Mathematical prediction
From My first scientific paper
This course joins two parts of the problem statements in Machine Learning. The first part comes from the structure of the measured data. The data come from Physics, Chemistry and Biology and have intrinsic algebraic structure. This stricture is part of the theory that stands behind the measurement. The second part comes from errors of the measurement. The stochastic nature errors request the statistical methods of analysis. So this course joins algebra and statistics. It is devoted to the problem of predictive model selection.
The course holds two semesters: Fall 2020 and Spring 2021. It contains lectures and practical works. Out of schedule cuts off half the score. The scoring, max:
- Questionnaires during lectures (3)
- Two application projects (2+2)
- The final exam: problems with discussion (3)
Schedule
Date | N | Subject | Link | |
September | 3 | 1 | Probabilistic models | |
10 | 2 | Models: regression, encoders, and neural networks | ||
17 | 3 | Processes: bayesian regression, generative and discriminative models | ||
24 | 4 | Functional data analysis: decomposition of processes | ||
31 | 5 | Spatiotemporal models | ||
October | 8 | 6 | Convolutional models | |
15 | 7 | Talks for the fist part of lab-projects | The talk template | |
22 | 8 | Graph convolutions and spectrum | ||
29 | 9 | Fourier transform and phase retrieval problem | ||
4 | 10 | Radon transform and tomography reconstruction | ||
November | 12 | 11 | Tensor decomposition and decoding problem | |
19 | 12 | Statistics on riemannian spaces | ||
26 | 13 | Statistics on stratified spaces | ||
December | 3 | 14 | Talks for the second part of lab-projects | The talk template |
4 | 15 | Exam: problems and discussion | List of problems |
Topics
- Forward and inverse problems, kernel regularisation
- Karhunen–Loeve theorem, FPCA
- Parametric and non-parametric models
- Reproductive kernel Hilbert space
- Integral operators and Mercer theorem Convolution theorem
- Graph convolution
- Manifolds and local models
L3 courses towards machine learning
- Functional analysis
- Differential geometry
References
- Functional data analysis by James Ramsay, Bernard Silverman, 2020
- Riemannian geometric statistics in medical image analysis. Edited by Xavier Pennec, Stefan Sommer, and Tom Fletcher, 2020
- Manifolds, tensors and forms by Paul Renteln, 2014
- Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators by Tailen Hsiing, Randall Eubank, 2013
- At the Interface of Algebra and Statistics by Tai-Danae Bradley, 2020