Flow-based Alignment Approaches for Probability Measures in Different Spaces

Tam Le, Nhat Ho, Makoto Yamada
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:3934-3942, 2021.

Abstract

Gromov-Wasserstein (GW) is a powerful tool to compare probability measures whose supports are in different metric spaces. However, GW suffers from a computational drawback since it requires to solve a complex non-convex quadratic program. In this work, we consider a specific family of cost metrics, namely, tree metrics for supports of each probability measure, to develop efficient and scalable discrepancies between the probability measures. Leveraging a tree structure, we propose to align flows from a root to each support instead of pair-wise tree metrics of supports, i.e., flows from a support to another support, in GW. Consequently, we propose a novel discrepancy, named Flow-based Alignment (FlowAlign), by matching the flows of the probability measures. FlowAlign is computationally fast and scalable for large-scale applications. Further exploring the tree structure, 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-metrics. We also derive tree-sliced variants of the proposed discrepancies for applications without prior knowledge about tree structures for probability measures, computed by averaging FlowAlign/DepthAlign using random tree metrics, adaptively sampled from supports of probability measures. Empirically, we test our proposed approaches against other variants of GW baselines on a few benchmark tasks.

Cite this Paper


BibTeX
@InProceedings{pmlr-v130-le21b, title = { Flow-based Alignment Approaches for Probability Measures in Different Spaces }, author = {Le, Tam and Ho, Nhat and Yamada, Makoto}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {3934--3942}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/le21b/le21b.pdf}, url = {https://proceedings.mlr.press/v130/le21b.html}, abstract = { Gromov-Wasserstein (GW) is a powerful tool to compare probability measures whose supports are in different metric spaces. However, GW suffers from a computational drawback since it requires to solve a complex non-convex quadratic program. In this work, we consider a specific family of cost metrics, namely, tree metrics for supports of each probability measure, to develop efficient and scalable discrepancies between the probability measures. Leveraging a tree structure, we propose to align flows from a root to each support instead of pair-wise tree metrics of supports, i.e., flows from a support to another support, in GW. Consequently, we propose a novel discrepancy, named Flow-based Alignment (FlowAlign), by matching the flows of the probability measures. FlowAlign is computationally fast and scalable for large-scale applications. Further exploring the tree structure, 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-metrics. We also derive tree-sliced variants of the proposed discrepancies for applications without prior knowledge about tree structures for probability measures, computed by averaging FlowAlign/DepthAlign using random tree metrics, adaptively sampled from supports of probability measures. Empirically, we test our proposed approaches against other variants of GW baselines on a few benchmark tasks. } }
Endnote
%0 Conference Paper %T Flow-based Alignment Approaches for Probability Measures in Different Spaces %A Tam Le %A Nhat Ho %A Makoto Yamada %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-le21b %I PMLR %P 3934--3942 %U https://proceedings.mlr.press/v130/le21b.html %V 130 %X Gromov-Wasserstein (GW) is a powerful tool to compare probability measures whose supports are in different metric spaces. However, GW suffers from a computational drawback since it requires to solve a complex non-convex quadratic program. In this work, we consider a specific family of cost metrics, namely, tree metrics for supports of each probability measure, to develop efficient and scalable discrepancies between the probability measures. Leveraging a tree structure, we propose to align flows from a root to each support instead of pair-wise tree metrics of supports, i.e., flows from a support to another support, in GW. Consequently, we propose a novel discrepancy, named Flow-based Alignment (FlowAlign), by matching the flows of the probability measures. FlowAlign is computationally fast and scalable for large-scale applications. Further exploring the tree structure, 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-metrics. We also derive tree-sliced variants of the proposed discrepancies for applications without prior knowledge about tree structures for probability measures, computed by averaging FlowAlign/DepthAlign using random tree metrics, adaptively sampled from supports of probability measures. Empirically, we test our proposed approaches against other variants of GW baselines on a few benchmark tasks.
APA
Le, T., Ho, N. & Yamada, M.. (2021). Flow-based Alignment Approaches for Probability Measures in Different Spaces . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:3934-3942 Available from https://proceedings.mlr.press/v130/le21b.html.

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