The dually flat structure for singular models

Naomichi Nakajima, Toru Ohmoto*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


The dually flat structure introduced by Amari–Nagaoka is highlighted in information geometry and related fields. In practical applications, however, the underlying pseudo-Riemannian metric may often be degenerate, and such an excellent geometric structure is rarely defined on the entire space. To fix this trouble, in the present paper, we propose a novel generalization of the dually flat structure for a certain class of singular models from the viewpoint of Lagrange and Legendre singularity theory—we introduce a quasi-Hessian manifold endowed with a possibly degenerate metric and a particular symmetric cubic tensor, which exceeds the concept of statistical manifolds and is adapted to the theory of (weak) contrast functions. In particular, we establish Amari–Nagaoka’s extended Pythagorean theorem and projection theorem in this general setup, and consequently, most of applications of these theorems are suitably justified even for such singular cases. This work is motivated by various interests with different backgrounds from Frobenius structure in mathematical physics to Deep Learning in data science.

Original languageEnglish
Pages (from-to)31-64
Number of pages34
JournalInformation Geometry
Issue number1
Publication statusPublished - 2021 Jul
Externally publishedYes


  • Canonical divergence
  • Caustics
  • Dually flat structure
  • Hessian geometry
  • Legendre duality
  • Singularity Theory
  • Wavefronts

ASJC Scopus subject areas

  • Geometry and Topology
  • Statistics and Probability
  • Applied Mathematics
  • Computer Science Applications
  • Computational Theory and Mathematics


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