The dually flat structure for singular models

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, such an excellent geometric rarely defined on entire space. To fix this trouble, present paper, we propose a novel generalization of for certain class singular models from viewpoint Lagrange Legendre singularity theory—we introduce quasi-Hessian manifold endowed with possibly degenerate particular symmetric cubic tensor, which exceeds concept statistical manifolds adapted to theory (weak) contrast functions. particular, establish Amari–Nagaoka’s extended Pythagorean theorem projection general setup, consequently, most applications these theorems are suitably justified even cases. This work motivated various interests different backgrounds Frobenius mathematical physics Deep Learning data science.