2026/2027
Линейная алгебра для наук о данных
Статус:
Маго-лего
Кто читает:
Департамент математики
Где читается:
Факультет экономических наук
Охват аудитории:
для своего кампуса
Преподаватели:
Пионтковский Дмитрий Игоревич
Язык:
русский
Кредиты:
6
Контактные часы:
52
Программа дисциплины
Аннотация
In the lecture course, we consider some topics of linear algebra beyond the standard first year course which are extremely important for applications. Mostly, these are applications to data analysis and machine learning, as well as to economics and statistics. We begin with inversions of rectangle matrices, that is, we discuss pseudo-inverse matrices (and their connections to the linear regression model). Among others, we discuss iteration methods (and their using in models of random walk on a graph applied to Internet search such as PageRank algorithm), matrix decompositions (such as SVD) and methods of dimension decreasing (with their connection to some image compression algorithms), and the theory of matrix norms and perturbation theory (for error estimates in matrix computations). The course includes also symbolic methods in systems of algebraic equations, approximation problems, Chebyshev polynomials, matrix functions such as exponents etc. We plan to invite some external lecturers who successfully apply linear algebra in their work. The students are also be invited to give their own talks on additional topics of applied or theoretical linear algebra.
Цель освоения дисциплины
- The aim the course is to provide both theoretical background and practical experience of solutions of linear algebra problems which appear in computer science, data analysis, mathematical modelling, machine learning, and economical models. The course covers 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 by students. A number of of these algorithms are in the core of modern machine learning and data analysis.
Планируемые результаты обучения
- Students would be able • to apply Chebyshev polynomials to various function interpolation problems, • to use dot product in spaces of functions and orthogonal families of polynomials to for approximation problems.
- The students will be able to use pseudoinverse matrix for simple solution of linear systems, to construct the least squared approximations, and apply it to the linear regression model.
- The students will be able to find the interpolation polynomials, to construct an interpolation of a function by polynomial splines, and to apply the Bezier curves for sketching pictures and diagrams.
- The students will be able to use metrics and norms in mathematical modelling.
- Students would be able to evaluate errors of approximate calculations such as the linear system solution and the eigenvalue calculation.
- Students would be able to apply iterative methods for linear system solutions and to evaluate the error of the result.
- The students would be able to locate and to find the eigenvalues of a matrix.
- The students would be able to find iteratively the Perron-Frobenius eigenvalue and to apply PageRank and similar methods to measure the entries of computer and social networks.
- The students would be able to find the value matrix functions and to apply them to systems of differential equations.
- Students would be able to use SVD and low rank approximation for the dimensionality reduction and in the Principal Component Analisys.
Содержание учебной дисциплины
- Pseudoinverse matrix, least squares, and linear regression
- Polynomial interpolation
- Metrics and norms
- Chebyshev polynomials and polynomial approximations
- Error evaluations. Elements of the perturbation theory.
- Iterative methods for systems of linear equations
- The eigenvalue problem
- Nonnegative matrices and PageRank.
- Functions of matrices
- Low rank approximation and the dimensionality reduction
Элементы контроля
- Final testTake-home test. After the assessment, an oral interview about the submitted solutions may be scheduled. The final grade will then be based on the results of the interview.
- Midterm TestTake-home test. After the assessment, an oral interview about the submitted solutions may be scheduled. The final grade will then be based on the results of the interview.
Промежуточная аттестация
- 2026/2027 4th moduleFinal grade = 0.5 (Test1 + Test 2) + (activity score) + (project bonus), where Test1 and Test 2 states for the midterm and the final tests, activity score states for the addition 1 or 2 scores for a few of the most active students, and project bonus (up to 5 scores) are awarded for an oral presentation with student's project.