Intersections of complex networks in real world and infinite particle systems
| Reference No. | 20200028 |
|---|---|
| Type/Category | Grant for Young Researchers-Short-term Joint Research |
| Title of Research Project | Intersections of complex networks in real world and infinite particle systems |
| Principal Investigator | Shota Osada（Institute of Mathematics for Industry, Kyushu university・Postdoctoral Researcher） |
| Research Period |
February 22,2021. ～
March 19,2021. |
| Keyword(s) of Research Fields | Large-scale interacting systems, universality, phase transitions and critical phenomena, mean field theory, logarithmic Sobolev inequality, spectral gap, entropy |
| Abstract for Research Report |
This joint research aims to develop approaches to real-world problems using stochastic analysis of infinite particle systems. Our purposes the following: (1) Theoretical study of function spaces on configuration spaces. (2) Theoretical study of features of infinite particle systems. (3) Application of the above (1) and (2) to complex networks. Infinite particle systems are a general term for dynamic/static stochastic models that finite or infinite particles interact in space. An infinite number of particles is an object of infinite dimension, which is difficult to analyze. However, by imposing the symmetry that each particle is identical and considering a small symmetric subspace in the infinite dimension (= configuration space), we can analyze the infinite-dimensional space. A problem with the analysis of configuration spaces is that the fundamental tools of analysis, such as Lebesgue measure and differential calculus, cannot be directly applied to infinite-dimensional spaces. In this research, as a stepping stone to the functional analysis on configuration spaces, we expand the stochastic analysis methods, mainly performed in the framework of L^2 spaces, to various function spaces. In particular, by choosing an appropriate function space for each problem, we construct new methods to evaluate the system size dependence of important features, such as spectral gap and entropy. These quantities are used to elucidate the relaxation phenomena of large-scale interacting systems, where a particle system converges from a non-equilibrium state to an equilibrium state. By considering the parameter components that appear in machine learning as particles, we try to construct an optimization theory using infinite particle systems. In particular, we expect to develop and advance infinite-particle system methods for nonparametric Bayesian theory, regularization theory, and nonconvex function optimization used in deep learning by focusing on function spaces. |
| Organizing Committee Members (Workshop) Participants (Short-term Joint Usage) |
Hayate Suda（Graduate school of mathematical sciences, The university of Tokyo・PhD student） Yuta Arai（Graduate school of Science and Engineering, Chiba university・PhD student） Shota Osada（Institute of Mathematics for Industry, Kyushu university・ Postdoctoral Researcher） Kohei Hayashi（Graduate school of mathematical sciences, The university of Tokyo・PhD student） Takahiro Mori（Research Institute for Mathematical Sciences, Kyoto University・PhD student） Yoshinori Kamijima（Department of mathematics, Graduate school of science, Hokkaido university・PhD student） |
| Adviser | Tomoyuki Shirai (IMI, Kyushu University / Professor) |