552, 0.2533, 0.2516]]])
某个轴计算 A 元素的累积总和
A.cumsum(axis = 0)
tensor([[[ 0., 1., 2., 3.],
[ 4., 5., 6., 7.],
[ 8., 9., 10., 11.],
[12., 13., 14., 15.],
[16., 17., 18., 19.]],
[[20., 22., 24., 26.],
[28., 30., 32., 34.],
[36., 38., 40., 42.],
[44., 46., 48., 50.],
[52., 54., 56., 58.]]])
点积是相同位置的按元素乘积的和
y = torch.ones(4,dtype = torch.float32)
x,y,torch.dot(x,y)
(tensor([0., 1., 2., 3.]), tensor([1., 1., 1., 1.]), tensor(6.))
矩阵向量积，此例子中使用了广播机制
A = torch.arange(20).reshape(5,4)
A = A.float()
A,x
(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.]))
A.shape,x.shape,torch.mv(A,x)
(torch.Size([5, 4]), torch.Size([4]), tensor([ 14., 38., 62., 86., 110.]))
矩阵乘法
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.]])
获取向量L2范数：向量的长度
u = torch.tensor([3.0,-4.0])
torch.norm(u)
tensor(5.)
L1范数
torch.abs(u).sum()
tensor(7.)
矩阵的弗罗贝尼乌斯范数
torch.norm(torch.ones((4,9)))
tensor(6.)