2024 Faculty Courses School of Computing Department of Mathematical and Computing Science Graduate major in Mathematical and Computing Science
Topics on Mathematical and Computing Science EB
- Academic unit or major
- Graduate major in Mathematical and Computing Science
- Instructor(s)
- Yumiharu Nakano / Toshio Mikami
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MCS.T415
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 2Q
- Syllabus updated
- Mar 14, 2025
- Language
- Japanese
Syllabus
Course overview and goals
E. Schrödinger published a paper on a probabilistic approach to quantum mechanics (Schrödinger's problem) in 1931-1932.
Schrödinger's problem began in the world of probability theory with the study of Bernstein processes and evolved into the study of h-path processes. By considering the continuous limit of the h-path process, Mikami gave another proof of the construction of the Nelson process, which plays an important role in E. Nelson's stochastic dynamics. He also gave a probabilistic proof of Monge's problem (optimal transport problem) of finding the optimal way to transport an object by the zero-noise limit of the h-path process. This led to the study of stochastic optimal transport problems, including the optimal transport problem as a special case. Schrödinger's problem is called entropy regularization of optimal transport problems in data science and has been actively studied in recent years. In this lecture, students will learn an introduction to stochastic optimal transport problems,
a generalization of the optimal transport problem, through optimal transport problems and Schrödinger's problem.
Various topics will also be discussed as time permits.
Course description and aims
Learn the basic theory of optimal transport problems and Schrödinger's problem and its application to data science.
Keywords
Stochastic optimal transport problem, Schrödinger's problem, Schrödinger's functional equation, optimal transport problem, entropy regularization, Sinkhorn algorithm, zero noise limit, ergodic problem
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
classroom lecture
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Overview of Schrödinger's problem (in probability theory) (Tue 6/11, 3-8 Period) | Give an overview of Schrödinger's problem |
| Class 2 | Introduction to optimal transport problems, basic properties of optimal transport problems: focus on finite probability vectors (Tue 6/11, 3-8 Period) | Outline and explain the basic properties of optimal transport problems in the framework of finite probability vector |
| Class 3 | Optimal Transport Problems in One Dimension (Tue 6/11, 3-8 Period) | Explain 1-dimensional optimal transport problems |
| Class 4 | Entropy regularization of discrete optimal transport problems: Schrödinger's functional equation and Sinkhorn algorithm (Fri 6/14, 3-8 Period) | Explain Schrödinger's functional equation and the Sinkhorn algorithm as they appear in the entropy regularization of a discrete optimal transport problem. |
| Class 5 | Zero-noise limit of entropy regularization of discrete optimal transport problems (Fri 6/14, 3-8 Period) | Explain the zero noise limit of the entropy regularization of the discrete optimal transport problem |
| Class 6 | Ergodic problem of entropy regularization of discrete optimal transport problems (Fri 6/14, 3-8 Period) | Explain the ergodic problem of entropy regularization for discrete optimal transport problems |
| Class 7 | Basic Properties of Schrödinger's Problem (Tue 6/18, 5-8 Period) | Explain the basic properties of Schrödinger's problem |
| Class 8 | Schrödinger's generalized functional equation and Sinkhorn algorithm (Tue 6/18, 5-8 Period) | Explain Schrödinger's generalized functional equation and Sinkhorn algorithm |
| Class 9 | Ergodic problem of Schrödinger's problem for the Markov chain (Fri 6/21, 5-8 Period) | Explain the ergodic problem of Schrödinger's problem for the Markov chain |
| Class 10 | Schrödinger's Problem and SDE (Fri 6/21, 5-8 Period) | Explain Schrödinger's problem and how it relates to SDE |
| Class 11 | Schrödinger's Problem and Stochastic Control (Wed 6/26, 5-8 Period) | Explain the connection between Schrödinger's problem and stochastic control |
| Class 12 | Zero-noise limit of Schrödinger's problem and optimal transport problem (Wed 6/26, 5-8 pm) | Explain the connection between the zero-noise limit of Schrödinger's problem and the optimal transport problem |
| Class 13 | Short-time behavior of Schrödinger's problem (Fri 6/28, 5-8 Period) | Explain the short-time behavior of Schrödinger's problem |
| Class 14 | Schrödinger's Problem of Ergodic Problems (Fri 6/28, 5-8 Period) | Explain the ergodic problem of Schrödinger's problem |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
No textbook is set.
Reference books, course materials, etc.
・確率力学としての最適輸送問題, 日本数学会「数学」第58巻, 第4号 (2006) 364--382.
https://www.jstage.jst.go.jp/article/sugaku1947/58/4/58_4_364/_pdf/-char/ja
・確率最適輸送問題入門, システム制御情報学会誌「システム/制御/情報」第64巻第7号・数学的視点からの確率制御と推定特集号，252-257 (2020).
https://www.jstage.jst.go.jp/article/isciesci/64/7/64_252/_pdf/-char/ja
・Chen, Y. , Georgiou, T. T., Pavon, M.:
Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schrödinger bridge.
SIAM Review 63, no. 2, 249–313 (2021)
・Mikami, T.:
Stochastic Optimal Transportation: Stochastic Control with Fixed Marginals.
Springer Briefs in Mathematics, Springer, Singapore (2021)
・Peyre, G., Cuturi, M.:
Computational Optimal Transport: With Applications to Data Science.
Now Publishers, Boston (2019)
・Rachev, S. T., Rüschendorf, L.:
Mass Transportation Problems, Vol. I: Theory, Vol. II: Application.
Springer, Heidelberg (1998)
・Schweizer, B., Sklar, A.:
Probabilistic Metric Space.
Dover Publications, New York (2005)
・Villani, C.:
Topics in Optimal Transportation.
Amer. Math. Soc., Providence, RI (2003)
Evaluation methods and criteria
Attendance (60%) and report (40%).
Related courses
- MCS.T212 ： Fundamentals of Probability
- MCS.T332 ： Data Analysis
- MCS.T312 ： Markov Analysis
- MCS.T410 ： Applied Probability
- MCS.M422 ： Statistical Mechanics for Information Processing
- MCS.T223 ： Mathematical Statistics
- MCS.T333 ： Information Theory
- MCS.T403 ： Statistical Learning Theory
- MCS.T419 ： Stochastic differential equations
Prerequisites
Knowledge of calculus and elementary probability theory is recommended. (Lectures will review/study advanced math knowledge as we go along.)