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Multidimensional Cost Geometry
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DOI：10.3390/axioms15050378.png)
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Abstract
En 中文
In this paper, we study the geometric structure induced by the canonical reciprocal cost function and its natural n-dimensional extension. In logarithmic coordinates, the potential depends only on the linear combination S=alpha & centerdot;t, and the associated Hessian metric has rank one at every point. The geometry is intrinsically degenerate and effectively one-dimensional, with an (n-1)-dimensional null distribution. On the other hand, when the same function is expressed in the original x-coordinates, the corresponding Hessian is generically nondegenerate and defines a pseudo-Riemannian metric away from explicit singular hypersurfaces. We further analyze affine and Levi-Civita geodesics and compare their behavior. In particular, affine geodesics in logarithmic coordinates are globally defined, while in x-coordinates their behavior is restricted by the domain and the singular set. Finally, we relate the construction to symmetrized Itakura-Saito and Bregman divergences, and give a Fisher-Rao realization of the logarithmic Hessian metric.
Keywords:
Hessian geometry
degenerate metric
affine structure
Levi-Civita connection
geodesics
gradient paths
reciprocal cost
Journal
A
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1.6
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593
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