[人工智能]torch 的 3种矩阵乘法运算

torch.tensor * torch.tensor

当操作符是最最最自然的 “*” 时，执行的时 element-wise 乘法，操作数会 broadcast。
更多细节请见Tensor unsqueeze 以 broadcast

torch.mm（就是执行矩阵乘法，1维不能作参数）

就是执行矩阵乘法。
torch.mm(input, mat2, *, out=None) → Tensor
Performs a matrix multiplication of the matrices input and mat2.
If input is a (n $\times$ m) tensor, mat2 is a (m $\times$ p) tensor, out will be a (n $\times$ p)tensor.

例1: [3,3] [3,3]

import torch
Mat1 = torch.tensor([[1, 6, 7],
                      [2, 5, 8],
                      [3, 4, 9]])
Mat2 = torch.tensor([[9, 6, 3],
                      [8, 5, 2],
                      [7, 4, 1]])

Mat3 = torch.mm(Mat1, Mat2, out=None)
print(Mat3)

输出

tensor([[106,  64,  22],
        [114,  69,  24],
        [122,  74,  26]])

例2 [1,2] [2,3]

import torch
Mat1 = torch.tensor([[1, 3]])
print("Mat1's shape: ",Mat1.shape)
Mat2 = torch.tensor([[6, 4, 2],
                      [5, 3, 1]])
print("Mat2's shape: ",Mat2.shape)
Mat3 = torch.mm(Mat1, Mat2, out=None)
print(Mat3)

输出：

Mat1's shape:  torch.Size([1, 2])
Mat2's shape:  torch.Size([2, 3])
tensor([[21, 13,  5]])

例3 与例2对比，只相差一个括号

将 Mat1 修改为

Mat1 = torch.tensor([1, 3])

输出：

Mat1's shape:  torch.Size([2])
Mat2's shape:  torch.Size([2, 3])
    Mat3 = torch.mm(Mat1, Mat2, out=None)
RuntimeError: self must be a matrix

例4 [2,2] [3,3] 报错

import torch
Mat1 = torch.tensor([[1, 3],
                      [2, 4]] )

Mat2 = torch.tensor([[6, 4, 2],
                      [5, 3, 1],
                     [7,8,9]])

Mat3 = torch.mm(Mat1, Mat2, out=None)
print(Mat3)

输出：

    Mat3 = torch.mm(Mat1, Mat2, out=None)
RuntimeError: mat1 and mat2 shapes cannot be multiplied (2x2 and 3x3)

torch.mutual

1 维tensor 和 1 维 tensor（向量点乘）

import torch
Vec1 = torch.tensor([6,4,2])
print("Vec1's shape: ",Vec1.shape)
Vec2 = torch.tensor([5,3,1])
print("Vec2's shape: ",Vec2.shape)
Vec3 = torch.matmul(Vec1, Vec2)
print("Vec3: ",Vec3)
print("Vec3's shape: ",Vec3.shape,"\n")

输出

Vec1's shape:  torch.Size([3])
Vec2's shape:  torch.Size([3])
Vec3:  tensor(44)
Vec3's shape:  torch.Size([]) 

注意，Vec3 的 shape 是 torch.Size([]) 。
但是如果直接 print(torch.tensor([44]).shape)
会得到 torch.Size([1]) 而不是 torch.Size([])

2 维tensor 和 2 维 tensor

Mat1 = torch.tensor([[1, 3]])
print("Mat1's shape: ",Mat1.shape)
Mat2 = torch.tensor([[6, 4, 2],
                      [5, 3, 1]])
print("Mat2's shape: ",Mat2.shape)
Mat3 = torch.matmul(Mat1, Mat2)
print("Mat3: ",Mat3)
print("Mat3's shape: ",Mat3.shape,"\n")

Mat4 = torch.matmul(Mat2, Mat1)
print("Mat4: ",Mat4)
print("Mat4's shape: ",Mat4.shape,"\n")

输出：

Mat1's shape:  torch.Size([1, 2])
Mat2's shape:  torch.Size([2, 3])

Mat3:  tensor([[21, 13,  5]])

Mat3's shape:  torch.Size([1, 3]) 

Traceback (most recent call last):
  File "D:/Test2022.py", line 29, in <module>
    Mat4 = torch.matmul(Mat2, Mat1)
RuntimeError: mat1 and mat2 shapes cannot be multiplied (2x3 and 1x2)

注意，上面的 Mat3 结果和 torch.mm 计算出来的例 2 的结果一样的。
说明 2 维 tensor 与 2 维 tensor，torch.matmul 函数也是执行矩阵乘法。
注意 Mat4 的报错。

1维-2 维，2维-1维

import torch

# first argument 1D and second argument 2D
mat1_1 = torch.tensor([3, 6, 2])

mat1_2 = torch.tensor([[1, 2, 3],
                       [4, 3, 8],
                       [1, 7, 2]])

out_1 = torch.matmul(mat1_1, mat1_2)
print("\n1D-2D matmul :\n", out_1)

# first argument 2D and second argument 1D
mat2_1 = torch.tensor([[1, 2, 3],
                       [4, 3, 8],
                       [1, 7, 2]])

mat2_2 = torch.tensor([3, 6, 2])

# assigning to output tensor
out_2 = torch.matmul(mat2_1, mat2_2)

print("\n2D-1D matmul :\n", out_2)

输出：

1D-2D matmul :
 tensor([29, 38, 61])

2D-1D matmul :
 tensor([21, 46, 49])

第一种情况1D-2D matmul 可以用
$1\times 3$ 和 $3\times 3$ 的矩阵乘法来理解。
第二种情况可以用
$3\times 3$ 和 $3\times 1$ 的矩阵乘法来理解。

批矩阵乘法

import torch

# first argument 1D and second argument 2D
Mat1 = torch.tensor([[[1, 4,-9],
                       [2,-5,8],
                       [3,6,-7]],
                      [[2, 4, 6],
                       [1, 3, 5],
                       [7, 8, 9]]])
print("Mat1's shape: ",Mat1.shape)
Mat2 = torch.tensor([[[1, 2, 3],
                       [4, 3, 8],
                       [1, 7, 2]],
                      [[1, 7, 2],
                       [3, 2, 3],
                       [1, 1, 2]]])
print("Mat2's shape: ",Mat2.shape)
Out1 = torch.matmul(Mat1, Mat2)
print("\n3D-3D matmul :\n", Out1)
print("Out1's shape: ",Out1.shape)

输出：

Mat1's shape:  torch.Size([2, 3, 3])
Mat2's shape:  torch.Size([2, 3, 3])

3D-3D matmul :
 tensor([[[  8, -49,  17],
         [-10,  45, -18],
         [ 20, -25,  43]],

        [[ 20,  28,  28],
         [ 15,  18,  21],
         [ 40,  74,  56]]])
Out1's shape:  torch.Size([2, 3, 3])

Process finished with exit code 0

注，输入的两个 tensor 的 shape 都是
$$[2,3,3]$$
输出的 tensor 的shape 也是
$$[2,3,3]$$
实际上是 2 个
$$3\times 3$$
的矩阵对应相乘，拼成一个
$$[2,3,3]$$
的输出