发表机构
Rice University（莱斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Gromov-Wasserstein问题，提出自适应KL-BAPG算法，结合固定惩罚预热和递增惩罚阶段，在保证可行性的同时实现高效计算，显著降低可行性差距和残差。
AI 中文摘要
Gromov-Wasserstein（GW）问题在不要求共享特征空间或已知对应关系的情况下比较结构化分布，但其非凸目标和耦合边际约束使得计算具有挑战性。Bregman交替投影梯度（BAPG）使用廉价的交替行和列更新，但其固定惩罚松弛留下了持续的可行性差距。我们提出自适应KL-BAPG（A-KL-BAPG），它结合了有限固定惩罚的预热阶段和带保护的递增惩罚阶段。在每个尾部迭代中，该方法重用BAPG的交替更新，并回溯延迟幂步骤，直到满足Sinkhorn启发的投影直径保护。我们证明了每次迭代回溯的有限终止性，并展示了可行性差距渐近消失。我们进一步建立了加权平方修正残差的最佳迭代$O(1/\log N)$界，并在支持正则性条件下，证明了原始GW问题存在平稳累积点。这使A-KL-BAPG区别于固定惩罚BAPG，后者仅对松弛问题给出平稳性保证。实验表明，相对于BAPG变体、基于投影的方法和任务特定基线，A-KL-BAPG在准确性、目标值、可行性和平稳性方面实现了有利的平衡。对于合成和真实图对齐问题，它紧密匹配固定惩罚KL-BAPG的准确性和目标值，同时将边际可行性差距减少62-99%，投影平稳性残差减少28-98%。异构域适应实验显示出类似模式：A-KL-BAPG在保持可比目标准确性和目标值的同时，比固定惩罚KL-BAPG实现了更好的可行性和平稳性。
英文摘要
The Gromov-Wasserstein (GW) problem compares structured distributions without requiring a shared feature space or known correspondences, but its nonconvex objective and coupled marginal constraints make computation challenging. Bregman alternating projected gradient (BAPG) uses inexpensive alternating row and column updates, yet its fixed-penalty relaxation leaves a persistent feasibility gap. We propose Adaptive KL-BAPG (A-KL-BAPG), which combines a finite fixed-penalty burn-in with a guarded increasing-penalty phase. At each tail iteration, the method reuses BAPG's alternating updates and backtracks a delayed-power step until a Sinkhorn-inspired projective-diameter safeguard is satisfied. We prove finite termination of the backtracking at each iteration and show that the feasibility gap vanishes asymptotically. We further establish a best-iterate $O(1/\log N)$ bound for the weighted squared corrected residual and, under a support regularity condition, the existence of a stationary accumulation point for the original GW problem. This distinguishes A-KL-BAPG from fixed-penalty BAPG, whose stationarity guarantees are given for the relaxed problem. Experiments show that A-KL-BAPG achieves a favorable balance of accuracy, objective value, feasibility, and stationarity relative to BAPG variants, projection-based methods, and task-specific baselines. For synthetic and real graph alignment problems, it closely matches the accuracy and objective value of fixed-penalty KL-BAPG while reducing the marginal feasibility gap by 62-99% and the projected stationarity residual by 28-98%. Heterogeneous domain adaptation experiments show a similar pattern: A-KL-BAPG maintains comparable target accuracy and objective values while achieving better feasibility and stationarity than fixed-penalty KL-BAPG.
Comments34 pages, 5 figures, 4 tables
