Applied Singularity Theory in Information Science and Stractural Design

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K18714
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: Hokkaido University
• Principal Investigator: Ohmoto Toru 北海道大学, 理学研究院, 教授 (20264400)
• Project Period (FY): 2018-06-29 – 2022-03-31
• Keywords: 特異点論 / ルジャンドル変換 / 接触幾何学 / 情報幾何学 / 応用幾何学 / Line Geometry

Outline of Final Research Achievements

Based on Legendre singularity theory, we generalize the theory of dual flat structures (Amari-Nagaoka) which takes an important role in information geometry. This leads to a new theoretical approach to singular models in Deep Learning. We also provide a singularity-theoretic approach to applied geometry in structural design. This is a new development of local theory in classical projective differential geometry which has long been forgotten for hundred years after Darboux and Wilczynski.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究課題は，機械学習分野あるいはアーキテクチュラル幾何など，数学の外の世界に向けた応用特異点論の新しい萌芽的研究である．応用分野の諸課題に《特異点》は本質的に現れるにも関わらず，本格的な研究は未開拓であった．その端緒をひらく本研究の学術的意義は大きい．また，大学院生や若手研究者を積極的に共同研究に巻き込むことで，次世代に向けた新しい学際的研究の雛形を提示するものと考える．