Poincare embeddings for Learning Hierarchical Representations
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Talked at ABEJA Paper Reading Meetup July 4, 2017
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Poincare embeddings for Learning Hierarchical Representations
- 1. Poincaré Embeddings for Learning Hierarchical Representations July 4, 2017 Tatsuya Shirakawa ABEJA Inc.
- 3. CONFIDENTIAL Today’s Paper Paper Stats • Guys from FAIR • Sumitted to arXiv at May 26, 2017 https://arxiv.org/abs/1705.08039 • Sumitted to NIPS2017? Key Contributions • Introducing hyperbolic geometry (Poincaré disk model) into word/graph embeddings paradigm • Automatically capture hierarchical structure of data • Achieved incredible better results than previous works.
- 4. CONFIDENTIAL 1. Problems 2. Hyperbolic Geometry 3. Poincaré Embeddings (and Some Incredible Results) Agenda 4
- 6. CONFIDENTIAL Find good representation(embedding) of items such that underlying hierarchical relation structure are well reconstructed The Problem
- 7. CONFIDENTIAL Embed nouns in WordNets such that related nouns are close in embedded space Taxonomy Embedding 7 http://www.nltk.org/book_1ed/ch02.html
- 8. CONFIDENTIAL Embed nodes in given graph such that missing links are well-reconstructed Graph Link Prediction 8 http://ml.cs.tsinghua.edu.cn/~jiaming/publications/
- 10. CONFIDENTIAL • Geometry with negative curvature • Many models (realizations): - Poincaré half space model - Poincaré disk model … each is isometric Hyperbolic Geometry 10
- 11. CONFIDENTIAL • Defined on upper half space with metric • Distance btw points is Poincaré Half Space Model 11
- 12. CONFIDENTIAL 12 Tree representation in H https://arxiv.org/abs/1006.5169 • Tree structure is well represented in Poincaré half space
- 13. CONFIDENTIAL • A realization of hyperbolic geometry • Defined on equipped with metric of • Distance btw points is Poincaré Disk Model 13 M.C. Escher's Circle Limit III, 1959
- 14. CONFIDENTIAL (for simplicity: 2-dim, identify as ) Relation to Poincaré Half Space Model 14 https://arxiv.org/abs/1006.5169
- 15. CONFIDENTIAL • Euclidean space is too narrow to embed hierarchical (tree) structures Why not Euclidean Space? 15 Surface Area / # of leaf nodes Volume / # of nodes Euclidean Ball O(R^n) O(R^n) b-ary tree O(b^R) O(b^R) ※ R=radius of ball or depth of tree
- 16. CONFIDENTIAL • b-array tree can be interpreted as discrete analogue of Poincaré disk Why Hyperbolic Space? 16
- 17. CONFIDENTIAL • Hyperbolic space is far more appropriate than Euclidean space to represent hierarchical structure • Many equivalent models - Poincaré half space model - Poincaré disk model … Conclusion Here 17
- 18. CONFIDENTIAL • R. Kleinberg, “Geographic routing using hyperbolic spaces”, 2007 • M. Boguna et al., “Sustaining the internet with hyperbolic mapping”, 2010 • P. D. Hoff et al., “Latent space approaches to social network analysis”, 2016 • A. B. Adcock et al., “Tree-like structure in large social and information networks’, 2013 • D. Krioukov et al., “Hyperbolic geometry of complex networks”, 2010 Prior Works around hyperbolic geometry applications 18
- 20. CONFIDENTIAL 1. Parametrize each item in Poincaré ball 2. Optimize them by Riemannian optimization under metric of Proposed Method
- 21. CONFIDENTIAL 1. Compute stochastic (Euclidean) gradient 2. Correct metric 3. Apply GD 4. Project onto space Riemannian SGD 21
- 22. CONFIDENTIAL Embed nouns in WordNets such that related nouns are close in embedded space Taxonomy Embedding 22 http://www.nltk.org/book_1ed/ch02.html
- 23. CONFIDENTIAL Maximize Reconstruction setting: - D is full relations Prediction setting - D is subset of full relations Objective Function 23 randomly chosen 10 negative samples
- 25. CONFIDENTIAL 25
- 26. CONFIDENTIAL Embed nodes in given graph such that missing links are well-reconstructed Graph Link Prediction 26 http://ml.cs.tsinghua.edu.cn/~jiaming/publications/
- 27. CONFIDENTIAL Minimize the cross entropy of probability Objective Function 27
- 29. CONFIDENTIAL • Poincaré embeddings automatically capture hierarchical structure from data • Riemannian SGD provides the way to optimize Poincaré embeddings • Achieved quite good results on word/graph embedding tasks Summary 29