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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

15.2 Linear functions in multiple variables

One of the most important functions in mathematics is the linear function. In one variable, it takes the form l(x) = ax + b, where a and b are arbitrary real numbers.

We’ve seen linear functions several times already. For instance, Theorem 77 gives that differentiation is equivalent to finding the best linear approximation.

Linear functions, that is, functions of the form

 ∑n f (x1,...,xn) = b+ aixi, b,ai ∈ ℝ i=1

are as important in multiple variables as in one.

To build up a deep understanding, we’ll take a look at the simplest case: a line on the two-dimensional plane.

PIC

Figure 15.5: A line on the plane

Given its normal vector m = (m1,m2) and its arbitrary point v0, the vector x is on the line if and only if m and x v0 is orthogonal, that is, if

⟨m, x − v0⟩ = 0 (15.1)

holds. (15.1) is called the normal vector equation of the line.

By using the bilinearity of the inner product and writing out m,x⟩...

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