MetricEmbedding: Accelerate Metric Nearness by Tropical Inner Product

Muyang Cao, Jiajun Yu, Xin Du, Gang Pan, Wei Wang
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:6675-6698, 2025.

Abstract

The Metric Nearness Problem involves restoring a non-metric matrix to its closest metric-compliant form, addressing issues such as noise, missing values, and data inconsistencies. Ensuring metric properties, particularly the $O(N^3)$ triangle inequality constraints, presents significant computational challenges, especially in large-scale scenarios where traditional methods suffer from high time and space complexity. We propose a novel solution based on the tropical inner product (max-plus operation), which we prove satisfies the triangle inequality for non-negative real matrices. By transforming the problem into a continuous optimization task, our method directly minimizes the distance to the target matrix. This approach not only restores metric properties but also generates metric-preserving embeddings, enabling real-time updates and reducing computational and storage overhead for downstream tasks. Experimental results demonstrate that our method achieves up to 60$\times$ speed improvements over state-of-the-art approaches, and efficiently scales from $1e4 \times 1e4$ to $1e5 \times 1e5$ matrices with significantly lower memory usage.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-cao25i, title = {{M}etric{E}mbedding: Accelerate Metric Nearness by Tropical Inner Product}, author = {Cao, Muyang and Yu, Jiajun and Du, Xin and Pan, Gang and Wang, Wei}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {6675--6698}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/cao25i/cao25i.pdf}, url = {https://proceedings.mlr.press/v267/cao25i.html}, abstract = {The Metric Nearness Problem involves restoring a non-metric matrix to its closest metric-compliant form, addressing issues such as noise, missing values, and data inconsistencies. Ensuring metric properties, particularly the $O(N^3)$ triangle inequality constraints, presents significant computational challenges, especially in large-scale scenarios where traditional methods suffer from high time and space complexity. We propose a novel solution based on the tropical inner product (max-plus operation), which we prove satisfies the triangle inequality for non-negative real matrices. By transforming the problem into a continuous optimization task, our method directly minimizes the distance to the target matrix. This approach not only restores metric properties but also generates metric-preserving embeddings, enabling real-time updates and reducing computational and storage overhead for downstream tasks. Experimental results demonstrate that our method achieves up to 60$\times$ speed improvements over state-of-the-art approaches, and efficiently scales from $1e4 \times 1e4$ to $1e5 \times 1e5$ matrices with significantly lower memory usage.} }
Endnote
%0 Conference Paper %T MetricEmbedding: Accelerate Metric Nearness by Tropical Inner Product %A Muyang Cao %A Jiajun Yu %A Xin Du %A Gang Pan %A Wei Wang %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-cao25i %I PMLR %P 6675--6698 %U https://proceedings.mlr.press/v267/cao25i.html %V 267 %X The Metric Nearness Problem involves restoring a non-metric matrix to its closest metric-compliant form, addressing issues such as noise, missing values, and data inconsistencies. Ensuring metric properties, particularly the $O(N^3)$ triangle inequality constraints, presents significant computational challenges, especially in large-scale scenarios where traditional methods suffer from high time and space complexity. We propose a novel solution based on the tropical inner product (max-plus operation), which we prove satisfies the triangle inequality for non-negative real matrices. By transforming the problem into a continuous optimization task, our method directly minimizes the distance to the target matrix. This approach not only restores metric properties but also generates metric-preserving embeddings, enabling real-time updates and reducing computational and storage overhead for downstream tasks. Experimental results demonstrate that our method achieves up to 60$\times$ speed improvements over state-of-the-art approaches, and efficiently scales from $1e4 \times 1e4$ to $1e5 \times 1e5$ matrices with significantly lower memory usage.
APA
Cao, M., Yu, J., Du, X., Pan, G. & Wang, W.. (2025). MetricEmbedding: Accelerate Metric Nearness by Tropical Inner Product. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:6675-6698 Available from https://proceedings.mlr.press/v267/cao25i.html.

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