Mathematics of Machine Learning (eBook)

Master linear algebra, calculus, and probability for machine learning

Tivadar Danka (Autor)

eBook Download: EPUB

2025
730 Seiten
Packt Publishing (Verlag)
978-1-83702-786-6 (ISBN)

Infos

Build a solid foundation in the core math behind machine learning algorithms with this comprehensive guide to linear algebra, calculus, and probability, explained through practical Python examplesPurchase of the print or Kindle book includes a free PDF eBook Key FeaturesMaster linear algebra, calculus, and probability theory for MLBridge the gap between theory and real-world applicationsLearn Python implementations of core mathematical conceptsBook DescriptionMathematics of Machine Learning provides a rigorous yet accessible introduction to the mathematical underpinnings of machine learning, designed for engineers, developers, and data scientists ready to elevate their technical expertise. With this book, you ll explore the core disciplines of linear algebra, calculus, and probability theory essential for mastering advanced machine learning concepts. PhD mathematician turned ML engineer Tivadar Danka known for his intuitive teaching style that has attracted 100k+ followers guides you through complex concepts with clarity, providing the structured guidance you need to deepen your theoretical knowledge and enhance your ability to solve complex machine learning problems. Balancing theory with application, this book offers clear explanations of mathematical constructs and their direct relevance to machine learning tasks. Through practical Python examples, you ll learn to implement and use these ideas in real-world scenarios, such as training machine learning models with gradient descent or working with vectors, matrices, and tensors. By the end of this book, you ll have gained the confidence to engage with advanced machine learning literature and tailor algorithms to meet specific project requirements. What you will learnUnderstand core concepts of linear algebra, including matrices, eigenvalues, and decompositionsGrasp fundamental principles of calculus, including differentiation and integrationExplore advanced topics in multivariable calculus for optimization in high dimensionsMaster essential probability concepts like distributions, Bayes' theorem, and entropyBring mathematical ideas to life through Python-based implementationsWho this book is forThis book is for aspiring machine learning engineers, data scientists, software developers, and researchers who want to gain a deeper understanding of the mathematics that drives machine learning. A foundational understanding of algebra and Python, and basic familiarity with machine learning tools are recommended. ]]>

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Introduction

Why do I have to learn mathematics? - This is a question I am asked daily.

Well, you don’t have to. But you should!

On the surface, advanced mathematics doesn’t impact software engineering and machine learning in a production setting. You don’t have to calculate gradients, solve linear equations, or find eigenvalues by hand. Basic and advanced algorithms are abstracted away into libraries and APIs, performing all the hard work for you.

Nowadays, implementing a state-of-the-art deep neural network is almost equivalent to instantiating an object in PyTorch, loading the pre-trained weights, and letting the data blaze through the model. Just like all technological advances, this is a double-edged sword. On the one hand, frameworks that accelerate prototyping and development enable machine learning in practice. Without them, we wouldn’t have seen the explosion in deep learning that we witnessed in the last decade.

On the other hand, high-level abstractions are barriers between us and the underlying technology. User-level knowledge is only sufficient when one is treading on familiar paths. (Or until something breaks.)

If you are not convinced, let’s do a thought experiment! Imagine moving to a new country without speaking the language and knowing the way of life. However, you have a smartphone and a reliable internet connection.

How do you start exploring?

With Google Maps and a credit card, you can do many awesome things there: explore the city, eat in excellent restaurants, and have a good time. You can do the groceries every day without speaking a word: just put the stuff in your basket and swipe your card at the cashier.

After a few months, you’ll also start to pick up some language – simple things like saying greetings or introducing yourself. You are off to a good start!

There are built-in solutions for everyday tasks that just work – food ordering services, public transportation, etc. However, at some point, they will break down. For instance, you need to call the delivery person who dropped off your package at the wrong door. You need to call help if your rental car breaks down.

You may also want to do more. Get a job, or perhaps even start your own business. For that, you need to communicate with others effectively.

Learning the language when you plan to live somewhere for a few months is unnecessary. However, if you want to stay there for the rest of your life, it is one of the best investments you can make.

Now, replace the country with machine learning and the language with mathematics.

The fact is that algorithms are written in the language of mathematics. To get proficient with algorithms, you have to speak it.

What is this book about?

”There is a similarity between knowing one’s way about a town and mastering a field of knowledge; from any given point one should be able to reach any other point. One is even better informed if one can immediately take the most convenient and quickest path from one point to the other.”

— George Pólya and Gábor Szegő, in the introduction of the legendary book Problems and Theorems in Analysis

The above quote is one of my all-time favorites. For me, it says that knowledge rests on many pillars. Like a chair has four legs, a well-rounded machine learning engineer also has a broad skill set that enables them to be effective in their job. Each of us focus on a balanced constellation of skills, and mathematics is a great addition for many. You can start machine learning without advanced mathematics, but at some point in your career, getting familiar with the mathematical background of machine learning can help you bring your skills to the next level.

There are two paths to mastery in deep learning. One starts from the practical parts and the other starts from theory. Both are perfectly viable, and eventually, they intertwine. This book is for those who started on the practical, application-oriented path, like data scientists, machine learning engineers, or even software developers interested in the topic.

This book is not a 100% pure mathematical treatise. At points, I will make some shortcuts to balance between clarity and mathematical correctness. My goal is to give you the “Eureka!” moments and help you understand the bigger picture instead of preparing you for a PhD in mathematics.

Most machine learning books I have read fall into one of two categories.

1. Focus on practical applications, but unclear and imprecise with mathematical concepts.
2. Focus on theory, involving heavy mathematics with almost no real applications.

I want this book to offer the best of both approaches: a sound introduction of basic and advanced mathematical concepts, keeping machine learning in sight at all times.

My goal is not only to cover the bare fundamentals but to give a breadth of knowledge. In my experience, to master a subject, one needs to go both deep and wide. Covering only the very essentials of mathematics would be like a tightrope walk. Instead of performing a balancing act every time you encounter a mathematical subject in the future, I want you to gain a stable footing. Such confidence can bring you very far and set you apart from others.

During our journey, we are going to follow a roadmap that takes us through

1. linear algebra,
2. calculus,
3. multivariable calculus,
4. and probability theory.

We are going to begin our journey with linear algebra. In machine learning, data is represented by vectors. Training a learning algorithm is the same as finding more descriptive representations of data through a series of transformations.

Linear algebra is the study of vector spaces and their transformations.

Simply put, a neural network is just a function that maps the data to a high-level representation. Linear transformations are the fundamental building blocks of these. Developing a good understanding of them will go a long way, as they are everywhere in machine learning.

While linear algebra shows how to describe predictive models, calculus has the tools to fit them to the data. When you train a neural network, you are almost certainly using gradient descent, a technique rooted in calculus and the study of differentiation.

Besides differentiation, its “inverse” is also a central part of calculus: integration. Integrals express essential quantities such as expected value, entropy, mean squared error, etc. They provide the foundations for probability and statistics.

However, when doing machine learning, we deal with functions with millions of variables. In higher dimensions, things work differently. This is where multivariable calculus comes in, where differentiation and integration are adapted to these spaces.

With linear algebra and calculus under our belt, we are ready to describe and train neural networks. However, we lack the understanding of extracting patterns from data. How do we draw conclusions from experiments and observations? How do we describe and discover patterns in them? These are answered by probability theory and statistics, the logic of scientific thinking. In the final chapter, we extend the classical binary logic and learn to deal with uncertainty in our predictions.

How to read this book

Mathematics follows a definition-theorem-proof structure that might be difficult to follow at first. If you are unfamiliar with such a flow, don’t worry. I’ll give a gentle introduction right now.

In essence, mathematics is the study of abstract objects (such as functions) through their fundamental properties. Instead of empirical observations, mathematics is based on logic, making it universal. If we want to use the powerful tool of logic, the mathematical objects need to be precisely defined. Definitions are presented in boxes like this below.

Definition 1. (An example definition)

Definitions appear like this.

Given a definition, results are formulated as if A, then B statements, where A is the premise, and B is the conclusion. Such results are called theorems. For instance, if a function is differentiable, then it is also continuous. If a function is convex, then it has global minima. If we have a function, then we can approximate it with arbitrary precision using a single-layer neural network. You get the pattern. Theorems are the core of mathematics.

We must provide a sound logical argument to accept the validity of a proposition, one that deduces the conclusion from the premise. This is called a proof, responsible for the steep learning curve of mathematics. Contrary to other scientific disciplines, proofs in mathematics are indisputable statements, set in...

Produkt-Details

• Erscheint lt. Verlag: 30.5.2025
• Vorwort: Santiago Valdarrama
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Mathematik / Informatik ► Mathematik
• ISBN-10: 1-83702-786-2 / 1837027862
• ISBN-13: 978-1-83702-786-6 / 9781837027866