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机器学习: 线性代数 101

linear_algebra_for_machine_learning_101

Linear Algebra: Scalar, Vectors, Matrices, Tensors¶

Scalar: a single number or value.
Vector: an 1-Dimension array of numbers, either in a row or in a column, identified by an index.
Matrix: a 2-Dimensions array of numbers, where each elements is identified by two indeces. 4. Tensors: more than 2-Dimensions; In general, an array of numbers with a variable number of Dimensions is known as a tensor.
If a 2-Dimensions Matrix has shape (i,j), a 3-Dimensions Tensor would have shape (k,i,j); the number of Matrix (i,j) is k.
If a 2-Dimensions Matrix has shape (i,j), a 4-Dimensions Tensor would have shape (l,m,i,j); the number of Matrix (i,j) is (l,m).

In [68]:

import sys
import numpy as np

Defining a scalar¶

In [69]:

x = 6
x

Out[69]:

Defining a vector¶

In [95]:

x = np.array((1,2,3))
x

Out[95]:

array([1, 2, 3])

In [96]:

print ('Vector Dimensions: {}'.format(x.shape))
print ('Vector size: {}'.format(x.size))

Vector Dimensions: (3,)
Vector size: 3

Defining a matrix¶

In [97]:

x = np.array([[1,2,3],[4,5,6],[7,8,9]])
x

Out[97]:

array([[1, 2, 3],
       [4, 5, 6],
       [7, 8, 9]])

In [98]:

print ('Matrix Dimensions: {}'.format(x.shape))
print ('Matrix size: {}'.format(x.size))

Matrix Dimensions: (3, 3)
Matrix size: 9

Defining a matrix of a given dimension¶

In [99]:

x = np.ones((3,3))
x

Out[99]:

array([[1., 1., 1.],
       [1., 1., 1.],
       [1., 1., 1.]])

Defining a tensor of a given dimension¶

2-Dimensions Matrix (3,5), the number of 2-Dimensions Matrix (3,5) is (2,3)¶

In [100]:

x = np.ones((2,3,3,5))
x

Out[100]:

array([[[[1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.]],

        [[1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.]],

        [[1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.]]],


       [[[1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.]],

        [[1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.]],

        [[1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.],
         [1., 1., 1., 1., 1.]]]])

In [101]:

print ('Tensor Dimensions: {}'.format(x.shape))
print ('Tensor size: {}'.format(x.size))

Tensor Dimensions: (2, 3, 3, 5)
Tensor size: 90

Indexing¶

In [102]:

A = np.ones((5,5), dtype = np.int)
A

Out[102]:

array([[1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1]])

Indexing starts at 0¶

In [103]:

A[0,1] = 2
A

Out[103]:

array([[1, 2, 1, 1, 1],
       [1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1]])

In [104]:

A[:,0] = 3
A

Out[104]:

array([[3, 2, 1, 1, 1],
       [3, 1, 1, 1, 1],
       [3, 1, 1, 1, 1],
       [3, 1, 1, 1, 1],
       [3, 1, 1, 1, 1]])

In [105]:

A[:,:] = 5
A

Out[105]:

array([[5, 5, 5, 5, 5],
       [5, 5, 5, 5, 5],
       [5, 5, 5, 5, 5],
       [5, 5, 5, 5, 5],
       [5, 5, 5, 5, 5]])

2-Dimensions Matrix (5,5), the number of 2-Dimensions Matrix (5,5) is 6¶

In [109]:

A = np.ones((6,5,5), dtype = np.int)
A

Out[109]:

array([[[1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]],

       [[1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]],

       [[1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]],

       [[1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]],

       [[1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]],

       [[1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]]])

For higher dimensions, simply add an index; Assign first row a new value¶

In [108]:

A[:,0,0] = 3
A

Out[108]:

array([[[3, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]],

       [[3, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]],

       [[3, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]],

       [[3, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]],

       [[3, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]],

       [[3, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1]]])

Matrix operation¶

In [110]:

A = np.array([[1,2], [3,4]])

print(A)
print ('Matrix Dimensions: {}'.format(A.shape))
print ('Matrix size: {}'.format(A.size))

[[1 2]
 [3 4]]
Matrix Dimensions: (2, 2)
Matrix size: 4

In [111]:

B = np.ones((2,2), dtype = np.int)

print(B)
print ('Matrix Dimensions: {}'.format(B.shape))
print ('Matrix size: {}'.format(B.size))

[[1 1]
 [1 1]]
Matrix Dimensions: (2, 2)
Matrix size: 4

Element wise sum¶

In [112]:

C = A + B

print(C)
print ('Matrix Dimensions: {}'.format(C.shape))
print ('Matrix size: {}'.format(C.size))

[[2 3]
 [4 5]]
Matrix Dimensions: (2, 2)
Matrix size: 4

Element wise subtraction¶

In [85]:

C = A - B

print(C)
print ('Matrix Dimensions: {}'.format(C.shape))
print ('Matrix size: {}'.format(C.size))

[[0 1]
 [2 3]]
Matrix Dimensions: (2, 2)
Matrix size: 4

Element wise multiplication¶

In [113]:

C = np.dot(A, B)

print(C)
print ('Matrix Dimensions: {}'.format(C.shape))
print ('Matrix size: {}'.format(C.size))

[[3 3]
 [7 7]]
Matrix Dimensions: (2, 2)
Matrix size: 4

Matrix transpose¶

In [88]:

# matrix transpose
A = np.array(range(9))
A = A.reshape(3,3)

print(A)
print ('Matrix Dimensions: {}'.format(A.shape))
print ('Matrix size: {}'.format(A.size))

[[0 1 2]
 [3 4 5]
 [6 7 8]]
Matrix Dimensions: (3, 3)
Matrix size: 9

In [89]:

B = A.T

print(B)
print ('Matrix Dimensions: {}'.format(B.shape))
print ('Matrix size: {}'.format(B.size))

[[0 3 6]
 [1 4 7]
 [2 5 8]]
Matrix Dimensions: (3, 3)
Matrix size: 9

In [90]:

C = B.T

print(C)
print ('Matrix Dimensions: {}'.format(C.shape))
print ('Matrix size: {}'.format(C.size))

[[0 1 2]
 [3 4 5]
 [6 7 8]]
Matrix Dimensions: (3, 3)
Matrix size: 9

In [91]:

A = np.array(range(10))
A = A.reshape(2,5)

print(A)
print ('Matrix Dimensions: {}'.format(A.shape))
print ('Matrix size: {}'.format(A.size))

[[0 1 2 3 4]
 [5 6 7 8 9]]
Matrix Dimensions: (2, 5)
Matrix size: 10

In [92]:

B = A.T

print(B)
print ('Matrix Dimensions: {}'.format(B.shape))
print ('Matrix size: {}'.format(B.size))

[[0 5]
 [1 6]
 [2 7]
 [3 8]
 [4 9]]
Matrix Dimensions: (5, 2)
Matrix size: 10

Tensor

In [94]:

# tensor
A = np.ones((3,3,3,3,3,3,3,3,3,3), dtype = np.int)

print ('Matrix Dimensions: {}'.format(A.shape))
print ('Matrix size: {}'.format(A.size))

Matrix Dimensions: (3, 3, 3, 3, 3, 3, 3, 3, 3, 3)
Matrix size: 59049

In [ ]:

Deep Learning

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方俊贤; Ken Fang; A9 Atlas 工作室

2019, 08. 一种深度学习的算法, 预测微服务持续部署、持续发布后对产品整体质量的影响, 获得国家知识财产局专利; 符合专利法实施细则第 44 条的规定。
专利号: 201910652769.4

主要专长: 运用深度学习的算法模型分析开发者的行为, 以提升软件开发的效率与质量、微服务化的架构设计、探索性测试、有价值的产品特性挖掘、使用者行为 (场景) 分析、领域驱动设计。

曾任职于: 腾讯科技 (深圳) 有限公司专家项目经理; 雅各布森软件 (北京) 有限公司首席谘询顾问; Rational; Telelogic; Borland; 联华电子; 京元电子。

有二十多年半导体、电信产业、军事研究单位与互联网的产品研发与咨询服务等的经验。

于 Illinois Institute of Technology, Chicago, USA 获得电子计算机科学硕士

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机器学习: 线性代数 101 - Cloud-Native 产品级敏捷

机器学习: 线性代数 101

Linear Algebra: Scalar, Vectors, Matrices, Tensors¶

Defining a scalar¶

Defining a vector¶

Defining a matrix¶

Defining a matrix of a given dimension¶

Defining a tensor of a given dimension¶

2-Dimensions Matrix (3,5), the number of 2-Dimensions Matrix (3,5) is (2,3)¶

Indexing¶

Indexing starts at 0¶

2-Dimensions Matrix (5,5), the number of 2-Dimensions Matrix (5,5) is 6¶

For higher dimensions, simply add an index; Assign first row a new value¶

Matrix operation¶

Element wise sum¶

Element wise subtraction¶

Element wise multiplication¶

Matrix transpose¶

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About Joyk

机器学习: 线性代数 101 - Cloud-Native 产品级敏捷

机器学习: 线性代数 101

Linear Algebra: Scalar, Vectors, Matrices, Tensors¶

Defining a scalar¶

Defining a vector¶

Defining a matrix¶

Defining a matrix of a given dimension¶

Defining a tensor of a given dimension¶

2-Dimensions Matrix (3,5), the number of 2-Dimensions Matrix (3,5) is (2,3)¶

Indexing¶

Indexing starts at 0¶

2-Dimensions Matrix (5,5), the number of 2-Dimensions Matrix (5,5) is 6¶

For higher dimensions, simply add an index; Assign first row a new value¶

Matrix operation¶

Element wise sum¶

Element wise subtraction¶

Element wise multiplication¶

Matrix transpose¶

发表评论 取消回复

通过电子邮件订阅博客

Copyright All rights reserved Theme: Blog Expert by Themeinwp

Recommend

About Joyk

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