# Support Topology and Gradient Mixing in Sinkhorn Layers

arXiv: https://arxiv.org/abs/2609.07954 · HTML: https://mlsanity.com/paper/2609.07954/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 15 references that are in the map.
- First posted on arXiv 7 Sep 2026.

## Builds on (most cited first, 5 of 5)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A Unified Framework for Implicit Sinkhorn Differentiation](https://mlsanity.com/paper/2205.06688/) (2022)
- [An explicit analysis of the entropic penalty in linear programming](https://mlsanity.com/paper/1806.01879/) (2018)
- [The derivatives of Sinkhorn-Knopp converge](https://mlsanity.com/paper/2207.12717/) (2022)
- [Dobrushin ergodicity coefficient for Markov operators on cones, and beyond](https://mlsanity.com/paper/1302.5226/) (2013)

## Abstract (arXiv)

Sparse Sinkhorn layers use a fixed support graph to restrict transport between tokens. How does this graph control gradient propagation through the scaling iterations. We develop a fixed-support calculus showing that each row-column cycle induces a row-stochastic operator on column-potential perturbations modulo constants. Its transpose propagates zero-mass reverse-mode cotangents. The finite-cycle operator uses two distinct half-step transport plans; at a balanced fixed point it reduces to a two-step walk determined by a single plan. We derive the accompanying score and marginal source terms and use Dobrushin contraction and minorization to bound homogeneous and source-driven tail cotangents. Our main result characterizes when support and marginals guarantee one-step contraction uniformly over finite scores: every feasible face of the transportation polytope must have pairwise two-hop column overlap. Otherwise, suitable score directions make the contraction coefficient arbitrarily close to one. We extend this analysis to ordered support schedules and derive certificates for partition heat-bath layers, coordinate sweeps, forced shared mass, and register-augmented supports. These results provide mathematical criteria for support design in differentiable transport layers, with guarantees restricted to the fixed-support quotient-gradient component.