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2.3 线性代数

发布时间 2023-05-24 10:26:26作者: AncilunKiang

2.3.1 标量

import torch

标量由只有一个元素的张量表示，可进行熟悉的算数运算。

x = torch.tensor(3.0)
y = torch.tensor(2.0)
x, y, x+y, x*y, x/y, x**y

(tensor(3.), tensor(2.), tensor(5.), tensor(6.), tensor(1.5000), tensor(9.))

2.3.2 向量

向量通过只有一个轴的张量表示。与普通的Python数组一样可通过索引访问，也可以通过调用Python的内置函数 len 来访问张量长度。

x = torch.arange(4)
x, x[3], len(x), x.shape

(tensor([0, 1, 2, 3]), tensor(3), 4, torch.Size([4]))

2.3.3 矩阵

矩阵通过具有两个轴的张量表示，可通过行索引和列索引访问矩阵中的标量元素，也可进行诸如转置之类的操作。

A = torch.arange(20).reshape(5, 4)
A, A.T

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

B = torch.tensor([[1, 2, 3], [2, 0, 4], [3, 4, 5]])  # 对称矩阵
B, B.T, B == B.T

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

2.3.4 张量

本节张量指的是代数对象，我的理解是多轴的数组即为张量。

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]]])

2.3.5 张量算法的基本性质

任何按元素的一元运算都不会改变其操作数的形状，同样，给定具有相同形状的任意两个张量进行任何按元素二元运算的结果都将是相同形状的张量。

具体而言，两个矩阵按元素乘法称为哈达玛积（Hadamard product）(数学符号 \(\odot\))

A = torch.arange(20, dtype=torch.float32).reshape(5, 4)
B = A.clone()  # 开辟新内存，复制一个A，若仅是B=A则只是引用
A, A+B, 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.]]),
 tensor([[  0.,   1.,   4.,   9.],
         [ 16.,  25.,  36.,  49.],
         [ 64.,  81., 100., 121.],
         [144., 169., 196., 225.],
         [256., 289., 324., 361.]]))

将张量加上或乘以一个标量也不会改变张量的形状，而是张量的每个元素都将与标量进行相加。

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]))

2.3.6 降维

默认情况下，调用求和函数 sum 会沿着所有的轴降低张量的维度，使它变成一个标量。当然也可以指定张量沿哪个轴求和降维。

x = torch.arange(4, dtype=torch.float32)
x, x.sum()  # 降为标量

(tensor([0., 1., 2., 3.]), tensor(6.))

A.shape, A.sum()  # 任意形状均可

(torch.Size([5, 4]), tensor(190.))

A_sum_axis0 = A.sum(axis=0)  # 沿0轴求和降维
A_sum_axis0, A_sum_axis0.shape

(tensor([40., 45., 50., 55.]), torch.Size([4]))

A_sum_axis1 = A.sum(axis=1)  # 沿1轴求和降维
A_sum_axis1, A_sum_axis1.shape

(tensor([ 6., 22., 38., 54., 70.]), torch.Size([5]))

A.sum(axis=[0, 1])  # 同时沿着0轴和1轴求和，等效于 A.sum()

tensor(190.)

与求和有关的量是平均值，可以通过总和除以元素数求得，也可以直接调用 mean 函数。同样，计算均值也可以沿指定轴方向降低维度。

A.mean(), A.sum()/A.numel()

(tensor(9.5000), tensor(9.5000))

A.mean(axis=0), A.sum(axis=0)/A.shape[0]

(tensor([ 8.,  9., 10., 11.]), tensor([ 8.,  9., 10., 11.]))

有时候需要非降维求和，即指定求和但是不降维，然后可以通过广播机制计算 A/sum_A。

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

(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]]))

也可以调用 cumsum 函数沿某个轴计算元素的累积总和，如 axis=0 则是按行累加。

A, 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.]]),
 tensor([[ 0.,  1.,  2.,  3.],
         [ 4.,  6.,  8., 10.],
         [12., 15., 18., 21.],
         [24., 28., 32., 36.],
         [40., 45., 50., 55.]]))

2.3.7 点积

点积即为按元素相乘再求和。可以调用 dot 函数求点积，等效于先做乘法再求和。

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

(tensor([0., 1., 2., 3.]), tensor([1., 1., 1., 1.]), tensor(6.), tensor(6.))

2.3.8 矩阵-向量积

矩阵-向量积即矩阵每行分别与向量做点积。注意，矩阵的列维数需与向量维数相同。

A.shape, x.shape, A, x, torch.mv(A, x)

(torch.Size([5, 4]),
 torch.Size([4]),
 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.]),
 tensor([ 14.,  38.,  62.,  86., 110.]))

2.3.9 矩阵-矩阵乘法

矩阵-矩阵乘法即一矩阵每行分别与另一矩阵每列做点积。注意，前者的列维度需与后者行维度相同。莫要把矩阵乘法与哈达玛积混淆。

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

(tensor([[ 0.,  1.,  2.,  3.],
         [ 4.,  5.,  6.,  7.],
         [ 8.,  9., 10., 11.],
         [12., 13., 14., 15.],
         [16., 17., 18., 19.]]),
 tensor([[1., 1., 1.],
         [1., 1., 1.],
         [1., 1., 1.],
         [1., 1., 1.]]),
 tensor([[ 6.,  6.,  6.],
         [22., 22., 22.],
         [38., 38., 38.],
         [54., 54., 54.],
         [70., 70., 70.]]))

2.3.10 范数

向量的范数（norm）表示一个向量的大小，此处的大小不涉及维度，而是分量的大小。

在线性代数中，向量的范数是将向量映射到标量的函数 \(f\)。给定任意向量 \(x\)，向量的范数具有以下三个性质：

  1. 如果按常数因子 \(\alpha\) 缩放向量则其范数也会按相同的常数因子的绝对值缩放：

\[ f(\alpha x)=|\alpha|f(x) \]

  1. 三角不等式：

\[ f(x+y)\le f(x)+f(y) \]

  1. 非负性：

\[f(x)\ge 0 \]

范数很像距离的度量。事实上，欧几里得距离是一个 \(L_2\) 范数：假设 \(n\) 维向量 \(x\) 中的元素是 \(x_1,\dots,x_n\)，其 \(L_2\) 范数是向量元素平方和的平方根：

\[||x||_2=\sqrt{\sum^n_{i=1}x^2_i} \]

其中，再 \(L_2\) 范数中常常省略下标 2，即 \(||x||_2\) 等同于 \(||x||\)。

可以调用 norm 函数计算向量的 \(L_2\) 范数。

u = torch.tensor([3.0, -4.0])
torch.norm(u)

tensor(5.)

深度学习中也会经常遇到 \(L_1\) 范数，它表示为向量元素的绝对值之和：

\[||x||_1=\sum^n_{i=1}|x_i| \]

与 \(L_2\) 范数相比，\(L_1\) 范数受异常值的影响较小，可以调用 abs 函数和 sum 函数计算 \(L_1\) 范数。

torch.abs(u).sum()

tensor(7.)

\(L_2\) 范数和 \(L_1\) 范数都是更一般的 \(L_p\) 范数的特例：

\[||x||_p=(\sum^n_{i=1}|x_i|^p)^{1/p} \]

类似于向量的 \(L_2\) 范数，矩阵 \(X\in \mathbb{R}^{m\times n}\) 的弗罗贝尼乌斯范数（Frobenius norm）是矩阵元素平方和的平方根：

\[||X||_F=\sqrt{\sum^m_{i=1}\sum^n_{j=1}x^2_{ij}} \]

弗罗贝尼乌斯范数具有向量范数的所有性质，他就像是矩阵型向量的 \(L_2\) 范数。

可以调用 norm 函数计算矩阵的弗罗贝尼乌斯范数。

torch.norm(torch.ones((4, 9)))

tensor(6.)

练习

（1）证明一个矩阵 \(A\) 的转置的转置是 \(A\)，即 \((A^T)^T=A\)

A = torch.randn(3, 4)
A, A.T, A.T.T, A==A.T.T, torch.equal(A, A.T.T)

(tensor([[-0.4384, -0.5538, -2.5270,  1.3256],
         [-0.4584,  0.5911,  1.3676, -0.7333],
         [ 0.5668, -1.3604,  1.3320, -0.5259]]),
 tensor([[-0.4384, -0.4584,  0.5668],
         [-0.5538,  0.5911, -1.3604],
         [-2.5270,  1.3676,  1.3320],
         [ 1.3256, -0.7333, -0.5259]]),
 tensor([[-0.4384, -0.5538, -2.5270,  1.3256],
         [-0.4584,  0.5911,  1.3676, -0.7333],
         [ 0.5668, -1.3604,  1.3320, -0.5259]]),
 tensor([[True, True, True, True],
         [True, True, True, True],
         [True, True, True, True]]),
 True)

（2）给出两个矩阵 \(A\) 和 \(B\)，证明“它们转置的和”等于“它们和的转置”，即 \(A^T+B^T=(A+B)^T\)。

A, B = torch.randn(3, 4), torch.randn(3, 4)
A, B, A.T, B.T, A.T+B.T, (A+B).T, A.T+B.T==(A+B).T, torch.equal(A.T+B.T, (A+B).T)

(tensor([[ 0.8526, -0.1816,  0.1884, -0.5057],
         [ 0.1776,  0.6299, -0.1878, -0.2197],
         [-0.3169, -0.6792,  1.4165, -0.8142]]),
 tensor([[ 1.3895,  0.9179, -1.6885,  0.7068],
         [-0.8290,  0.6529, -0.6209, -0.1764],
         [-1.5397,  0.3814,  0.0838,  0.5798]]),
 tensor([[ 0.8526,  0.1776, -0.3169],
         [-0.1816,  0.6299, -0.6792],
         [ 0.1884, -0.1878,  1.4165],
         [-0.5057, -0.2197, -0.8142]]),
 tensor([[ 1.3895, -0.8290, -1.5397],
         [ 0.9179,  0.6529,  0.3814],
         [-1.6885, -0.6209,  0.0838],
         [ 0.7068, -0.1764,  0.5798]]),
 tensor([[ 2.2421, -0.6513, -1.8566],
         [ 0.7363,  1.2828, -0.2978],
         [-1.5000, -0.8088,  1.5002],
         [ 0.2011, -0.3961, -0.2345]]),
 tensor([[ 2.2421, -0.6513, -1.8566],
         [ 0.7363,  1.2828, -0.2978],
         [-1.5000, -0.8088,  1.5002],
         [ 0.2011, -0.3961, -0.2345]]),
 tensor([[True, True, True],
         [True, True, True],
         [True, True, True],
         [True, True, True]]),
 True)

（3）给定任意方阵 \(A\)，\(A+A^T\) 总是对称的吗?为什么?

A = torch.randn(4, 4)
A, A.T, A+A.T, A+A.T==(A+A.T).T, torch.equal(A+A.T, (A+A.T).T)

(tensor([[-0.1174, -0.9523,  2.9669, -1.2442],
         [ 0.3419, -0.7263,  1.0194, -0.0063],
         [-1.2912, -0.4803,  0.6785,  1.3618],
         [-0.0641,  0.9961, -2.2250,  1.8944]]),
 tensor([[-0.1174,  0.3419, -1.2912, -0.0641],
         [-0.9523, -0.7263, -0.4803,  0.9961],
         [ 2.9669,  1.0194,  0.6785, -2.2250],
         [-1.2442, -0.0063,  1.3618,  1.8944]]),
 tensor([[-0.2348, -0.6105,  1.6758, -1.3084],
         [-0.6105, -1.4527,  0.5390,  0.9898],
         [ 1.6758,  0.5390,  1.3571, -0.8632],
         [-1.3084,  0.9898, -0.8632,  3.7888]]),
 tensor([[True, True, True, True],
         [True, True, True, True],
         [True, True, True, True],
         [True, True, True, True]]),
 True)

（4）本节中定义了形状为(2, 3, 4)的张量 \(X\)。len(X)的输出结果是什么？

X = torch.arange(24).reshape(2, 3, 4)
X, len(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]]]),
 2)

（5）对于任意形状的张量 \(X\)，len(X)是否总是对应于 \(X\) 特定轴的长度?这个轴是什么?

import random

for i in range(10):
    shape = []
    for j in range(random.randint(1, 9)):  # 生成随机shape
        shape.append(random.randint(1, 9))
    X = torch.zeros(shape)
    print(f'shape is {X.shape}, len(X) is {len(X)}')

# 足以见得len()显示的是0轴的长度

shape is torch.Size([1, 9, 3, 3, 2, 7, 2, 5, 5]), len(X) is 1
shape is torch.Size([8, 5, 7, 1, 3, 5, 5]), len(X) is 8
shape is torch.Size([5, 6, 3, 6, 5, 4, 9, 3]), len(X) is 5
shape is torch.Size([2, 3, 8, 9]), len(X) is 2
shape is torch.Size([7, 3, 3, 3, 8]), len(X) is 7
shape is torch.Size([3, 6, 6, 8, 6, 4, 5]), len(X) is 3
shape is torch.Size([2, 2]), len(X) is 2
shape is torch.Size([4, 9]), len(X) is 4
shape is torch.Size([6, 2, 6, 8, 3]), len(X) is 6
shape is torch.Size([6, 1, 2, 5]), len(X) is 6

（6）运行 A/A.sum(axis=1)，看看会发生什么。请分析一下原因。

A, A.sum(axis=1)
# A/A.sum(axis=1)形状不同则无法计算

(tensor([[-0.1174, -0.9523,  2.9669, -1.2442],
         [ 0.3419, -0.7263,  1.0194, -0.0063],
         [-1.2912, -0.4803,  0.6785,  1.3618],
         [-0.0641,  0.9961, -2.2250,  1.8944]]),
 tensor([0.6530, 0.6286, 0.2688, 0.6014]))

'''
参见2.3.6及1.2.1.练习（2）
可以保留维度然后依据广播机制进行运算
'''
sum_A = A.sum(axis=1, keepdim=True)
A, sum_A, A.shape, sum_A.shape, A/sum_A

(tensor([[-0.1174, -0.9523,  2.9669, -1.2442],
         [ 0.3419, -0.7263,  1.0194, -0.0063],
         [-1.2912, -0.4803,  0.6785,  1.3618],
         [-0.0641,  0.9961, -2.2250,  1.8944]]),
 tensor([[0.6530],
         [0.6286],
         [0.2688],
         [0.6014]]),
 torch.Size([4, 4]),
 torch.Size([4, 1]),
 tensor([[-0.1798, -1.4585,  4.5439, -1.9056],
         [ 0.5439, -1.1555,  1.6217, -0.0101],
         [-4.8027, -1.7867,  2.5239,  5.0654],
         [-0.1067,  1.6564, -3.6999,  3.1501]]))

（7）考虑一个具有形状(2, 3, 4)的张量，在轴0、1、2上的求和输出是什么形状?

X = torch.randn([2, 3, 4])
X, X.shape, X.sum(axis=0), X.sum(axis=0).shape, X.sum(axis=1), X.sum(axis=1).shape, X.sum(axis=2), X.sum(axis=2).shape

(tensor([[[-0.1514, -0.2254, -1.1703, -0.4737],
          [-0.0562,  1.1885,  0.2306,  0.5065],
          [ 0.2709,  1.2751, -2.2213, -0.7125]],
 
         [[ 0.3613,  0.7764,  0.8976,  0.1476],
          [-0.1609,  0.3369, -0.9397, -1.1766],
          [ 0.3401, -0.6927, -1.8565,  0.9088]]]),
 torch.Size([2, 3, 4]),
 tensor([[ 0.2099,  0.5510, -0.2727, -0.3261],
         [-0.2171,  1.5254, -0.7091, -0.6701],
         [ 0.6111,  0.5825, -4.0777,  0.1963]]),
 torch.Size([3, 4]),
 tensor([[ 0.0634,  2.2382, -3.1609, -0.6797],
         [ 0.5405,  0.4206, -1.8986, -0.1202]]),
 torch.Size([2, 4]),
 tensor([[-2.0207,  1.8694, -1.3877],
         [ 2.1828, -1.9403, -1.3002]]),
 torch.Size([2, 3]))

（8）为 linalg.norm 函数提供3个或更多轴的张量，并观察其输出。对于任意形状的张量这个函数计算得到什么?

A, B, C = torch.randn([2, 3, 4]), torch.randn([2, 3, 4, 5]), torch.randn([2, 3, 4, 5, 6])
A, B, C, torch.norm(A), torch.norm(B), torch.norm(C)  # 计算的是所有元素平方和的平方根，也就是L2范数咯

(tensor([[[-1.4605, -0.2626, -0.3146,  0.5800],
          [-0.3007, -1.1624,  1.4690,  0.8512],
          [-1.5118, -1.2382, -0.6615,  0.6414]],
 
         [[ 1.4322, -0.6534, -0.0602, -0.5418],
          [-0.3674, -0.5996, -0.0298,  0.1505],
          [ 1.8959,  1.4191,  1.4638, -1.2170]]]),
 tensor([[[[-1.3103,  0.2245,  0.5714, -0.6395, -0.3428],
           [-1.9542, -0.2442, -0.4077,  0.1311,  1.2586],
           [ 0.9246,  0.3465, -0.1761, -0.3634,  0.8664],
           [-0.2300, -0.1846, -0.9212,  0.6557,  0.4497]],
 
          [[-0.5540,  0.0068,  0.6770,  0.0562,  0.1607],
           [-1.6862, -0.6131, -2.0522, -1.1732,  1.4401],
           [ 1.8759, -1.3954, -1.9391, -1.1330, -0.8850],
           [-0.3903, -1.3384,  0.3776, -0.8520, -0.2333]],
 
          [[-0.1714, -0.6471,  1.0929,  0.0403, -0.3621],
           [-1.2680, -0.6445,  0.9973, -1.6713,  0.9688],
           [ 0.0406, -0.3986, -0.8694, -0.3931, -0.2176],
           [-0.9195, -0.7479,  0.8918, -0.3224,  0.2345]]],

​ [[[-0.2234, -0.4964, 1.0276, -0.9460, 1.5755],
​ [-1.4506, -0.4309, 0.1666, 1.1141, 0.5686],
​ [-0.6848, -0.5883, 0.2217, 0.5959, 1.0737],
​ [-1.2661, 0.2323, -0.3551, -0.2596, 1.2994]],
​
​ [[ 1.0310, 2.4682, 0.1451, 1.6385, -1.0438],
​ [ 1.3069, -0.7536, 1.4158, 1.2705, -1.5419],
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tensor(4.9148),
tensor(10.2012),
tensor(26.9727))