Multidimensional cost geometry

TL;DR

This paper explores the geometric structure of a reciprocal cost function and its extension, revealing an intrinsic degeneracy and connections to divergences and Fisher-Rao metrics.

Contribution

It introduces a detailed analysis of the cost geometry in both logarithmic and original coordinates, highlighting degeneracy, geodesic behavior, and links to divergence measures.

Findings

The Hessian metric is rank one in log-coordinates, indicating a degenerate, effectively one-dimensional geometry.

Affine geodesics in log-coordinates are globally defined, unlike in original coordinates.

Connections are established between the geometry, divergences like Itakura-Saito, and Fisher-Rao metrics.

Abstract

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 $S=\alpha\cdot t$, 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.