Teaching Philosophy

Vadim, 2023

I aim to show the power of mathematical methods in the practical application of Machine Learning and to show Machine Learning scientific research principles to graduate and undergraduate students. I join students, their senior mates, and experienced researchers in productive research groups.

Practice: My first scientific paper

I plan to continue and expand my course, "My first scientific paper." The goal of the course is to introduce students to the technologies of scientific research. The course teaches how to plan, perform, and present research results. It provides formats acknowledged by other researchers. Each student works with an advisor and a consultant to learn how to formally state research problems, find adequate references, and generate novel and significant ideas for problem-solving. The expected outcome of the course is a research paper submitted to a peer-reviewed journal.

The course has been successfully delivered during the last eight years. Each year 15-30 students perform their research projects. Each project ends with its scientific paper, code, presentation, and video. The course has its repository with over ​500 projects and its ​YouTube channel​. I believe this course contributes to educational science and citizen science.

Mathematical forecasting

My specialization is Mathematical forecasting. I present a new view of this field of knowledge: the forecasting problems deal not with the vector spaces but with the vector fields. The main subject is vector and tensor fields over time and space. The modeling data are spatial-time series: audio-video streams, brain signals, and images, biomedical live measurements, wearable device sensor signals, and other signals in biology and physics. The practical applications for study and labwork are brain-computer interface, human motion, and human health monitoring. The course is organized into eight sections: autoregressive models, tensor decomposition, canonic correlation analysis, continuous-time analysis, dynamic systems, spatial-time alignment, metrics learning, and diffusion-graphical models. Each section runs labwork with various practical applications in Python.

Theoretical foundations

To provide theoretical foundations for study and practice, I am planning to deliver lecture courses on Geometric Deep Learning, Differential Geometry for Data Analysis, Functional Data Analysis, and Bayesian Model Selection. My course Fundamental theorems of Machine learning boosts the quality of bachelor's and master's thesis works.

Functional Data Analysis combines continuous and infinite-dimensional spaces and statistical hypotheses to select a forecasting model using convergence analysis, tensor decompositions, neural differential equations, and flows. My new course Physics-Informed Learning uses methods of mathematical modeling and theoretical physics to analyze signals and images. It includes Riemannian geometry for fMRI, Neural PDE for data flows, differential forms for BCI, and methods of geometric algebra for signals from wearable devices.

Organizational activities

I have experience serving as an editor of scientific journals and a member of MS, BS, and Ph.D. committees, so I gladly continue these activities to help undergraduate and graduate students.

I plan to continue working with Ph.D. students to produce high-quality applied and theoretical research papers and theses. I gladly work with research-track undergraduate and graduate students.

Development of educational technologies

The last works, devoted to machine learning applications, have a profound theoretical background. They received prizes at well-recognized conferences like NIPS and ICML. These works use advanced and tensor calculus, differential geometry, geometric algebra, and theoretical physics. Skills to use these tools in applied projects and to create machine learning models with notions of continuous fields and infinite spaces boost the quality of student works and give our students the power to apply the new modeling technologies in the future decade. The next scheme will fit project-based education. Introduce a practical application, run the code, analyze the modeling results, and then explain the model's theory with hypothesis and axioms. It delivers the experiment with code and the necessary language to exchange ideas and communicate the results.