2023/2024

# Linear Algebra for Data Science

Type:
Mago-Lego

Delivered by:
Department of Mathematics

When:
1, 2 module

Open to:
students of all HSE University campuses

Instructors:
Dmitri Piontkovski

Language:
English

ECTS credits:
6

Contact hours:
54

### Course Syllabus

#### Abstract

In the course we consider different linear algebra applications and methods, which can be applied to problems arising in other areas of science. These applications are very important and usually beyond the standard undergraduate courses. The course consists of both theoretical background and practical experience of solutions of linear algebra problems which can be used in computer science, data analysis, machine learning, mathematical modeling, and economical models. We discuss some topics of matrix analysis and numerical methods of linear algebra as well as some elements of functional analysis and mathematical statistics. We provide a number of useful algorithms which can be implemented and used in practice. In the course students are invited to give talks on the topics of choice related to applied or theoretical linear algebra. We also plan to invite some external lecturers who successfully apply linear algebra in their work.

#### Learning Objectives

- Upon completion of this course students would be able to better understand and master the methods of applied linear algebra used in computer science, data analysis, machine learning, mathematical modeling, and economical models.

#### Expected Learning Outcomes

- To apply Chebyshev polynomials to various function interpolation problems
- To apply the First and Second Gershgorin’s theorems to evaluate the eigenvalues of a given matrix
- To apply the iterative method of finding the Perron eigenvector in the PageRank algorithm
- To calculate and use SVD and full rank decompositions
- To calculate functions of square matrices, including matrix powers, exponents, polynomials, and logarithms using either the Jordan form or the Lagrange−Sylvester) polynomials
- To calculate persistence diagram for a given dataset using methods from TDA.
- To calculate the matrix norms induced by the most important vector norms
- To evaluate matrix norms using the spectral radius and the singular radius of a matrix
- To evaluate the number of iterations needed to find a solution with a given precision
- To find a norm of given vector by using a unit circle associated to the norm only
- To find matrix norms compatible with a given vector norm
- To find the condition number of a matrix and to use it to evaluate the error of a solution of a system of linear equations
- To find the pseudoinverse matrix and to use to find the minimal length solution of an indefinite linear system
- To find various norms of the same vectors and to find a vector norm which is the most appropriate for a give problem
- To interpolate an unknown function by polynomials and by polynomial splines
- To interpolate an unknown function by polynomials and by polynomial splines,
- To understand the least squares method and to find the least square solution of a system of linear equations
- To understand the linear productive model and to use the Leontiev inverse for evaluating the direct and indirect multi-sector transactions in the global economy
- To use advanced effective methods for finding the characteristic polynomials and the eigenvalues of a matrix
- To use Bézier curve for elegant smoothing of a polygon
- To use dot product in spaces of functions and orthogonal families of polynomials to for approximation problems
- To use Hermite polynomial to interpolate functions with known values of the derivatives
- To use iterative methods for finding an eigenvector and an eigenvalue of a matrix, including the Perron eigenvector and the Perron-Frobenius eigenvalue of anindecomposable nonnegative matrix
- To use PageRank for influence ranking of a social network of a known configuration
- To use the condition number to evaluate the error of an approximatecalculation of a matrix inverse
- To use the linear regression model to make simple prognoses
- To use the various iteration methods (including the Jacobi method and the Gauss-Seidel method) for solutions of (large) systems of linear equations

#### Course Contents

- Topic 1. Pseudoinverse matrix and the least squares method
- Topic 2. Matrix decomposition.
- Topic 3. Metrics and norms. Matrix norms.
- Topic 4. Elements of perturbation theory.
- Topic 5. Linear algebra and optimization.
- Topic 6. Introduction to topological data analisys (TDA).

#### Interim Assessment

- 2023/2024 2nd moduleMIN(10, (Test 1 + Test 2 +Bonus Score)/2), where Test 1 and Test 2 are the grades for the intermediate and the final task , and Bonus Score include the grade for the (optional) oral presentation and, in extraordinary cases, for the activity during the classes.

#### Bibliography

#### Recommended Core Bibliography

- F. Aleskerov, H. Ersel, D. Piontkovski. Linear Algebra for Economists. Springer, 2011

#### Recommended Additional Bibliography

- David A. Cox, John Little, Donal O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. - Springer International Publishing, Switzerland, 2015. Print ISBN: 978-3-319-16720-6. Online ISBN: 978-3-319-16721-3.