机器学习必备：基础线性代数讲解

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"《机器学习大师系列：线性代数基础》是一本专为机器学习从业者设计的教程，旨在帮助他们快速掌握线性代数基础知识。作者强调线性代数在机器学习中的核心地位，它是数据的数学语言，无论是现代统计学的表述，还是机器学习方法的描述，都依赖于向量和矩阵运算。从线性回归到主成分分析等方法，无一不是线性代数的体现。然而，作者指出，许多初学者在学习机器学习时过于强调先从线性代数开始，这可能造成学习路径冗长且效率不高。实际上，更适合开发者的方法是采用“结果导向”的学习策略，先通过实践获取结果，再逐步深化理论理解。作者认为，当从业者最终回到线性代数的学习时，他们往往涉猎了大量与机器学习实际应用不那么直接的知识。线性代数本身是一个广泛的学科，与工程、物理学和量子物理等领域紧密相连，其中包含众多理论和推导，对机器学习模型的实际提升和理解帮助有限。尽管如此，对于机器学习而言，只有特定的线性代数子集是必需的，掌握了基础后，可以进一步深入研究。本书的特色在于用Python来探索数据的数学语言，作者Jason Brownlee在书中分享了实用的教学内容，并提醒读者，所有信息仅供教育目的，使用者需自行承担行动责任。同时，作者对编辑Sarah Martin、技术编辑Arun Koshy和Andrei Cheremskoy表示感谢，并声明本书版权以及版本信息。《机器学习大师系列：线性代数基础》提供了一个高效且针对性强的线性代数入门指南，帮助读者在机器学习的道路上少走弯路，专注于最核心的数学工具。"

About Further Reading

Each lesson includes a list of further reading resources. This may i n cl ud e:

Books and book chapters.

API documentation.

Articles a n d Web p a g es.

Wherever possible, I try to list and link to the relevant API documentation for key functi on s

used in each lesson so you can learn more about them. I have tr i ed to link to books on Amazon

so that you can learn more about t h em. I don’t know everything, and if you discover a good

resource related to a given lesson, please let me know so I can update the book.

About Getting Help

You might need help along the way. Don’t worry; you are not alone.

Help with a Technique?

If you need help with the technica l aspects of a sp e ci ﬁc

operation or technique, see the Further Reading sections at the end of each lesson.

Help with NumPy?

If you need help with using the NumPy library, see the list of

resources in the Further Reading section at the end of each lesson, and also see Appendix

Help with your workst a ti on?

If you need help set t i n g up your envi r on m ent, I would

recommend using Anaconda and following my tutorial in Appendix B.

Help with the math?

Iprovidedalistoflocationswhereyoucansearchforanswers

and ask questions about linear algebra math in Appendix A. You can also see Appendix D

for a crash course on math notation.

Help in general? You can shoot me an email. My details are in Appendix A.

Summary

Are you ready? Let’s dive in!

Next up you will discover a gentl e introduction to the ﬁeld of linear algebra.

1.3. Num er i ca l Linear Algeb ra 3

of numbers called matrices, to create new columns and arrays of numbers. Linear algebra is the

study of lines and planes, vector spaces and mappings that are required for linear transforms.

It is a relatively young ﬁeld of study, having initially been formalized in the 1800s in order

to ﬁnd unknowns in systems of linear equations. A linear equation is just a series of terms and

mathematical operations where some terms are unknown; for example:

y =4⇥ x +1 (1.1)

Equations like this are linear in that they describe a line on a two-dim ensi on al graph. The

line comes from plugging in di↵erent values into the unknown

to ﬁnd out what the equation

or model does to the value of

. We can line up a system of equations with the same form with

two or more unknowns; for ex am p l e:

y =0.1 ⇥ x

+0.4 ⇥ x

y =0.3 ⇥ x

+0.9 ⇥ x

y =0.2 ⇥ x

+0.3 ⇥ x

···

(1.2)

The column of

values can be taken as a column vector of outputs from the equation. The

two columns of integer values are the data columns, say

and

, and can be taken as a matrix

. The two unknown values

and

can be taken as the coeﬃcients of the equation and

together form a vect or of unknowns b to be solved. We can write this compactly using linear

algebra notation as:

y = A · b (1.3)

Problems of this form are generally challenging to solve because there are more unknowns

(here we have 2) than there are equations to solve (here we have 3). Further, there is often no

single lin e that can satisfy all of the equations without error. Systems describing problems we

are often interested in (such as a linear regression) can have an inﬁnite number of solutions.

This g i ves a small taste of the very core of linear algebra that interests us as machine learning

practitioners. Much of the rest of the operations are about making thi s problem and problems

like it easier to understand and solve.

1.3 Numerical Linea r Algebra

The appli cat i on of linear algebra in computers is often called numerical linear algebra.

“numerical” linear algebra is really applied linear algebra.

—Pageix,Numerical Linear Algebra,1997.

It is more than just the implementation of linear algebra operations in code libraries; it also

includes the careful handling of the problems of applied mathematics, such as working with th e

limited ﬂoating point precision of digital comput er s. Compu t er s are good at performing linear

algebra calculatio n s, and mu ch of the dependence on Gra p h i ca l Processing Units (GPUs) by

modern machine learning methods such as deep learning is because of their ability to compute

linear algebra operations fast.

1.4. Li n ea r Algebra and Statistics 4

Eﬃcient implementations of vector and matrix operation s were or i gin al l y implemented in

the FORTRAN programming language i n the 1970s a n d 1980s and a lot of code, or code ported

from those implementations, underlies much of the linear algebra performed using modern

programming languages, such as Python. Three popular open source nume ri c al linear algeb ra

libraries that implement these funct i on s are:

Linear Al g eb r a Package, or LAPACK.

Basic Lin ea r Algebra Subp r o gr a m s, or BLAS (a standard for linear algebra libraries).

Automatically Tuned Linear Algebra Software, or ATLAS.

Often, when you are calculating linear algebra operations directly or indirectly via higher-

order algorithms, your code is very likely dipping down to use one of these, or similar linear

algebra libraries. The name of one of more of these underlying libraries may be familiar to you

if you have installed or compiled any of Python’s numerical libraries such a s SciPy and NumPy.

1.4 Linear Algebra and Statis t ics

Linear a l g eb r a is a valuable tool in other branches of mat h ema t i cs, especially statistics.

Usually students studying statistics are expected to have seen at least one semester

of linear algebra (or applied algebra) at the undergraduate level.

—Pagexv,Linear Algebra and Matrix Analysis for Statistics,2014.

The impact of linear algebra is important to consider, given the foundat i on al relationship

both ﬁelds have with the ﬁeld of applied machine learning. Some clear ﬁngerprints of linear

algebra on statistics and statistical methods include:

Use of vector and matrix notation, especially with multivariate statistics.

Solutions to least squares and weighted least squares, such as for linear regression.

Estimates of mean and variance of data matrices.

The covariance matrix that plays a key role in multinomial Gaussian distri bu t i on s.

Principal component analysis for data reduction that draws many of these elements

together.

As you can see, modern statistics and data analysis, at least as far as t h e interests of a

mach i n e learning practitioner are concerned, depend on the u n d er st a n d i n g and tools o f linear

algebra.

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