{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/computationally-efficient-tree-variants-of","title":"Flow-based Alignment Approaches for Probability Measures in Different Spaces","arxiv_id":"1910.04462","date":"2019-10-10","proceeding":null,"authors":["Tam Le","Nhat Ho","Makoto Yamada"],"abstract":"Gromov-Wasserstein (GW) is a powerful tool to compare probability measures whose supports are in different metric spaces. GW suffers however from a computational drawback since it requires to solve a complex non-convex quadratic program. We consider in this work a specific family of cost metrics, namely \\textit{tree metrics} for a space of supports of each probability measure, and aim for developing efficient and scalable discrepancies between the probability measures. By leveraging a tree structure, we propose to align \\textit{flows} from a root to each support instead of pair-wise tree metrics of supports, i.e., flows from a support to another, in GW. Consequently, we propose a novel discrepancy, named Flow-based Alignment (\\FlowAlign), by matching the flows of the probability measures. We show that \\FlowAlign~shares a similar structure as a univariate optimal transport distance. Therefore, \\FlowAlign~is fast for computation and scalable for large-scale applications. By further exploring tree structures, we propose a variant of \\FlowAlign, named Depth-based Alignment (\\DepthAlign), by aligning the flows hierarchically along each depth level of the tree structures. Theoretically, we prove that both \\FlowAlign~and \\DepthAlign~are pseudo-distances. Moreover, we also derive tree-sliced variants, computed by averaging the corresponding \\FlowAlign~/ \\DepthAlign~using random tree metrics, built adaptively in spaces of supports. Empirically, we test our proposed discrepancies against other baselines on some benchmark tasks.","url_abs":"https://arxiv.org/abs/1910.04462v5","url_pdf":"https://arxiv.org/pdf/1910.04462v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"computationally-efficient-tree-variants-of","repo_url":"https://github.com/lttam/Kmeans-FlowTreeGW-Barycenter","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[{"method_slug":"test","method_name":"Test"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}