Roadmap of math disciplines for machine learning, part1

## Instead of the preface.

Suppose, sitting in the evening in a warm chair, you suddenly had a crazy thought: “Hmm, why don't I find out instead of random selection of hyperparameters of the model, and why it all works?” 3r33232.

stepik.org/course/716

stepik.org/course/711

I did not look at the courses from the lecture hall of the Moscow Institute of Physics and Technology for analysis, but for the sake of completeness I will also give: 3r33275

lectoriy.mipt.ru/course/Maths-Basic_Analysis

lectoriy.mipt.ru/course/Maths-Basic_Analysis-2sem

#### Practice.

To practice and apply the knowledge gained is not that “optional”, but strictly MANDATORY, otherwise the whole theory will hang on you as a dead weight, and you will quickly go to the bottom, even without realizing it.

I propose to consider the following options:

*Demidovich, problem sets with courses MIT 3r-3266. (https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/index.htm)*

Sustainable bread for Data Science and in general for science in general. Unfortunately, people have learned to solve only linear equations and their systems well; for equations of degree 2 and above, there are all very nontrivial theories (commutative algebra, algebraic geometry, and others like them). Therefore, in data analysis, linear models are mainly used (or generalized linear models, such as logistic regressions, perceptrons, etc.).

There are many books on linear algebra in Russian. The problem is that they are written either for mathematicians, or there are depressingly many indexes in them (and there is no forest behind the trees). Often the emphasis in university courses is on Jordan form; other standard forms are often not mentioned; there is Gauss and stupid Kramer, but rarely what happens about LU, about SVD.

The concept of vector and vector space; the concept of a linear operator; communication of operators and matrices; matrix expansions (LU, SVD at least); eigenvectors and eigenvalues; orthogonal, unitary operators; symmetric and Hermitian operators; quadratic forms, leading to the main axes.

3r3183. ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/index.htm

The best thing about this course is the absence of “complex” and rather confusing theorems of linear algebra, all kinds of dual spaces, a large number of problems in the book, a practice-oriented approach (not “what it is”, but “how to calculate it”). More sensible courses on linear algebra, I have not yet met.

## Linear algebra.

Sustainable bread for Data Science and in general for science in general. Unfortunately, people have learned to solve only linear equations and their systems well; for equations of degree 2 and above, there are all very nontrivial theories (commutative algebra, algebraic geometry, and others like them). Therefore, in data analysis, linear models are mainly used (or generalized linear models, such as logistic regressions, perceptrons, etc.).

There are many books on linear algebra in Russian. The problem is that they are written either for mathematicians, or there are depressingly many indexes in them (and there is no forest behind the trees). Often the emphasis in university courses is on Jordan form; other standard forms are often not mentioned; there is Gauss and stupid Kramer, but rarely what happens about LU, about SVD.

#### What you need to know from linear algebra?

The concept of vector and vector space; the concept of a linear operator; communication of operators and matrices; matrix expansions (LU, SVD at least); eigenvectors and eigenvalues; orthogonal, unitary operators; symmetric and Hermitian operators; quadratic forms, leading to the main axes.

#### Literature.

**Bring it on**: [i] OCW-MIT Gilbert Strang course on linear algebra + his book 3r3-33266. .3r3183. ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/index.htm

The best thing about this course is the absence of “complex” and rather confusing theorems of linear algebra, all kinds of dual spaces, a large number of problems in the book, a practice-oriented approach (not “what it is”, but “how to calculate it”). More sensible courses on linear algebra, I have not yet met.

**Hurt me plenty**: [i] Axler "Linear algebra done right"; Gelfand "Lectures on linear algebra 3r-3266. "; MIPT course lectoriy.mipt.ru/course/LinearAlgebra ; [i] Kostrikin “Introduction to Algebra, Part 2”, Tyrtyshnikov “Matrix Analysis and Linear Algebra”.The problem with books and courses from this level of complexity is that they are theoretical-oriented. There are linear functionals and dual spaces, but there is no projection matrix on the subspace and no practical ways of calculating eigenvalues. Most likely, courses from this level will have to be complemented by strong practice; for example, numerical methods of linear algebra.

About the last book separately. In my opinion, this is one of the most successful Russian-language books on linear algebra in the sense that it is not very much divorced from practice; while it contains all sorts of "advanced" topics. To some extent, it can replace Strang’s lectures completely, but it should be supplemented with simple tasks to “tamper with the hand”. There are some problems in this book, but they are rather severe.

**Nightmare**:

*Kostrikin-Manin "Linear algebra and geometry", Shafarevich-Remizov "Linear algebra and geometry."*

In general, there is a lot of good literature in Russian, especially at the last level, but it suffers from excessive complexity.

#### Practice.

As in the first case, the practice is obligatory. Go SVD - learn the compression of images. Pass through matrix multiplications — study the fast Fourier transform, the Strassen algorithm; Solve many problems (for example, from 3r33265. Kostrikin's problem books or Proskuryakova 3r-3666.); write your LU decomposition, Gauss. For the most obstinate, I can offer wonderful books on numerical methods of linear algebra, such as 3r-3265. Trefethen, Bau "NUMERICAL LINEAR ALGEBRA"; Horn, Johnson The Matrix Analysis, 3-333266. . These books will be useful, firstly, for "filling" hands; secondly, it will immediately become clear that many theoretical methods are broken up into pieces about the prose of life (machine accuracy, instability of methods, work with sparse matrices).

## Discrete Math.

Another whale of modern CS. Here we will mainly be interested in combinatorics and fundamentals of graph theory.

#### What you need to know from combinatorics and graph theory?

Binomial coefficients, their asiptotics; graphs; trees; search in depth and width; recursive relations and their solutions;

#### Literature.

**Bring it on**:

*Anderson, J. Discrete Mathematics and Combinatorics; Haggarty, Schlipf J., Whitesides S. “Discrete Mathematics for Programmers”, Ore O. “Graphs and Their Applications” 3-333266. .*

The first two books are excellent Talmuds in discrete mathematics, covering almost all the questions that need to be known.

The first two books are excellent Talmuds in discrete mathematics, covering almost all the questions that need to be known.

**Hurt me plenty**: [i] Graham, Knut, Patashnik "Concrete Mathematics", Harari "Graph Theory", Ore "Graph Theory".**Nightmare**:

*Sachkov “Introduction to combinatorial methods of discrete mathematics”, Omelchenko “Graph theory”.*

#### Practice.

As a rule, a large number of tasks are included in combinatorics textbooks; they need to be resolved. In fact, all combinatorics is the art of solving various problems, rather than some kind of unified theory.

It may be interesting

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#### weber

Author**10-12-2018, 16:38**

Publication Date
#### Mathematics / Machine learning

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