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Scalars are regular numbers, such as 7, 82, and 93,454. They only have a magnitude and are used to represent time, speed, distance, length, mass, work, power, area, volume, and so on.

Vectors, on the other hand, have magnitude and direction in many dimensions. We use vectors to represent velocity, acceleration, displacement, force, and momentum. We write vectors in bold—such as a instead of a—and they are usually an array of multiple numbers, with each number in this array being an element of the vector.

We denote this as follows:

Here, shows the vector is in n-dimensional real space, which results from taking the Cartesian product of n times; shows each element is a real number; i is the position of each element; and, finally, is a natural number, telling us how many elements are in the vector.

As with regular numbers, you can add and subtract vectors. However, there are some limitations.

Let's take the vector we saw earlier (x) and add it with another vector (y), both of which are in , so that the following applies:

However, we cannot add vectors with vectors that do not have the same dimension or scalars.

Note that when in , we reduce to 2-dimensions (for example, the surface of a sheet of paper), and when n = 3, we reduce to 3-dimensions (the real world).

We can, however, multiply scalars with vectors. Let λ be an arbitrary scalar, which we will multiply with the vector , so that the following applies:

As we can see, λ gets multiplied by each x_i in the vector. The result of this operation is that the vector gets scaled by the value of the scalar.

For example, let , and . Then, we have the following:

While this works fine for multiplying by a whole number, it doesn't help when working with fractions, but you should be able to guess how it works. Let's see an example.

Let , and . Then, we have the following:

There is a very special vector that we can get by multiplying any vector by the scalar, 0. We denote this as 0 and call it the zero vector (a vector containing only zeros).

Key benefits

Understand linear algebra, calculus, gradient algorithms, and other concepts essential for training deep neural networks

Learn the mathematical concepts needed to understand how deep learning models function

Use deep learning for solving problems related to vision, image, text, and sequence applications

Description

Most programmers and data scientists struggle with mathematics, having either overlooked or forgotten core mathematical concepts. This book uses Python libraries to help you understand the math required to build deep learning (DL) models. You'll begin by learning about core mathematical and modern computational techniques used to design and implement DL algorithms. This book will cover essential topics, such as linear algebra, eigenvalues and eigenvectors, the singular value decomposition concept, and gradient algorithms, to help you understand how to train deep neural networks. Later chapters focus on important neural networks, such as the linear neural network and multilayer perceptrons, with a primary focus on helping you learn how each model works. As you advance, you will delve into the math used for regularization, multi-layered DL, forward propagation, optimization, and backpropagation techniques to understand what it takes to build full-fledged DL models. Finally, you’ll explore CNN, recurrent neural network (RNN), and GAN models and their application. By the end of this book, you'll have built a strong foundation in neural networks and DL mathematical concepts, which will help you to confidently research and build custom models in DL.

What you will learn

Understand the key mathematical concepts for building neural network models

Discover core multivariable calculus concepts

Improve the performance of deep learning models using optimization techniques

Cover optimization algorithms, from basic stochastic gradient descent (SGD) to the advanced Adam optimizer

Understand computational graphs and their importance in DL

Explore the backpropagation algorithm to reduce output error

Cover DL algorithms such as convolutional neural networks (CNNs), sequence models, and generative adversarial networks (GANs)

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Davd Suzuki Oct 24, 2024

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Kabeer Oct 09, 2020

The book is a good introductory text to get into the subject. Really clarifies concepts in a very simple and intuitive way. Recommended in particular for anyone who has struggled with math in the past too.

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Matthew Emerick Jun 29, 2020

About This BookThis book is the latest in a collection of attempts to teach aspiring and experience machine learning practitioners the mathematics underlying their field. Knowing this will help anyone truly understand how their algorithms work in more detail and can assist in strengthening their predictions or troubleshooting when the results don't match up.It begins with the basics of linear algebra, calculus, and statistics, moves on to optimization and graph theory, and then digs deeper into more specific mathematics. The second section takes the foundation and looks at different kinds of simpler neural networks. The final section pushes even further into deep learning and makes sure that the reader can explain why their deep learning models do what they do.Who is This For?From the Preface, this book is for anyone who wants to go from the basic mathematical concepts to the exact mathematical structures used in calculating the results for deep learning. It does not expect the reader to know the math, though it would be helpful, but it does expect the reader to know the basics of how machine learning works.With how difficult most people find mathematics, I know that many machine learning practitioners won't want to delve deep into these fundamentals, if they even are curious at all. But every field has a foundation, and I would argue that mathematics is that foundation for machine learning. After all, machine learning is fundamentally a mathematically transformation of the input to the output.Why Was This Written?There are only a handful of books written on this subject, of which I own most of them. Most focus only on the basics of linear algebra, calculus, and statistics. Another perspective on comparing these books is the approach taken. I can think of only one other book written specifically for the mathematics of deep learning, and that one assumes the reader thinks more like a mathematician. This book, instead, sees the reader as a developer with hands on experience who wants to know why their algorithms work. I feel that this is an important contribution to anyone's collection.OrganizationThree three section organization works well here. The author progresses from the basics to the core subject matter indicated by the title of the book without leaving anything important out. Remember, this book isn't about learning everything about these mathematical subjects, but about knowing enough to better understand deep learning. You're aren't going to focus on proving that the math works, but will trust that the math works and apply it.Within each chapter, the author deftly introduces each topic, builds on the subject from the basics up to the more advanced subtopics you should know about, and then gives a summary to refresh the earlier material and bring it all together. There is a complete preface, which I recommend you read through before starting, and a further reading section with other offerings from the publisher on deep learning.Did This Book Succeed?I think the author did very well. It gives the reader all of the mathematics they will need to progress on their machine learning and deep learning journeys. Where I think this book fell short, as it is written for practitioners, is that it could have expanded by another 100-150 pages and added in code to show how a programming language can efficiently calculate the math. This could help the reader write better code. If the author decides to write a second edition, I hope that they will consider this.Rating and Final ThoughtsI give this book a 5 out of 5. It is complete, readable, and helpful. It does what it says it is going to do very well. I commend the author and hope to see more from them. If you are considering buying this book, it is a good addition if you want to be a better machine learning practitioner.

Amazon Customer Oct 09, 2020

I have been a data scientist for several years now and because of the hype I decided to try and learn deep learning. I decided to buy this book and it does a very good job of explaining how the different fields of math such as linear algebra, vector calculus, probability, etc come together to create various neural networks in a very clear and simple manner that anyone can understand. And it gives several good walk-throughs of forward and backward propagation in various NNs and shows comparisons between architectures and possible use cases as well.

TD59 Feb 24, 2021

I teach Machine Learning in graduate school. Many times, students ask me for a book that provides a quick refresh of key math principles. While there are many great books to choose from, I usually put Hands-on Mathematics high on my list.

Hands-On Mathematics for Deep Learning: Build a solid mathematical foundation for training efficient deep neural networks

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