AAAI 2014 Tutorial Nevin L. Zhang HKUST3 Latent Tree Models Part I: Non-Technical Overview (25 minutes) Part II: Definition and Properties (25 minutes) Part III: Learning Algorithms (110 minutes, 30 minutes break half way) Part IV: Applications (50 minutes)
AAAI 2014 Tutorial Nevin L. Zhang HKUST4 Part I: Non-Technical Overview Latent tree models What can LTMs be used for: Discovery of co-occurrence/correlation patterns Discovery of latent variable/structures Multidimensional clustering Examples Danish beer survey data Text data
AAAI 2014 Tutorial Nevin L. Zhang HKUST5 Latent Tree Models (LTMs) Tree-structured probabilistic graphical models Leaves observed (manifest variables) Discrete or continuous Internal nodes latent (latent variables) Discrete Each edge is associated with a conditional distribution One node with marginal distribution Defines a joint distributions over all the variables (Zhang, JMLR 2004)
AAAI 2014 Tutorial Nevin L. Zhang HKUST6 Latent Tree Analysis (LTA) Learning latent tree models: Determine Number of latent variables Numbers of possible states for latent variables Connections among nodes Probability distributions From data on observed variables, obtain latent tree model
AAAI 2014 Tutorial Nevin L. Zhang HKUST7 LTA on Danish Beer Market Survey Data 463 consumers, 11 beer brands Questionnaire: For each brand: Never seen the brand before (s0); Seen before, but never tasted (s1); Tasted, but do not drink regularly (s2) Drink regularly (s3). Page 7 (Mourad et al. JAIR 2013)
AAAI 2014 Tutorial Nevin L. Zhang HKUST8 Why variables grouped as such? Responses on brands in each group strongly correlated. GronTuborg and Carlsberg: Main mass-market beers TuborgClas and CarlSpec: Frequent beers, bit darker than the above CeresTop, CeresRoyal, Pokal, …: minor local beers In general, LTA partitions observed variables into groups such that Variables in each group are strongly correlated, and The correlations among each group can be properly be modeled using one single latent variable Page 8
AAAI 2014 Tutorial Nevin L. Zhang HKUST9 Multidmensional Clustering Each Latent variable gives a partition of consumers. H1: Class 1: Likely to have tasted TuborgClas, Carlspec and Heineken, but do not drink regularly Class 2: Likely to have seen or tasted the beers, but did not drink regularly Class 3: Likely to drink TuborgClas and Carlspec regularly H0 and H2 give two other partitions. In general, LTA is a technique for multiple clustering. In contrast, K-Means, mixture models give only one partition. Page 9
AAAI 2014 Tutorial Nevin L. Zhang HKUST10 Unidimensional vs Multidimensional Clustering l Grouping of objects into clusters such that objects in the same cluster are similar while objects from different clusters are dissimilar. Page 10 l Result of clustering is often a partition of all the objects.
AAAI 2014 Tutorial Nevin L. Zhang HKUST11 How to Cluster Those?
AAAI 2014 Tutorial Nevin L. Zhang HKUST12 How to Cluster Those? Style of picture
AAAI 2014 Tutorial Nevin L. Zhang HKUST13 How to Cluster Those? Type of object in picture
AAAI 2014 Tutorial Nevin L. Zhang HKUST14 Multidimensional Clustering Complex data usually have multiple facets and can be meaningfully partitioned in multiple ways. Multidimensional clustering / Multi-Clustering LTA is a model-based method for multidimensional clustering. Other methods: http://www.siam.org/meetings/sdm11/clustering.pdf
AAAI 2014 Tutorial Nevin L. Zhang HKUST15 LTA produces a partition of observed variables. For each cluster of variables, it produces a partition of objects. Clustering of Variables and Objects
AAAI 2014 Tutorial Nevin L. Zhang HKUST16 1041 web pages collected from 4 CS departments in 1997 336 words Binary Text Data: WebKB
AAAI 2014 Tutorial Nevin L. Zhang HKUST17 Latent Tree Model for WebKB Data (Liu et al. MLJ 2013) 89 latent variables
AAAI 2014 Tutorial Nevin L. Zhang HKUST21 Words in each group tend to co-occur. On binary text data, LTA partitions word variables into groups such that Words in each group tend to co-occur and The correlations can be properly be explained using one single latent variable Why variables grouped as such? LTA is a method for identifying co-occurrence relationships.
AAAI 2014 Tutorial Nevin L. Zhang HKUST22 LTA is an alternative approach to topic detection Y66=4: Object Oriented Programming (oop) Y66=2: Non-oop programming Y66=1: programming language Y66=3: Not on programming Multidimensional Clustering More on this in Part IV
AAAI 2014 Tutorial Nevin L. Zhang HKUST23 Summary Latent tree models: Tree-structured probabilistic graphical models Leaf nodes: observed variables Internal nodes: latent variable What can LTA be used for: Discovery of co-occurrence patterns in binary data Discovery of correlation patterns in general discrete data Discovery of latent variable/structures Multidimensional clustering Topic detection in text data Probabilistic modelling
Key References: Anandkumar, A., Chaudhuri, K., Hsu, D., Kakade, S. M., Song, L., & Zhang, T. (2011). Spectral methods for learning multivariate latent tree structure. In Twenty-Fifth Conference in Neural Information Processing Systems (NIPS-11). Anandkumar, A., Ge, R., Hsu, D., Kakade, S.M., and Telgarsky, M. Tensor decompositions for learning latent variable models. In Preprint, 2012a. Anandkumar, A., Hsu, D., and Kakade, S. M. A method of moments for mixture models and hidden Markov models. In An abridged version appears in the Proc. Of COLT, 2012b. Choi, M. J., Tan, V. Y., Anandkumar, A., & Willsky, A. S. (2011). Learning latent tree graphical models. Journal of Machine Learning Research, 12, 1771–1812. Friedman, N., Ninio, M., Pe’er, I., & Pupko, T. (2002). A structural EM algorithm for phylogenetic inference.. Journal of Computational Biology, 9(2), 331–353. Harmeling, S., & Williams, C. K. I. (2011). Greedy learning of binary latent trees. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(6), 1087–1097. Hsu, D., Kakade, S., & Zhang, T. (2009). A spectral algorithm for learning hidden Markov models. In The 22nd Annual Conference on Learning Theory (COLT 2009).
AAAI 2014 Tutorial Nevin L. Zhang HKUST25 Key References: E. Mossel, S. Roch, and A. Sly. Robust estimation of latent tree graphical models: Inferring hidden states with inexact parameters. Submitted. http://arxiv.org/abs/1109.4668, 2011. Mourad, R., Sinoquet, C., & Leray, P. (2011). A hierarchical Bayesian network approach for linkage disequilibrium modeling and data-dimensionality reduction prior to genomewide association studies. BMC Bioinformatics, 12, 16. Mourad R., Sinoquet C., Zhang N. L., Liu T. F. and Leray P. (2013). A survey on latent tree models and applications. Journal of Artificial Intelligence Research, 47, 157-203, 13 May 2013. doi:10.1613/jair.3879. Parikh, A. P., Song, L., & Xing, E. P. (2011). A spectral algorithm for latent tree graphical models. In Proceedings of the 28th International Conference on Machine Learning (ICML-2011). Saitou, N., & Nei, M. (1987). The neighbor-joining method: A new method for reconstructing phylogenetic trees.. Molecular Biology and Evolution, 4(4), 406–425. Song, L., Parikh, A., & Xing, E. (2011). Kernel embeddings of latent tree graphical models. In Twenty- Fifth Conference in Neural Information Processing Systems (NIPS-11). Tan, V. Y. F., Anandkumar, A., & Willsky, A. (2011). Learning high-dimensional Markov forest distributions: Analysis of error rates. Journal of Machine Learning Research,12, 1617–1653.
AAAI 2014 Tutorial Nevin L. Zhang HKUST26 Key References: T. Chen and N. L. Zhang (2006). Quartet-based learning of shallow latent variables. In Proceedings of the Third European Workshop on Probabilistic Graphical Model,59-66, September 12-15, 2006. Chen, T., Zhang, N. L., Liu, T., Poon, K. M., & Wang, Y. (2012). Model-based multidimensional clustering of categorical data. Artificial Intelligence, 176(1), 2246–2269. Liu, T. F., Zhang, N. L., Liu, A. H., & Poon, L. K. M. (2013). Greedy learning of latent tree models for multidimensional clustering. Machine Learning, doi:10.1007/s10994-013-5393-0. Liu, T. F., Zhang, N. L., and Chen, P. X. (2014). Hierarchical latent tree analysis for topic detection. ECML, 2014 Poon, L. K. M., Zhang, N. L., Chen, T., & Wang, Y. (2010). Variable selection in modelbased clustering: To do or to facilitate. In Proceedings of the 27th International Con-ference on Machine Learning (ICML- 2010). Wang, Y., Zhang, N. L., & Chen, T. (2008). Latent tree models and approximate inference in Bayesian networks. Journal of Articial Intelligence Research, 32, 879–900. Wang, X. F., Guo, J. H., Hao, L. Z., Zhang, N.L., & P. X. Chen (2013). Recovering discrete latent tree models by spectral methods. Wang, X. F., Zhang, N. L. (2014). A Study of Recently Discovered Equalities about Latent Tree Models using Inverse Edges. PGM 2014. Zhang, N. L. (2004). Hierarchical latent class models for cluster analysis. The Journal of Machine Learning Research, 5, 697–723. Zhang, N. L., & Kocka, T. (2004a). Effective dimensions of hierarchical latent class models. Journal of Articial Intelligence Research, 21, 1–17.
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