Singular statistics and computational algebraic analysis of machine learning models

2017 Fiscal Year Final Research Report

• Project/Area Number: 15KT0102
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 特設分野
• Research Field: Mathematical Sciences in Search of New Cooperation
• Research Institution: University of Tsukuba
• Principal Investigator: Tajima Shinichi 筑波大学, 数理物質系, 教授 (70155076)
• Co-Investigator(Kenkyū-buntansha): 小原 功任 金沢大学, 数物科学系, 准教授 (00313635) ; 照井 章 筑波大学, 数理物質系, 准教授 (80323260) ; 渋田 敬史 九州産業大学, 理工学部, 講師 (40648200)
• Co-Investigator(Renkei-kenkyūsha): Hasegawa Makoto 東京電機大学, 工学部, 教授 (80303171)
• Project Period (FY): 2015-07-10 – 2018-03-31
• Keywords: 特異点 / アルゴリズム / semi-algebraic set / local Euler obstruction / Grothendieck residue / Samuel multiplicity / Matlis duality

Outline of Final Research Achievements

We study and analyze the singular statistical structure of machine learning models and investigate algorithms from the point of view of algebraic analysis. Main results of our research are (i) algorithms for computing reductions and Hilbert-Samuel multiplicities, (ii) an algorithm for computing Matlis duality of modules, (iii) exact eigenproblems, (iv) algorithms for computing Grothendieck local residues, (v) algorithms for computing b-functions and relevant holonomic D-modules via Poincare-Birkhoff-Witt algebra, (vi) an algorithm for computing the local Euler obstruction of a hypersurface.

Free Research Field

解析学基礎