  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.
