Transport Adjoint Matching

Sampling from high-dimensional discrete distributions specified only through unnormalized mass functions is a fundamental problem in statistical physics, combinatorial optimization, and probabilistic inference. Adjoint matching provides an effective framework for learning nonequilibrium neural samplers, but the translation-based discrete Schr\"odinger-bridge construction relies on additive reference dynamics and deterministic cyclic endpoint correspondences, restricting its applicability when such structure is unavailable. We introduce Transport Adjoint Matching (TAM), which represents the optimal-control potential ratio through transport-compatible couplings between reference endpoint kernels, thereby removing the need for deterministic endpoint correspondences. For coordinate-local reference dynamics satisfying a shared-fiber marginal condition, we further derive a fiberwise Rao–Blackwellized target that marginalizes both the auxiliary transport variable and the local endpoint symbol. The resulting target preserves the exact population adjoint identity, requires only known local endpoint weights, local energy differences, and corrector potential ratios, and avoids explicit transport-matrix construction. We evaluate TAM on high-dimensional Ising and four-state Potts models across three temperature regimes, and further test a genuinely non-additive Potts reference for which the deterministic cyclic-shift construction is unavailable. TAM achieves accurate sampling across the standard benchmarks and remains effective under the non-additive reference.