无监督对齐异构数据，通过不平衡最优传输实现跨域映射
Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization
- 基于不平衡最优传输与格罗莫夫-沃瑟斯坦边际惩罚的联合嵌入方法
- 可将不同域数据映射到统一度量空间，且支持欧几里得与非欧空间
- 适用于无对应关系的数据对齐，尤其适合多模态或结构差异大的场景
我们提出一种新的无监督方法，用于对齐异构数据集，将两个不同域的数据在无已知对应关系的情况下映射到共同的度量空间。该方法基于带有格罗莫夫-沃瑟斯坦边际惩罚的不平衡最优传输问题，可视为近期提出的联合多维标度方法的对偶形式。我们证明了目标泛函存在最小值，并且当惩罚参数趋于无穷时，对应的最小化序列收敛至嵌入沃瑟斯坦距离的最小值。模型可重写为二次、多边际、不平衡最优传输问题，其双凸松弛可通过块坐标下降法求解。我们在欧几里得及非欧空间中提供了联合嵌入的数值示例。
原文摘要 · Abstract (English)
We propose a new approach for unsupervised alignment of heterogeneous datasets, which maps data from two different domains without any known correspondences to a common metric space. Our method is based on an unbalanced optimal transport problem with Gromov-Wasserstein marginal penalization. It can be seen as a counterpart to the recently introduced joint multidimensional scaling method. We prove that there exists a minimizer of our functional and that for penalization parameters going to infinity, the corresponding sequence of minimizers converges to a minimizer of the so-called embedded Wasserstein distance. Our model can be reformulated as a quadratic, multi-marginal, unbalanced optimal transport problem, for which a bi-convex relaxation admits a numerical solver via block-coordinate descent. We provide numerical examples for joint embeddings in Euclidean as well as non-Euclidean spaces.
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