Escape rates and exponential-rate losses for Fisher–Rao dynamics

Characterize, for dimensions n>2, a matching escape-rate result for the Fisher–Rao flow near permutation vertices and determine which losses attain the exponential convergence rate, analogously to the reduced-cost gap in linear programming.

Background

The paper compares two optimization dynamics on the positive Birkhoff polytope: exact gradient flow in Sinkhorn-logit coordinates and Fisher–Rao gradient flow. The authors prove that Fisher–Rao flow can approach a permutation vertex exponentially quickly for a particular linear loss, while logit gradient flow approaches or escapes vertices only at a polynomial rate.

For dimensions greater than two, the paper identifies two unresolved issues concerning escape from vertices under Fisher–Rao flow: obtaining a matching escape-rate theorem and identifying the losses for which the exponential rate is achieved. The latter is explicitly related to the reduced-cost gap that governs convergence in linear programming.

References

For $n>2$, two questions are open: a matching result for escape, and a characterization of the losses that attain the exponential rate.

— The Birkhoff Geometry of Manifold-Constrained Hyper-Connections: Two Channels, Vertex Viscosity, and Sinkhorn as a Retraction  (2610.06653 - Li et al., 5 Oct 2026) in Appendix, Section 6, subsection “Open problems,” item 2