# The Art of Scientific Research

From Research management course

This is a preparatory course for the main part of m1p.

## Contents

### Goals of the seminar

- Gather tools and train skills to run a scientific research
- Elaborate competencies of the scientific problem statement and reporting
- Fit your research society, find a high-quality scientific advisor, and select an important problem to engage

#### Organizers' goals

- Boost the performance of the MS thesis works, namely change
*the magical presentation*of the machine learning models for*the theoretical-based one* - Persuade the scientific advisers to set complex and well-elaborated problems with high-quality planning

### Outline of a seminar

- Test (five open or closed questions) with a brief analysis
- Theoretical part (15 minutes) and references to study
- Practice and homework handout
- Talks and discussion (20 minutes)

### Coursework and talks

- Module 1: A formal description of a problem (project), a two-page text plus a three-slide talk
- Module 2: An error analysis, a plan of a computational experiment with model selection plus a talk

### Homeworks

Since it is a preparatory course, the change of research subject for different HWs is welcome.

#### Module 1

- Select and read the reference paper
- Reconstruct its abstract
- Formulate highlights
- Collect a SOTA review
- Extract the principles of the paper
- Prepare one-slide talk

#### Module 2

- State the problem
- State statistical hypotheses
- Construct algebraic structures
- Gather the theory
- Select a model
- Plan the experiment

## The student's response-based syllabus

- Step 0: We start
- Prepare your tools
- Check the foundations
- How to measure impact
- Describe your system
- Write the abstract
- Write the intro
- Review the paper
- Deliver a message
- Your one-slide talk
- Blind management game
- List your ideas
- List the foundations
- Suggest an impactful theorem
- Review for your topic
- Good, bad, ugly: tell the difference
- Tell about a scientific society
- Reproducible computational experiment
- Computer-supported brainstorming
- Conferences and journals, reviews, and schedules
- Writing a grant proposal

### addendum

- Annotate and highlight (rules of annotation and highlighting applied)
- Write a review
- Boost a review by gathering your colleagues' efforts
- Make long and short lists of your ideas and solutions
- Select a topic from the list
- Find the data if you need something special, it takes time and efforts
- Structure of a thesis work and bureaucracy of thesis defense

## The theory to discuss

- Machine learning at one go
- Linear models (and processes) with time (regression, SVD, PCA, NN)
- Tensor indexing and decomposition, Tucker, HOSVD, TT (getting rid of time by constructing a state space)
- Types of optimization (what is the gradient and Jacoby matrix)
- Convolution and Fourier transform is a linear operator
- Kernel methods and RKHS
- Graph convolution, metric spaces (if possible)
- Canonical correlation analysis and autoencoders
- Bayesian inference and regularization, optimization
- Model selection
- Multimodeling (privilege, distilling, domain transfer)
- Introduction to sampling and generative models

- Goals for the next year are CaТ, NODE, SDE, Diffusion, Riemannian, Tensors as tensors, Advanced calculus, Clifford algebra, Homology

## Scoring

- Tests at the beginning of a seminar
- Talks at the end of a seminar
- Downloads of the homework
- The coursework

Weekly homework. All points are added up and scaled to [0,10]. Deadlines are strict. Normally there is no exam.

## Student's risks

Despite m1p (it flourishes over years), this is a new course, so:

- It might end abruptly, after one week, one month, or one module.
- There will be no resources to check and review your texts.
- Most likely there will be no possibilities to listen to all of your talks.
- So feedback is limited.

## Student prerequisites

Briefly: it is for 3rd year BS students.

- Discrete Analysis and Set Theory
- Calculus and Mathematical Analysis
- Probability and Statistics
- Algebra, Group theory is welcome
- Functional Analysis is welcome
- General Physics is highly welcome
- Machile learning by C.P. Bishop is a must!

## Main references

- (long reading 2196 pages) Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning by Jean Gallier and Jocelyn Quaintance, 2024. pdf, github
- Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges by M.M. Bronstein, J. Bruna, T. Cohen, P. Veličković, 2021. arxiv
- Deep Learning: Foundations and Concepts by C.M. Bishop, H. Bishop, 2024 version'06
- Mathematics for Physicists: Introductory Concepts and Methods by A. Altland. J. von Delf, 2017 pdf
- Mathematics for Machine Learning by M.P. Deisenroth, A.A. Faisal, C.S. Ong pdf
- Python for Probability, Statistics, and Machine Learning by J. Unpingco, 2016 github

- Cath-up references are in the Week 0 of the main course

## Dates

2024 on Sat 14:40 – 16:00 zoom | Sept 14 21 28 | Oct 5 12 19 || skip 26, 2 || Now 9 16 23 30 | Dec 7 14 |