分析切片 Wasserstein 距离优化时的临界点，揭示其稳定特性。
Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis
- 通过显式扰动计算，研究切片距离优化中的临界点性质。
- 发现稳定临界点无法集中在线段上，解释优化行为。
- 理论结合实验，适合研究生成模型与优化机制的读者。
本文研究将切片 Wasserstein 距离（SW）作为目标函数时的性质。由于其能捕捉概率分布的复杂几何结构且计算可行，SW 在最优传输与机器学习领域备受关注，广泛应用于生成建模和域自适应等任务。本文旨在对优化 SW 目标时产生的临界点进行严格分析。通过显式扰动计算，我们证明了 SW 的稳定临界点无法集中在线段上。该稳定性分析对于理解使用 SW 目标训练的模型的优化算法行为至关重要。此外，我们还探讨了 SW 目标本身的性质，揭示了临界点的存在性与收敛行为。数值实验验证了理论结果。
原文摘要 · Abstract (English)
In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to its ability to capture intricate geometric properties of probability distributions while remaining computationally tractable, making it a valuable tool for various applications, including generative modeling and domain adaptation. Our study aims to provide a rigorous analysis of the critical points arising from the optimization of the SW objective. By computing explicit perturbations, we establish that stable critical points of SW cannot concentrate on segments. This stability analysis is crucial for understanding the behaviour of optimization algorithms for models trained using the SW objective. Furthermore, we investigate the properties of the SW objective, shedding light on the existence and convergence behavior of critical points. We illustrate our theoretical results through numerical experiments.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。