Introduction to deep learning linear algebra (pytorch)

Teacher Li Mu's study notes

linear algebra

Scalar

Simple operation

c = a + b
c = a·b
c = sin a

length

$|a|=\{\begin{array}{ll}a& ifa>0\\ -a& otherwise\end{array}$
$|a+b|\le |a|+|b|$
$|a\cdot b|=|a|\cdot |b|$

vector

Simple operation

$c=a+bwhere{c}_i=a_i+b_i$
$c=\alpha \cdot bwhere{c}_i=\alpha b_i$
$c=sinawhere{c}_i=sin{a}_i$

length

The length of the vector , That is, sum the squares of each element of the vector and then open the root
$||a|{|}_2=[{\sum }_{i=1}^m{a}_i^2{]}^{\frac{1}{2}}$
$||a||\ge 0foralla$
Trigonometric Theorem $||a+b||\le ||a||+||b||$
If a Is a constant $||a\cdot b||=|a|\cdot ||b||$

Green is c

Point multiplication

$a^{T}b={\sum }_i{a}_i{b}_i$

orthogonal

$a^{T}b={\sum }_i{a}_i{b}_i=0$

matrix

Simple operation

Multiplication （ Matrix times vector ）

Multiplication （ Matrix times matrix ）

norm

$$c=A\cdot bhence||c||\le ||A||\cdot ||b||$$

• Depends on how you measure b and c The length of
  
  
• Common norm
• Matrix constant ： The minimum value that satisfies the above formula
• Frobenius norm
  $$||A|{|}_{Frob}=[\sum _{ij}{A}_{ij}^2{]}^{\frac{1}{2}}$$

Special matrix

• Orthogonal matrix
  All rows are orthogonal to each other
  All lines have a unit length \qquad $Uwith{\sum }_j{U}_{ij}{U}_{kj}={\delta }_{ik}$
  It can be written. $U{U}^T=1$

• Permutation matrix
  $Pwhere{P}_{ij}=1ifandonlyifj=\pi (i)$
  A permutation matrix is an orthogonal matrix

Eigenvectors and eigenvalues

Matrix is to distort space

• A vector that is not redirected by a matrix
  
• Symmetric matrices always find eigenvectors

Linear algebraic implementation

Scalar

Scalars are represented by tensors with only one element

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.]))

vector

You can think of vectors as a list of scalars

x = torch.arange(4)
x

tensor([0, 1, 2, 3])
Any element is accessed through the index of the tensor

x[3]

tensor(3)

index from 0 At the beginning , Mathematical index It's from 1 At the beginning

The length of the access tensor

len(x)

4
Tensor with only one axis , Character has only one element

x.shape

torch.size([4])

matrix

By specifying two components m and n To create a character named m × n Matrix

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]])

The transpose of the matrix

A.T

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

Symmetric matrix （symmetric matrix）A Equal to its transpose ：$A=A^T$

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]])


Just like a vector is a generalization of scalar , A matrix is a generalization of a vector , We can build data structures with more axes

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]]])


Given any two tensors with the same properties , Any structure settled by element duality will be a tensor of the same character .

A = torch.arange(20, dtype=torch.float32).reshape(5, 4)
B = A.clone()  #  By allocating new memory , take A A copy of is assigned to 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.]]))

The multiplication of two matrices by elements is called Hadamaji （Hadamard product）( Mathematical symbols ⊙)

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]))

Calculate the sum of its elements

x = torch.arange(4, dtype=torch.float32)
x, x.sum()

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

Specifies the axis of the summation tensor

A = torch.arange(20*2).reshape(2, 5, 4)

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, 2])	# Same as `A.sum()`

tensor(780)

It can be understood as axis=？ , Just get rid of that dimension

A quantity related to summation is Average （mean or average）

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

Keep the number of axes unchanged when calculating the sum or mean

sum_A = A.sum(axis=1, keepdims=True)
sum_A

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

Broadcast A Divide sum_A

A / sum_A

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]])

Calculation of an axis A Cumulative sum of elements

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.]])

Dot product is the sum of the product of elements in the same position

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.))

We can perform multiplication by element , Then sum to represent the dot product of the two vectors

torch.sum(x * y)

tensor(6.)

Matrix vector product Ax Is a length of m The column vector , Its $i^{th}$ Elements are dot products $a_i^{T}x$

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

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

We can put the matrix - Matrix multiplication AB Think of it as simply performing m Submatrix - Vector product , And spliced the results together , formation n X m matrix

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.]])

$L_2$ norm Is the square root of the sum of squares of vector elements ：
$$||x|{|}_2=\sqrt{\sum _{i=1}^n{x}_i^2}$$

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

tensor(5.)

norm Is the length of a vector or matrix

$L_1$ norm , It is expressed as the sum of the absolute values of the vector elements ：
$$||x|{|}_1=\sum _{i=1}^n|x_i|$$

torch.abs(u).sum()

tensor(7.)

Matrix Frobenius norm (Frobenius norm) Is the square root of the sum of squares of matrix elements ：
$$||X|{|}_F=\sqrt{\sum _{i=1}^m\sum _{j=1}^n{x}_{ij}^2}$$

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

tensor(6.)

Equivalence then pulls this matrix into a vector , Then do the norm of the vector

QA

Q1： Why should machine learning be represented by tensors ？
Statistics use tensors to express .

Q2：copy and clone The difference between ？
copy Memory may not be copied , There are shallow copy and deep copy
clone Memory must be copied

Q3：torch Do not distinguish between line vectors and column vectors ？
If you want to distinguish line vectors and column vectors , You need to use a matrix .
If you use vectors , A vector is a one-dimensional array for a computer .