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Fast Unbalanced Optimal Transport on a Tree
- 1. 1 KYOTO UNIVERSITY KYOTO UNIVERSITY Fast Unbalanced Optimal Transport on a Tree Ryoma Sato Kyoto University / RIKEN AIP
- 2. 2 / 13 KYOTO UNIVERSITY Self Introduction I am a second year master's student at Kyoto University I’m interested in algorithmic aspects of machine learning and data mining for structured data, including Graph neural networks: Ryoma Sato, Makoto Yamada, Hisashi Kashima. Approximation Ratios of Graph Neural Networks for Combinatorial Problems. NeurIPS 2019. Ryoma Sato, Makoto Yamada, Hisashi Kashima. Random Features Strengthen Graph Neural Networks. SDM 2021 Optimal transport: Ryoma Sato, Makoto Yamada, Hisashi Kashima. Fast Unbalanced Optimal Transport on a Tree. NeurIPS 2020. Today’s topic
- 3. 3 / 13 KYOTO UNIVERSITY Background: optimal transport is useful The optimal transport (OT) distance measures the discrepancy of two distributions. We consider discrete distributions in this presentation. The OT distance is the minimum cumulative distance that all masses need to travel from one distribution to another distribution In generative modeling, a mass is a sample. discrepancy of model sample distribution In text classification, a mass is a word. OT does not require the same support KL divergence OT exploits the underground geometry From Word Embeddings To Document Distances, ICML 2015
- 4. 4 / 13 KYOTO UNIVERSITY Background: sliced OT is computationally cheap OT is formulated as a linear program cubic cost Sliced OT projects distributions to random 1D spaces and computes OT there Greedy matching solves 1D OT exactly linear cost : distance matrix (input), : matching matrix (variable) : 1st mass vector (input), : 2nd mass vector (input) The leftmost mass should be matched to the leftmost mass The second leftmost mass should be matched to the second leftmost mass ... https://www.programmersought.com/article/67174999352/ https://analyticsindiamag.com/how-to-establish-domain-transferability-in-neural-models/
- 5. 5 / 13 KYOTO UNIVERSITY Background: unbalanced OT is robust OT is sensitive to outliers because transporting outliers becomes the dominating term Unbalanced OT (UOT) allows to discard and create masses by paying some penalties We can discard outliers robust to outliers UOT is also formulated by a linear program cubic cost
- 6. 6 / 13 KYOTO UNIVERSITY Background: UOT is difficult even in 1D spaces We want to make a cheap alternative of UOT as 1D OT But the greedy matching fails to solve 1D UOT Let’s consider the following instance with discard cost λ The following plan costs 3λ. The following plan costs 2λ + 2ε. Thus this is better. λ λ λ
- 7. 7 / 13 KYOTO UNIVERSITY Background: UOT is difficult even in 1D spaces Let’s consider the following instance with discard cost λ The following plan costs λ + 2ε. The following plan costs 2λ + 2ε. Thus this is worse. λ ε
- 8. 8 / 13 KYOTO UNIVERSITY Background: UOT is difficult even in 1D spaces Although these two instances share the leftmost part, the leftmost mass in the first instance should be discarded while that in the second instance should not The optimal UOT plan cannot be determined locally The optimal OT plan is determined locally Thus the greedy algorithm fails to solve 1D UOT We proposed how to solve 1D UOT efficiently λ λ λ λ ε
- 9. 9 / 13 KYOTO UNIVERSITY Algorithm: prune redundant plans Our proposed method determines assignments from left to right (as the greedy algorithm) Although there are exponentially many plans, most of them are redundant. We proved that only O(n) plans are non-redundant Only one plan is non-redundant (thus greedy is valid) in the standard OT not yet not yet non redundant redundant
- 10. 10 / 13 KYOTO UNIVERSITY Algorithm: we solve 1D UOT in O(n log2 n) time A naive algorithm requires cubic time even with this (non redundant plan) observation More algorithmic techniques are required for further speedup (skipped in this presentation) Dynamic programming Fast convex min-sum convolution Efficient data structure (BBST) Weighted union heuristics Finally, we derived a quasi-linear time algorithm which runs in O(n log2 n) time in the worst case
- 11. 11 / 13 KYOTO UNIVERSITY Algorithm: tree UOT generalizes 1D UOT Our method can be extended to tree spaces A 1D space (path) is a special case of tree spaces In text classification, the word space can be represented by a word tree. Each mass (word) travels on the word tree to a nearby (semantically similar) word. We can “tree-slice” high dimensional spaces instead of 1D-slicing, which captures richer structures http://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/117-hcpc-hierarchical-clustering-on-principal-components-essentials/ a → b
- 12. 12 / 13 KYOTO UNIVERSITY Experiments: our algorithm is empirically fast We confirmed that our algorithm could compute tree UOT with one million masses within one second We also confirmed that tree-slicing high dimensional spaces could approximate the original UOT problem
- 13. 13 / 13 KYOTO UNIVERSITY Conclusion: fast computation of tree UOT Sliced OT is a fast alternative of OT UOT is a robust variant of OT 1D UOT is more difficult than 1D OT We proposed an efficient algorithm for 1D UOT for the first time Our method can be extended to tree spaces Our method is empirically fast (1M masses in 1 sec)