Differentiable matching layers, often implemented via entropy-regularized Optimal Transport, serve as a critical approximate inference mechanism in structural prediction. However, recovering discrete permutations via annealing $ε\to 0$ is notoriously unstable. We identify a fundamental mechanism for this failure: \textbf{Premature Mode Collapse}. By analyzing the non-normal dynamics of the Sinkhorn fixed-point map, we reveal a theoretical \textbf{thermodynamic speed limit}. Under standard exponential cooling, the shift in the target posterior ($O(1)$) outpaces the contraction rate of the inference operator, which degrades as $O(1/ε)$. This mismatch inevitably forces the inference trajectory into spurious local basins. To address this, we propose \textbf{Efficient PH-ASC}, an adaptive scheduling algorithm that monitors the stability of the inference process. By enforcing a linear stability law, we decouple expensive spectral diagnostics from the training loop, reducing overhead from $O(N^3)$ to amortized $O(1)$. Our implementation and interactive demo are available at https://github.com/xxx0438/torch-sinkhorn-asc and https://huggingface.co/spaces/leon0923/torch-sinkhorn-asc-demo. bounded away from zero in generic training dynamics unless the feature extractor converges unrealistically fast.

翻译：可微匹配层（通常通过熵正则化最优传输实现）是结构预测中一种关键的近似推断机制。然而，通过退火 $ε\to 0$ 来恢复离散排列是众所周知的不稳定过程。我们识别了导致此失败的一个根本机制：\textbf{过早模态坍缩}。通过分析 Sinkhorn 不动点映射的非正规动力学，我们揭示了一个理论上的\textbf{热力学速度极限}。在标准的指数冷却方案下，目标后验的偏移（$O(1)$）超过了推断算子的收缩速率，而该收缩速率以 $O(1/ε)$ 退化。这种不匹配不可避免地迫使推断轨迹陷入虚假的局部盆地。为解决此问题，我们提出了\textbf{Efficient PH-ASC}，一种监测推断过程稳定性的自适应调度算法。通过强制执行线性稳定性定律，我们将昂贵的谱诊断从训练循环中解耦，将开销从 $O(N^3)$ 降低至摊销 $O(1)$。我们的实现和交互式演示可在 https://github.com/xxx0438/torch-sinkhorn-asc 和 https://huggingface.co/spaces/leon0923/torch-sinkhorn-asc-demo 获取。除非特征提取器以不切实际的速度收敛，否则在一般的训练动力学中，该值有界且远离零。