This short course will give a comprehensive introduction to the emerging interdisciplinary field of Information Geometry. Information Geometry provides a differential geometric characterization of probability models. Starting with the concept of divergence functions (such as Kullback-Leibler divergence and Bregman divergence), a manifold of the probability functions is constructed with Fisher information as the Riemannian metric and a family of affine connections with dualistic structures. The course explains dually-flat (“Hessian”) geometry of exponential family, with e- and m-projections, for maximum entropy inference, and their generalizations. The course also presents several new research directions, including deformation theory, which constructs a conformal Hessian structure based on deforming the exponential/logarithm function, and the link of information geometry to optimal transport, which is another framework for the manifold of probability measures and both are used in machine learning, neural network, statistical inference, signal processing, optimization, etc.
This course is offered to graduate students and advanced undergraduate students from Mathematics, Statistics, Information Science, Machine Learning, Computer Science and Engineering, Physics, Complex Systems, Neuroscience, etc. Prerequisites include advanced calculus, linear algebra, ordinary differential equation, and probability theory. Students who have taken courses in differential geometry/differentiable manifold will be in an advantaged position, but not required for two-thirds of the course materials.
This course will be conducted primarily in Chinese language (with lecture slides and reading materials in English). The main course will consist of 9 lectures, each with 90-minutes, spreading across July 4- Aug 1. There will also be 5 advanced lectures on geometric foundation of information geometry; audience are those who have taken differential geometry/differentiable manifold courses, through a screening process. Students who excel from the course will be invited to conduct research projects on information geometry in August under the direction of the instructor.
1) S. Amari“Information Geometry and Its Applications” Springer 2016 (Chapters 1,2,3,4,5,6)
2) S. Amari and H. Nagaoka “Method of Information Geometry” AMS/Oxford 2000 (Chapters 1,2,3,4)
3) Other readings and course materials to be distributed later.
Note: Please register through the following link. (Deadline: July 1, 2022)