1 Introduction
Restricted Boltzmann machines (RBMs) [36, 9, 20]
have recently attracted significant interest due to their versatility in a variety of unsupervised and supervised learning tasks
[35, 18, 25], and in building deep architectures [14, 31]. A RBM is a bipartite undirected model that captures the generative process in which a data vector is generated from a binary hidden vector. The bipartite architecture enables very fast data encoding and samplingbased inference; and together with recent advances in learning procedures, we can now process massive data with large models [13, 37, 2].This paper presents our contributions in developing RBM specifications as well as learning and inference procedures for multivariate ordinal data. This extends and consolidates the reach of RBMs to a wide range of usergenerated domains  social responses, recommender systems, product/paper reviews, and expert assessments of health and ecosystems indicators. Ordinal variables are qualitative in nature – the absolute numerical assignments are not important but the relative order is. This renders numerical transforms and realvalued treatments inadequate. Current RBMbased treatments, on the other hand, ignore the ordinal nature and treat data as unordered categories
[35, 40]. While convenient, this has several drawbacks: First, order information is not utilised, leading to more parameters than necessary  each category needs parameters. Second, since categories are considered independently, it is less interpretable in terms of how ordinal levels are generated. Better modelling should account for the ordinal generation process.Adapting the classic idea from [24], we assume that each ordinal variable is generated by an underlying latent utility, along with a threshold per ordinal level. As soon as the utility passes the threshold, its corresponding level is selected. As a result, this process would implicitly encode the order. Our main contribution here is a novel RBM architecture that accounts for multivariate, ordinal data. More specifically, we further assume that the latent utilities are Gaussian variables connected to a set of binary hidden factors (i.e., together they form a Gaussian RBM [14]). This offers many advantages over the standard approach that imposes a fully connected Gaussian random field over utilities [17, 15]
: First, utilities are seen as being generated from a set of binary factors, which in many cases represent the user’s hidden profile. Second, utilities are decoupled given the hidden factors, making parallel sampling easier. And third, the posteriors of binary factors can be estimated from the ordinal observations, facilitating dimensionality reduction and visualisation. We term our model Cumulative RBM (
)^{1}^{1}1The term ’cumulative’ is to be consistent with the statistical literature when referring to the ordinal treatment in [24]..This new model behaves differently from standard Gaussian RBMs since utilities are never observed in full. Rather, when an ordinal level of an input variable is observed, it poses an interval constraint over the corresponding utility. The distribution over the utilities now becomes a truncated multivariate Gaussian. This also has another consequence during learning: While in standard RBMs we need to sample for the freephase only (e.g., see [13]), now we also need to sample for the clampedphase.
As a result, we introduce a double persistent contrastive divergence (PCD) learning procedure, as opposed to the single PCD in
[37].The second contribution is in advancing these ordinal RBMs from modelling i.i.d. vectors to modelling matrices of correlated entries. These ordinal matrices are popular in multiusergenerated assessments: Each user would typically judge a number of items producing a userspecific data vector where intravector entries are inherently correlated. Since user’s choices are influenced by their peers, these intervector entries are no longer independent. The idea is borrowed from a recent work in [40] which models both the userspecific and itemspecific processes. More specifically, an ordinal entry is assumed to be jointly generated from userspecific latent factors and itemspecific latent factors. This departs significantly from the standard RBM architecture: we no longer map from a visible vector to an hidden vector but rather map from a visible matrix to two hidden matrices.
In experiments, we demonstrate that our proposed is capable of capturing the latent profile of citizens around the world. Our model is also competitive against stateoftheart collaborative filtering methods on largescale public datasets.
We start with the RBM structure for ordinal vectors in Section 2, and end with the general structure for ordinal matrices in Section 3. Section 4 presents experiments validating our ordinal RBMs in modelling citizen’s opinions worldwide and in collaborative filtering. Section 5 discusses related work, which is then followed by the conclusions.
2 Cumulative RBM for Vectorial Data
2.1 Model Definition
Denote by
the set of ordinal observations. For ease of presentation we assume for the moment that observations are homogeneous, i.e., observations are drawn from the same discrete ordered category set
where denotes the order in some sense. We further assume that each ordinal is solely generated from an underlying latent utility as follows [24](1) 
where are threshold parameters. In words, we choose an ordered category on the basis of the interval to which the underlying utility belongs.
The utilities are connected with a set of hidden binary factors so that the two layers of form a undirectional bipartite graph of Restricted Boltzmann Machines (RBMs) [36, 9, 14]. Binary factors can be considered as the hidden features that govern the generation of the observed ordinal data. Thus the generative story is: we start from the binary factors to generate utilities, which, in turn, generate ordinal observations. See, for example, Fig. 1 for a graphical representation of the model.
Let be the model potential function, which can be factorised as a result of the bipartite structure as follows
where and
are local potential functions. The model joint distribution is defined as
(2) 
where is the normalising constant.
We assume the utility layer and the binary factor layer form a Gaussian RBM^{2}^{2}2This is for convenience only. In fact, we can replace Gaussian by any continuous distribution in the exponential family. [14]. This translates into the local potential functions as follows
The ordinal assumption in Eq. (1) introduces hard constraints that we do not see in standard Gaussian RBMs. Whenever an ordered category is observed, the corresponding utility is automatically truncated, i.e., , where is the new domain of defined by as in Eq. (1). In particular, the utility is truncated from above if the ordinal level is the lowest, from below if the level is the largest, and from both sides otherwise. For example, the conditional distribution of the latent utility is a truncated Gaussian
(4) 
where is the normal density distribution of mean and standard deviation . The mean is computed as
(5) 
As a generative model, we can estimate the probability that an ordinal level is being generated from hidden factors
as follows(6) 
where , and
is the cumulative distribution function of the Gaussian. Given this property, we term our model by Cumulative Restricted Boltzmann Machine (
).Finally, the thresholds are parameterised so that the lowest threshold is fixed to a constant and the higher thresholds are spaced as with free parameter for .
2.2 Factor Posteriors
Often we are interested in the posterior of factors as it can be considered as a summary of the data . The nice thing is that it is now numerical and can be used for other tasks such as clustering, visualisation and prediction.
Like standard RBMs, the factor posteriors given the utilities are conditionally independent and assume the form of logistic units
(7) 
However, since the utilities are themselves hidden, the posteriors given only the ordinal observations are not independent:
(8) 
where and is the domain of the utility constrained by (see Eq. (1
)). Here we describe two approximation methods, namely Markov chain Monte Carlo (MCMC) and variational method (meanfield).
Mcmc.
Variational method.
We make the approximation
Minimising the KullbackLeibler divergence between
and its approximation leads the following recursive update(9)  
(10) 
where is the update index of the recursion, is the mean of utility with respect to , is the normalising constant, and . Finally, we obtain .
2.3 Prediction
An important task is prediction of the ordinal level of an unseen variable given the other seen variables, where we need to estimate the following predictive distribution
(11) 
Unfortunately, now are coupled due to the integration over making the evaluation intractable, and thus approximation is needed.
For simplicity, we assume that the seen data is informative enough so that . Thus we can rewrite Eq. (11) as
Now we make further approximations to deal with the exponential sum over .
Mcmc.
Given the sampling from described in Section 2.2, we obtain
where is the sample size, and is computed using Eq. (6).
Variational method.
2.4 Stochastic Gradient Learning with Persistent Markov Chains
Learning is based on maximising the data loglikelihood
where is defined in Eq. (2) and . Note that includes as a special case when the domain is the whole real space .
Recall that the model belongs to the exponential family in that we can rewrite the potential function as
where is a sufficient statistic, and is its associated parameter. Now the gradient of the loglikelihood has the standard form of difference of expected sufficient statistics (ESS)
where is a truncated Gaussian RBM and is the standard Gaussian RBM.
Put in common RBMterms, there are two learning phases: the clamped phase in which we estimate the ESS w.r.t. the empirical distribution , and the free phase in which we compute the ESS w.r.t. model distribution .
2.4.1 Persistent Markov Chains
The literature offers efficient stochastic gradient procedures to learn parameters, in which the method of [42] and its variants – the Contrastive Divergence of [13] and its persistent version of [37] – are highly effective in largescale settings. The strategy is to update parameters after short Markov chains. Typically only the free phase requires the MCMC approximation. In our setting, on the other hand, both the clamped phase and the free phase require approximation.
Since it is possible to integrate over utilities when the binary factors are known, it is tempting so sample only the binary factors in the RaoBlackwellisation fashion. However, here we take the advantage of the bipartite structure of the underlying RBM: the layerwise sampling is efficient and much simpler. Once the hidden factor samples are obtained, we integrate over utilities for better numerical stability. The ESSes are the averaged over all factor samples.
For the clamped phase, we maintain one Markov chain per data instance. For memory efficiency, only the binary factor samples are stored between update steps. For the free phase, there are two strategies:

Contrastive chains: one short chain is needed per data instance, but initialised from the clamped chain. That is, we discard those chains after each update.

Persistent chains: freephase chains are maintained during the course of learning, independent of the clampphase chains. If every data instance has the same dimensions (which they do not, in the case of missing data), we need to maintain a moderate number of chains (e.g., ). Otherwise, we need one chain per data instance.
At each step, we collect a small number of samples and estimate the approximate distributions and . The parameters are updated according to the stochastic gradient ascent rule
where is the learning rate.
2.4.2 Learning Thresholds
Thresholds appear only in the computation of as they define the utility domain . Let be the upper boundary of , and the lower boundary. The gradient of the loglikelihood w.r.t. boundaries reads
Recall from Section 2.1 that the boundaries and are the lowerthreshold and the upperthreshold , respectively, where
. Using the chain rule, we would derive the derivatives w.r.t. to
.2.5 Handling Heterogeneous Data
We now consider the case where ordinal variables do not share the same ordinal scales, that is, we have a separate ordered set for each variable . This requires only slight change from the homogeneous case, e.g., by learning separate set of thresholds for each variable.
3 for Matrix Data
Often the data has the matrix form, i.e., a list of column vectors and we often assume columns as independent. However, this assumption is too strong in many applications. For example, in collaborative filtering where each user plays the role of a column, and each item the role of a row, a user’s choice can be influenced by other users’ choices (e.g., due to the popularity of a particular item), then columns are correlated. Second, it is also natural to switch the roles of the users and items and this clearly destroys the i.i.d assumption over the columns.
Thus, it is more precise to assume that an observation is jointly generated by both the rowwise and columnwise processes [40]. In particular, let be the index of the data instance, each observation is generated from an utility . Each data instance (column) is represented by a vector of binary hidden factors and each item (row) is represented by a vector of binary hidden factors . Since our data matrix is usually incomplete, let us denote by the incidence matrix where if the cell is observed, and otherwise. There is a single model for the whole incomplete data matrix. Every observed entry is connected with two sets of hidden factors and . Consequently, there are binary factor units in the entire model.
Let denote all latent variables and all visible ordinal variables. The matrixvariate model distribution has the usual form
where is the normalising constant and is the product of all local potentials. More specifically,
where are the same as those defined in Eq. (3), respectively, and
The ordinal model is similar to that defined in Eq. (1) except for the thresholds, which are now functions of both the data instance and the item, that is and for .
3.1 Model Properties
It is easy to see that conditioned on the utilities, the posteriors of the binary factors are still factorisable. Likewise, given the factors, the utilities are univariate Gaussian
where is the domain defined by the thresholds at the level , and the mean structure is
(12) 
Previous inference tricks can be reused by noting that for each column (i.e., data instance), we still enjoy the Gaussian RBM when conditioned on other columns. The same holds for rows (i.e., items).
3.2 Stochastic Learning with Structured MeanFields
Although it is possible to explore the space of the whole model using Gibbs sampling and use the short MCMC chains as before, here we resort to structured meanfield methods to exploit the modularity in the model structure. The general idea is to alternate between the columnwise and the rowwise conditional processes:

In the columnwise process, we estimate itemspecific factor posteriors , where and use them as if the itemspecific factors are given. For example, the mean structure in Eq. (12) now has the following form
which is essentially the mean structure in Eq. (5) when is absorbed into . Conditioned on the estimated posteriors, the data likelihood is now factorisable , where denotes the observations of the th data instance.

Similarly, in the rowwise process we estimate dataspecific posteriors , where and use them as if the dataspecific factors are given. The data likelihood has the form , where denotes the observations of the th item.
At each step, we then improve the conditional data likelihood using the gradient technique described in Section 2.4, e.g., by running through the whole data once.
3.2.1 Online Estimation of Posteriors
The structured meanfields technique requires the estimation of the factor posteriors. To reduce computation, we propose to treat the trajectory of the factor posteriors during learning as a stochastic process. This suggests a simple smoothing method, e.g., at step :
where is the smoothing factor, and is a utility sample in the clamped phase. This effectively imposes an exponential decay to previous samples. The estimation of would be of interest in its own right, but we would empirically set and do not pursue the issue further.
4 Experiments
In this section, we demonstrate how can be useful in realworld data analysis tasks. To monitor learning progress, we estimate the data pseudolikelihood . For simplicity, we treat as if it is not in and replace by . This enables us to use the same predictive methods in Section 2.3. See Fig. 3(a) for an example of the learning curves. To sample from the truncated Gaussian, we employ methods described in [30], which is more efficient than standard rejection sampling techniques. Mapping parameters are initialised randomly, bias paramters are from zeros, and thresholds are spaced evenly at the begining.
4.1 Global Attitude Analysis: Latent Profile Discovery
In this experiments we validate the capacity to discover meaningful latent profiles from people’s opinions about their life and the social/political conditions in their country and around the world. We use the public worldwide survey by PewResearch Centre^{3}^{3}3http://pewresearch.org/ in 2008 which interviewed people from countries. After reprocessing, we keep ordinal responses per respondent. Example questions are: “(Q1) [..] how would you describe your day today—has it been a typical day, a particularly good day, or a particularly bad day?”, “(Q5) […] over the next 12 months do you expect the economic situation in our country to improve a lot, improve a little, remain the same, worsen a little or worsen a lot?”.
The data is heterogeneous since question types are different (see Section 2.5). For this we use a vectorbased with hidden units. After model fitting, we obtain a posterior vector , which is then used as the representation of the respondent’s latent profile. For visualisation, we project this vector onto the 2D plane using a localitypreserving dimensionality reduction method known as tSNE^{4}^{4}4Note that the tSNE does not do clustering, it tries only to map from the input to the 2D so that local properties of the data in preserved. [41]. The opinions of citizens of countries are depicted in Fig. 2. This clearly reveals how cultures (e.g., Islamic and Chinese) and nations (e.g., the US, China, Latin America) see the world.
4.2 Collaborative Filtering: Matrix Completion
We verify our models on three public rating datasets: MovieLens^{5}^{5}5http://www.grouplens.org/node/12 – containing million ratings by thousand users on nearly thousand movies; Dating^{6}^{6}6http://www.occamslab.com/petricek/data/ – consisting of million ratings by thousand users on nearly thousand profiles; and Netflix^{7}^{7}7http://netflixprize.com/ – millions ratings by thousand users on nearly thousand movies. The Dating ratings are on the point scale and the other two are on the star scale. We then transform the Dating ratings to the point scale for uniformity. For each data we remove those users with less than ratings, of which are used for tuning and stopping criterion, for testing and the rest for training. For MovieLens and Netflix, we ensure that rating timestamps are ordered from training, to validation to testing. For the Dating dataset, the selection is at random.
For comparison, we implement stateoftheart methods in the field, including: Matrix Factorisation (MF) with Gaussian assumption [34], MF with cumulative ordinal assumption [16] (without itemitem neighbourhood), and RBM with multinomial assumption [35].For prediction in the CRBM, we employ the variational method (Section 11). The training and testing protocols are the same for all methods: Training stops where there is no improvement on the likelihood of the validation data. Two popular performance metrics are reported on the test data: the rootmean square error (RMSE), the mean absolute error (MAE). Prediction for ordinal MF and RBMs is a numerical mean in the case of RMSE: , and an MAP estimation in the case of MAE: .
Fig. 3(a) depicts the learning curve of the vectorbased and matrixbased s, and Fig. 3(b) shows their predictive performance on test datasets. Clearly, the effect of matrix treatment is significant. Tables 1,2,3 report the performances of all methods on the three datasets. The (matrix) are often comparable with the best rivals on the RMSE scores and are competitive against all others on the MAE.
(a) Monitoring pseudolikelihood in training  (b) RMSE on test data 
RMSE  MAE  RMSE  MAE  RMSE  MAE  

Gaussian Matrix Fac.  0.914  0.720  0.911  0.719  0.908  0.716 
Ordinal Matrix Fac.  0.904  0.682  0.902  0.682  0.902  0.680 
Multinomial RBM  0.928  0.711  0.926  0.707  0.928  0.708 
Matrix Cumul. RBM  0.904  0.666  0.904  0.662  0.906  0.664 
RMSE  MAE  RMSE  MAE  RMSE  MAE  

Gaussian Matrix Fac.  0.852  0.596  0.848  0.592  0.840  0.586 
Ordinal Matrix Fac.  0.857  0.511  0.854  0.507  0.849  0.502 
Multinomial RBM  0.815  0.483  0.794  0.470  0.787  0.463 
Matrix Cumul. RBM  0.815  0.475  0.799  0.461  0.794  0.458 
RMSE  MAE  RMSE  MAE  

Gaussian Matrix Fac.  0.890  0.689  0.888  0.688 
Ordinal Matrix Fac.  0.904  0.658  0.902  0.657 
Multinomial RBM  0.894  0.659  0.887  0.650 
Matrix Cumul. RBM  0.893  0.641  0.892  0.640 
5 Related Work
This work partly belongs to the thread of research that extends RBMs for a variety of data types, including categories [35], counts [10, 33, 32], bounded variables [19] and a mixture of these types [38]. Gaussian RBMs have been only used for continuous variables [13, 25] – thus our use for ordinal variables is novel. There has also been recent work extending Gaussian RBMs to better model highly correlated input variables [28, 8]. For ordinal data, to the best of our knowledge, the first RBMbased work is [40], which also contains a treatment of matrixwise data. However, their work indeed models multinomial data with knowledge of orders rather than modelling the ordinal nature directly. The result is that it is overparameterised but less efficient and does not offer any underlying generative mechanism for ordinal data.
Ordinal data has been well investigated in statistical sciences, especially quantitative social studies, often under the name of ordinal regression, which refers to single ordinal output given a set of input covariates. The most popular method is by [24] which examines the levelwise cumulative distributions. Another wellknown treatment is the sequential approach, also known as continuation ratio [23], in which the ordinal generation process is considered stepwise, starting from the lowest level until the best level is chosen. For reviews of recent development, we refer to [22]
. In machine learning, this has attracted a moderate attention in the past decade
[12, 6, 7, 3], adding machine learning flavours (e.g., largemargins) to existing statistical methods.Multivariate ordinal variables have also been studied for several decades [1]
. The most common theme is the assumption of the latent multivariate normal distribution that generates the ordinal observations, often referred to as
multivariate probit models [5, 11, 17, 27, 15, 4]. The main problem with this setting is that it is only feasible for problems with small dimensions. Our treatment using RBMs offer a solution for largescale settings by transferring the loworder interactions among the Gaussian variables onto higherorder interactions through the hidden binary layer. Not only this offers much faster inference, it also enables automatic discovery of latent aspects in the data.For matrix data, the most wellknown method is perhaps matrix factorisation [21, 29, 34]. However, this method assumes that the data is normally distributed, which does not meet the ordinal characteristics well. Recent research has attempted to address this issue [26, 16, 39]. In particular, [26, 16] adapt cumulative models of [24], and [39] tailors the sequential models of [23] for task.
6 Conclusion
We have presented , a novel probabilistic model to handle vectorvariate and matrixvariate ordinal data. The model is based on Gaussian restricted Boltzmann machines and we present the model architecture, learning and inference procedures. We show that the model is useful in profiling opinions of people across cultures and nations. The model is also competitive against stateofart methods in collaborative filtering using largescale public datasets. Thus our work enriches the RBMs, and extends their use on multivariate ordinal data in diverse applications.
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