Latent Tree Models



A latent tree model, or LTM for short, is a tree-structured Bayesian network with discrete  variables. (There are variants with continuous variables.) The variables at leaf nodes are observed, and the variables at internal nodes are latent, i.e., not observed. The root is associated with a marginal distribution, and each of the other node is associated with a distribution of the node given its parent. The product of all the distributions defines a joint distribution over all the variables.

LTMs can be used for co-occurrence modelling and for multidimensional clustering (i.e., the partition of data in multiple ways using latent variables).

        Videos: Overview. Lectures: 1 - Basics & Algorithms, 2 C Algorithms & Applications, 3 C More Applications.

         N. L. Zhang, L. K. M. Poon (2017).  Latent Tree Analysis. AAAI Senior Member Track: 4891-4898.  ppt

        N. L. Zhang (2002). Hierarchical latent class models for cluster analysisAAAI-02, 230-237.

        N. L. Zhang (2004). Hierarchical latent class models for cluster analysisJournal of Machine Learning Research, 5(6):697-723, 2004.

        AAAI 2014 Tutorial:  Part I Part II,   Part III Part IV


Modelling word co-occurrences: LTMs can be used to model word co-occurrence patterns in documents. In the example below, the latent variable Z14 indicates the words video, card, and driver often co-occur in documents; Z15 represents the co-occurrence of the words dos and windows; Z17 represents the co-occurrence of image, graphics, display; and so on.

The states of the latent variables represent soft clusters of documents. They can be interpreted as topics and LTMs are hence used as a novel tool for hierarchical topic detection , which significantly outperforms alternative methods.

        P. Chen, N.L. Zhang, et al. Latent Tree Models for Hierarchical Topic DetectionArtificial Intelligence, 250:105-124, 2017.

        P. Chen, N.L. Zhang, et al. Progressive EM for Latent Tree Models and Hierarchical Topic DetectionAAAI 2016.

        T. Liu, N.L. Zhang, P. Chen. Hierarchical Latent Tree Analysis for Topic DetectionECML/PKDD (2) 2014: 256-272.

        Dedicated Webpage: http://home.cse.ust.hk/~lzhang/topic/ijcai2016/


Modelling co-consumption of items:  LTM can also be used to model co-consumption of items by users. In the example below, Z13 indicates that the movies Armageddon, Golden Eye and Con Air tend to be co-consumed, i.e., watched by the same viewers.  Z1148 reveals that the movies Tarzan, Rugrats, Mulan and Winnie the Pooh tend to be co-consumed.

The states of the latent variables represent soft clusters of users with different tastes, which are used in a novel method for item recommendation called conformative filtering. It significantly outperforms alternative collaborative filtering methods for implicit feedback.

        F. Khawar, N.L. Zhang, Y. Yu.  Conformative Filtering for Implicit Feedback Data. arXiv:1704.01889.


Modelling co-occurrences of symptoms: LTMs are a natural tool for modelling co-occurrences of symptoms on patients. The example below is from traditional Chinese medicine (TCM).  The left-most part of the model indicates that the symptoms cold limbs, cold lumbus and back, intolerance to cold, loose stool tend to co-occurrence. This co-occurrence pattern corresponds to the TCM concept of Yang Deficiency. The right-most part of the model indicates that the symptoms yellow urine, thirst, dry tongue, rapid pulse, tidal fever etc tend to co-occur. This pattern correspond to the TCM concept of Yin Deficiency.

The work shows that TCM concepts such as Yang Deficiency and Yin Deficiency are soft patient clusters that can be identified from clinic data. This is of fundamental importance to TCM because TCM patient class definitions are subjective and vague. Our work opens up the perspective of deriving TCM patient class definitions from clinic symptom distribution data.

        N. L. Zhang,  S. H. Yuan, T. Chen and  Y. Wang (2008).  Statistical Validation of TCM Theories. The Journal of Alternative and Complementary Medicine, 14(5):583-7.

         Z.X. Xu, N. L. Zhang, et al. (2013). Statistical Validation of Traditional Chinese Medicine Syndrome Postulates in the Context of Patients with Cardiovascular Disease.  The Journal of Alternative and Complementary Medicine. 18, 1-6.

        N. L. Zhang, C. Fu, et al. (2017). A data-driven method for syndrome type identification and classification in traditional Chinese medicine. Journal of Integrative Medicine, 15(2):110C123.

        Dedicated Webpage: http://www.cse.ust.hk/~lzhang/tcm/


Modelling co-occurrences (correlations) in survey data: The model below is learned from data from a survey about the Danish beer market. According to the model, people's view about Carlseberg and Groun Tuborg are strongly correlated. This makes sense because those are the two main market beers in Denmark. The group in the middle, CarlsSpec, Tuborgclass and Henneken are also frequent beers, but they are darker in taste as compared with the group on the right. The group on the left are local beers.

The states of the latent variables identify customers with different preferences and opinions. They are useful when making marketing strategies.

         R. Mourad, C. Sinoquet, N. L. Zhang, T.F. Liu and P. Leray (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


Multidimensional ClusteringClustering is a data analysis approach where the objective is to find `naturally occurring' groups. Early research work on clustering usually assumed that there was one true clustering of data. This assumption does not hold for complex data which are typically multifaceted and can be meaningfully clustered in many different ways. There is a growing interest in methods that produce multiple partitions of data with each partition being based on a different subset of attributes. We call such methods multi-partition clustering methods.



Analyzing data using LTMs can result in multiple discrete latent variables, each representing a soft partition of data. So LTMs can be used for multi-partition clustering. This potential was first pointed out in (Zhang 2002, 2004). Because latent variables can be viewed as latent attributes of data, we sometimes call LTM-based cluster analysis multidimensional clustering.

        T. Chen, N. L. Zhang, T. F. Liu, Y. Wang, L. K. M. Poon (2012). Model-based multidimensional clustering of categorical data. Artificial Intelligence. 176(1), 2246-2269.

        T.F, Liu, N. L. Zhang, P. X. Chen, A. H.Liu, L. K. M. Poon, and Yi Wang (2013). Greedy learning of latent tree models for multidimensional clustering. Machine Learning, doi:10.1007/s10994-013-5393-0.

        L. K. M. Poon, N. L. Zhang, T. Chen, and Y. Wang (2010). Variable selection in model-based clustering: To do or to facilitate. ICML-10.

         L.K.M. Poon, N.L. Zhang, T.F. Liu, A.H. Liu (2013). Model-Based Clustering of High-Dimensional Data: Variable Selection versus Facet Determination. International Journal of Approximate Reasoning. 54(1), 196-215



A link to Deep Learning: Hierarchical latent tree models (HLTM) and deep belief networks (DBNs) are similar in that they both define a distribution over a set of observed variables and they both use multiple layers of latent variables. On the other hand, there are obvious differences. One is tree-structured and learned from data, and the other is fully connected and manually specified. Those are two extremes. 

It would be interesting to explore the middle ground between the two extremes. One idea is to first learn an HLTM from data, use it as the skeleton for a deep model, and add additional links to improve model fit.

        Z. Chen, N. L. Zhang, et al. (2017). Sparse Boltzmann Machines with Structure Learning as Applied to Text Analysis. AAAI 2017: 1805-1811



        HLTA: https://github.com/kmpoon/hlta Use of latent tree models for hierarchical topic detection.

        Lantern: http://www.cse.ust.hk/~lzhang/ltm/softwares/Lantern.zip LTM GUI mainly for TCM research.

        BI: http://www.cse.ust.hk/~lzhang/ltm/softwares/BI.zip code for Liu et al. (MLJ 2013).

        EAST: http://www.cse.ust.hk/~lzhang/ltm/softwares/EAST.zip code for Chen et al. (AIJ 2012).