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Mathematics in Machine Learning

01TXGSM

A.A. 2021/22

2021/22

Mathematics in Machine Learning

The aim of this course is to introduce the students to a solid mathematical foundation of Machine Learning (ML) by blending learning theory, geometry, topology and statistics. Starting from introducing the algebraic and geometric structures used to represent and manipulate data, we will move to the more geometrical aspects of ML with a particular attention to the various concepts of dimension and learnability. Linear algebra-based methods will be thoroughly presented. At the same time, (generalized) linear models, their selection, regularization, validation and hyperparametric tuning will be presented in full detail from a rigorous statistical point of view as well as some Bayesian methods. These two aspects of the theory, the geometrical and the statistical one, will merged via case study on real and in silico data.

Mathematics in Machine Learning

The aim of this course is to introduce the students to a solid mathematical foundation of Machine Learning (ML) by blending learning theory, geometry, topology and statistics. Starting from introducing the algebraic and geometric structures used to represent and manipulate data, we will move to the more geometrical aspects of ML with a particular attention to the various concepts of dimension and learnability. Linear algebra-based methods will be thoroughly presented. At the same time, (generalized) linear models, their selection, regularization, validation and hyperparametric tuning will be presented in full detail from a rigorous statistical point of view as well as some Bayesian methods. These two aspects of the theory, the geometrical and the statistical one, will merged via case study on real and in silico data.

Mathematics in Machine Learning

The student will learn the basic concepts of machine and statistical learning from both the frequentist and the Bayesian viewpoint, the main techniques for multivariate data and the critical use of specialised software (R, SAS, BUGS, STAN, MATLAB, ORANGE, R, Python, Rapid Miner and the like), being able to tell the pros and cons.

Mathematics in Machine Learning

The student will learn the basic concepts of machine and statistical learning from both the frequentist and the Bayesian viewpoint, the main techniques for multivariate data and the critical use of specialised software (R, SAS, BUGS, STAN, MATLAB, ORANGE, R, Python, Rapid Miner and the like), being able to tell the pros and cons.

Mathematics in Machine Learning

Knowledge of basic probability theory and statistics; linear algebra, in particular SVD; basic of metric geometry and calculus are the prerequisites for this course.

Mathematics in Machine Learning

Knowledge of basic probability theory and statistics; linear algebra, in particular SVD; basic of metric geometry and calculus are the prerequisites for this course.

Mathematics in Machine Learning

• Mathematical representations of data: spaces (including Hilbert spaces), metrics, distances, dissimilarities and kernels. Geometry of very high dimensional spaces and the curse of dimensionality. • Learning theory, PAC, Rademacher and VC dimension. Trade-off Bias vs Model Variance and Model Complexity. • Cross validation, bootstrap and applications. • Linear algebra-based methods: Principal Component Analysis, Linear Discriminant Analysis, Independent Component Analysis and Stochastic projections (Johnson - Lindenstrauss Transform). • Linear Models (regression, ANOVA, DOE). • Generalized linear models (categorical data, logistic and multinomial regression). • Model and feature selection, hyperparameter tuning (e.g. lasso, AIC, BIC, ridge). • Bayesian networks (basic concepts, exact and MCMC-based computations).

Mathematics in Machine Learning

• Mathematical representations of data: spaces (including Hilbert spaces), metrics, distances, dissimilarities, and kernels. The geometry of very high dimensional spaces and the curse of dimensionality. • Learning theory, PAC, and VC dimension. Trade-off Bias vs Model Variance and Model Complexity. • Cross-validation, bootstrap, and applications. Ensemble methods. • Trees, Random Forest, Boosting. • Linear algebra-based methods: Principal Component Analysis, Linear Discriminant Analysis, Independent Component Analysis, and Stochastic projections (Johnson - Lindenstrauss Transform). • Linear Models (regression, ANOVA, DOE). • Generalized linear models (categorical data, logistic and multinomial regression). • Model and feature selection, hyperparameter tuning (e.g. lasso, AIC, BIC, ridge). • Bayesian networks (basic concepts, exact and MCMC-based computations).

Mathematics in Machine Learning

Mathematics in Machine Learning

Mathematics in Machine Learning

In the first part of the course the lectures are held with the support of slides. Exercises are presented and solved in the class as well. In the final part of the course the lessons will mainly consist in activities carried out at the computer lab under the guidance of the teacher. Technical discussions during class lectures will also help to assess the acquired level of knowledge and ability at the different stages of the course.

Mathematics in Machine Learning

In the first part of the course the lectures are held with the support of slides. Exercises are presented and solved in the class as well. In the final part of the course the lessons will mainly consist in activities carried out at the computer lab under the guidance of the teacher. Technical discussions during class lectures will also help to assess the acquired level of knowledge and ability at the different stages of the course.

Mathematics in Machine Learning

Slides of the lectures, examples of R and python scripts and exercises with solutions will be available in the website of the course. A list of suggested books will be also provided by the teacher during the first lecture.

Mathematics in Machine Learning

Slides of the lectures, examples of R and python scripts and exercises with solutions will be available on the website of the course. A list of suggested books: - An Introduction to Statistical Learning with Applications in R. James, G., Witten, D., Hastie, T., Tibshirani, R. Springer Verlag - Understanding machine learning: From theory to algorithms. Shalev-Shwartz, Shai, and Shai Ben-David. Cambridge university press, 2014. - Data Science and Machine Learning: Mathematical and Statistical Methods Dirk P. Kroese, Zdravko I. Botev, Thomas Taimre, Radislav Vaisman CRC Press, 2019 - 510 pages.

Mathematics in Machine Learning

Modalità di esame: Prova scritta (in aula); Prova orale facoltativa;

Mathematics in Machine Learning

The goal of the exam is to test the knowledge of the candidate about the topics included in the official program and to test their skills in analyzing data using the methods explained in the course. The exam consists of a written examination and an optional oral examination. The written examination consists of 3 exercises. Two of the exercises will be similar to those presented during the lectures and will consist of modeling some practical problems or providing a suitable model for a given dataset. The third will be conceptual. The length of the written exam is two hours, and during the test, it is allowed the use of textbooks, student notes or formularies provided by the teacher during the year. The maximum possible score will be 27/30. The oral exam is possible under request for those students that in the written exam get a positive mark (greater or equal to 18/30). It will be the discussion of a short technical presentation on the analysis of a data set performed by using the methods taught in the course. This will be software independent i.e. one can use Orange, R, Matlab, Rapidminer, Python, C++ etc. according to their knowledge or willingness. The students will be asked methodological and theoretical questions related to the methods used in their analysis. After the oral test, the mark obtained in the first part of the exam can be increased or decreased by no more than 6 points.

Mathematics in Machine Learning

Exam: Written test; Optional oral exam;

Mathematics in Machine Learning

The goal of the exam is to test the knowledge of the candidate about the topics included in the official program and to test their skills in analyzing data using the methods explained in the course. The exam consists of a written examination and an optional oral examination. The written examination consists of 3 exercises. Two of the exercises will be similar to those presented during the lectures and will consist of modeling some practical problems or providing a suitable model for a given dataset. The third will be conceptual. The length of the written exam is two hours, and during the test, it is allowed the use of textbooks, student notes, or formularies provided by the teacher during the year. The maximum possible score will be 27/30. The oral exam is possible under request by the student (or by the professor) for those students that in the written exam get a positive mark (greater or equal to 18/30). The students will be asked methodological and theoretical questions related to the course’s contents. After the oral test, the mark obtained in the first part of the exam can be increased or decreased by no more than 6 points.

Mathematics in Machine Learning

Modalità di esame: Prova orale obbligatoria;

Mathematics in Machine Learning

The goal of the exam is to test the knowledge of the candidate about the topics included in the official program and to test their skills in analyzing data using the methods explained in the course. The goal of the exam is to test the knowledge of the candidate about the topics included in the official program and to test their skills in analysing data using the methods explained in the course. The exam consists of two parts: first, the candidate will write a technical relation "tesina" on the analysis of a data set performed by using the methods taught in the course. Once the "tesina" is approved by the professor, then the student is allowed to present it in an oral exam (about 15.min) during which the professor will also ask questions on the theoretical aspects of the methods used in the tesina.

Mathematics in Machine Learning

Exam: Compulsory oral exam;

Mathematics in Machine Learning

The goal of the exam is to test the knowledge of the candidate about the topics included in the official program and to test their skills in analyzing data using the methods explained in the course. The exam will be oral: it will consist of 3 exercises/questions. Two of the exercises will be similar to those presented during the lectures and will consist of modeling some practical problems or providing a suitable model for a given dataset. The third question will be conceptual/theoretical.

Mathematics in Machine Learning

Modalità di esame: Prova orale obbligatoria;

Mathematics in Machine Learning

The goal of the exam is to test the knowledge of the candidate about the topics included in the official program and to test their skills in analyzing data using the methods explained in the course. The goal of the exam is to test the knowledge of the candidate about the topics included in the official program and to test their skills in analysing data using the methods explained in the course. The exam consists of two parts: first, the candidate will write a technical relation "tesina" on the analysis of a data set performed by using the methods taught in the course. Once the "tesina" is approved by the professor, then the student is allowed to present it in an oral exam (about 15.min) during which the professor will also ask questions on the theoretical aspects of the methods used in the tesina.

Mathematics in Machine Learning

Exam: Compulsory oral exam;

Mathematics in Machine Learning

The goal of the exam is to test the knowledge of the candidate about the topics included in the official program and to test their skills in analyzing data using the methods explained in the course. The exam will be oral: it will consist of 3 exercises/questions. Two of the exercises will be similar to those presented during the lectures and will consist of modeling some practical problems or providing a suitable model for a given dataset. The third question will be conceptual/theoretical.

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