Non-Conservative Sinkhorn: Optimal Transport with Dissipation

Optimal transport (OT) assumes mass is conserved, yet in many settings, including portfolio rebalancing with transaction costs, Mixture-of-Experts routing, and logistics with spoilage, mass dissipates in transit. We develop NC-Sinkhorn, the first scalable Sinkhorn-type solver for discrete non-conservative OT (NC-OT), where an edge-dependent preservation map $m_{ij}\le 1$ encodes edge-dependent mass loss while hard marginal constraints are maintained. Edge-dependent mass loss breaks the rank-one factorization that makes classical Sinkhorn efficient; we identify a dual consistency condition unique to NC-OT and enforce it via a coupled Newton step that preserves $O(NM)$ per-iteration cost. We prove convergence to the unique optimum with an $O({\delta }^{-2})$ iteration complexity bound for feasibility tolerance $\delta$ (empirically linear), and show that structured preservation maps admit progressively cheaper updates, collapsing to a closed-form $O(N+M)$ solution in the separable case. On benchmarks up to $N=5,000$, NC-Sinkhorn achieves over $600\times$ speedup versus LP solvers. We demonstrate two applications: portfolio rebalancing with transaction costs and taxes, where the separable bid–ask structure yields real-time solves even at $N=10,000$; and Mixture-of-Experts routing, where NC-OT provides exact quality weighted load balance without auxiliary losses or token dropping while matching or exceeding standard baselines.