Sinkhorn 层中的支撑拓扑与梯度混合
Support Topology and Gradient Mixing in Sinkhorn Layers
AI总结：
本文研究稀疏Sinkhorn层中固定支撑图对梯度传播的影响，提出固定支撑微积分，刻画一步收缩条件，为可微分传输层的支撑设计提供数学准则。
AI中文摘要：
稀疏 Sinkhorn 层使用固定的支撑图来限制令牌之间的传输。该图如何控制通过缩放迭代的梯度传播？我们发展了一种固定支撑微积分，表明每个行列环在模常数意义下对列势扰动诱导一个行随机算子。其转置传播零质量的逆向模式余切向量。有限环算子使用两个不同的半步传输计划；在平衡不动点处，它简化为由单一计划确定的两步游走。我们推导了伴随的得分和边际源项，并利用 Dobrushin 收缩和劣化来界定齐次和源驱动的尾部余切向量。我们的主要结果刻画了支撑和边际何时保证在有限得分上均匀的一步收缩：运输多面体的每个可行面必须具有成对的两跳列重叠。否则，适当的得分方向会使收缩系数任意接近于一。我们将此分析扩展到有序支撑调度，并为分区热浴层、坐标扫描、强制共享质量和寄存器增强支撑导出证书。这些结果为可微分传输层中的支撑设计提供了数学准则，其保证仅限于固定支撑商梯度分量。
英文摘要：
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.
