Multidimensional cost geometry
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916022215245824 |
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| author | Washburn, Jonathan Zlatanović, Milan Beltracchi, Philip |
| author_facet | Washburn, Jonathan Zlatanović, Milan Beltracchi, Philip |
| contents | 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 $S=α\cdot t$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06957 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multidimensional cost geometry Washburn, Jonathan Zlatanović, Milan Beltracchi, Philip Differential Geometry Mathematical Physics 53A15, 53B20, 53C21, 53C25 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 $S=α\cdot t$, 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. |
| title | Multidimensional cost geometry |
| topic | Differential Geometry Mathematical Physics 53A15, 53B20, 53C21, 53C25 |
| url | https://arxiv.org/abs/2604.06957 |