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Mathematics of Machine Learning

Mathematics of Machine Learning

By : Tivadar Danka
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Mathematics of Machine Learning

Mathematics of Machine Learning

By: Tivadar Danka

Overview of this book

Mathematics 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.
Table of Contents (36 chapters)
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2
Part 1: Linear Algebra
11
References
12
Part 2: Calculus
19
References
20
Part 3: Multivariable Calculus
24
References
25
Part 4: Probability Theory
29
References
30
Part 5: Appendix
31
Other Books You May Enjoy
32
Index

8.2 Benefits of the graph representation

Let’s talk about the concrete advantages that the graph representation offers. For one, the powers of the matrix correspond to walks in the graph. Say, for any let A = (ai,j)ni,j=1 ∈ ℝn×n . Its square is denoted by A2 = (a(2))ni,j=1 ∈ ℝn ×n i,j , where the elements  (2) ai,j are defined by

 n a(2) = ∑ a a . i,j i,k k,j k=1

(Note that the (2) in the superscript of ai,j(2) is not an exponent; this is just an index indicating that ai,(2) is the element of A2.)

Figure 8.5 shows the elements of the square matrix and its graph: all possible two-step walks are accounted for in the sum defining the elements of A2.

PIC

Figure 8.5: Powers of the matrix describe walks on its directed graph

There is much more to this connection; for instance, it gives us a deep insight into the structure of nonnegative matrices. To see how, let’s talk about the concept of strongly connected components.

8.2.1 The connectivity of graphs

Intuitively, we can think of connectivity as the ability to reach every node from the others. To formalize this...

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