Abstract:
A recommender system may be used to predict a user behavior that a user will give in relation to an item. In an embodiment such predictions are used to enable items to be recommended to users. For example, products may be recommended to customers, potential friends may be recommended to users of a social networking tool, organizations may be recommended to automated users or other items may be recommended to users. In an embodiment a memory stores a data structure specifying a bi-linear collaborative filtering model of user behaviors. In the embodiment an automated inference process may be applied to the data structure in order to predict a user behavior given information about a user and information about an item. For example, the user information comprises user features as well as a unique user identifier.

Description:
BACKGROUND 
       [0001]    Recommender systems are currently used by online providers of goods and services to select products and/or services that might be of interest to customers or other users. For example, a customer who has previously bought books and DVDs from an online provider may access a web site of that provider and be presented with information about two or three other products that the customer might be interested in purchasing. The recommender system can direct the customer towards useful other goods and services in this way. This aids both the user or customer as well as the online provider. The customer does not need to spend time searching for other goods and services and may be directed to goods and services that might not otherwise have been found. The online provider gains increased sales as well as improved customer satisfaction and likelihood of repeat business in the future. 
         [0002]    Recommender systems may be used in any situation in which it is required to select items for specified users. The users may be human users or automated users. The items may be goods, services, advertisements, people, other users, companies, institutions, or other entities. For example, a social networking site may use a recommender system to recommend users to other users. A web search engine may use a recommender system to recommend products and services to a user. An online provider of movies may use a recommender system to recommend movies to users. 
         [0003]    Some previous recommender systems have used a content-based approach whereby descriptions of both the user and the item are used. For example, for a user the descriptions may comprise feature vectors storing user details such as a user&#39;s age, sex, native language and the like. For an item, the descriptions may comprise feature vectors storing item details such as price, author, manufacturer and the like. 
         [0004]    Other previous recommender systems have used collaborative filtering approaches. In this case, abstract identifications of users and items are used (such as product codes and customer numbers). Ratings are observed for user-item pairs and used to form a matrix of such ratings for every user-item pair. The ratings indicate how useful an item is to a particular user. The ratings may be explicit, for example, where users are asked to provide the ratings. The ratings may be implicit, for example, they may be inferred from user behavior such as purchases or click data. Recommender systems using collaborative filtering approaches may suffer from a “cold start” problem whereby performance is poor early on when the matrix of ratings is under-populated. 
         [0005]    There is a desire to improve the relevance of results produced by recommender systems and to achieve this in a manner that reduces resource requirements (such as processor and memory requirements). There is also a need to made recommendations in real-time with respect to queries. That is, if a user makes a search query to find a particular product, service, other user, or item, the recommender system is desired to provide a recommendation in time for that recommendation to be provided together with the search results. There is also a desire to enable a recommendation system to be updated on-the-fly such that user feedback about recommended items is taken into account as soon as possible. Where large numbers of users and items are involved (which is normal for most recommender system applications, especially those using content-based approaches) these problems are particularly acute. For example, a recommender system may be required to be scalable to applications involving hundreds of thousands of users and billions of ratings. 
         [0006]    The embodiments described below are not limited to implementations which solve any or all of the problems mentioned above. 
       SUMMARY 
       [0007]    The following presents a simplified summary of the disclosure in order to provide a basic understanding to the reader. This summary is not an extensive overview of the disclosure and it does not identify key/critical elements of the invention or delineate the scope of the invention. Its sole purpose is to present some concepts disclosed herein in a simplified form as a prelude to the more detailed description that is presented later. 
         [0008]    A recommender system may be used to predict a user behavior that a user will make in relation to an item. In an embodiment such predictions are used to enable items to be recommended to users. For example, products may be recommended to customers, potential friends may be recommended to users of a social networking tool, organizations may be recommended to automated users or other items may be recommended to users. In an embodiment a memory stores a data structure specifying a bi-linear collaborative filtering model of user behavior. In the embodiment an automated inference process may be applied to the data structure in order to predict a user behavior given information about a user and information about an item. For example, the user information comprises user features as well as a unique user identifier. In some embodiments the data structure comprises a factor graph and the inference process comprises carrying out message passing over the factor graph using an assumed density filtering process. 
         [0009]    Many of the attendant features will be more readily appreciated as the same becomes better understood by reference to the following detailed description considered in connection with the accompanying drawings. 
     
    
     
       DESCRIPTION OF THE DRAWINGS 
         [0010]    The present description will be better understood from the following detailed description read in light of the accompanying drawings, wherein: 
           [0011]      FIG. 1  is a schematic diagram of an example recommender system; 
           [0012]      FIG. 2  is a schematic diagram of another example recommender system; 
           [0013]      FIG. 3  is a schematic diagram of another example recommender system; 
           [0014]      FIG. 4  is a block diagram of an example method of training a recommender system; 
           [0015]      FIG. 5  gives more detail of an example of part of the method of  FIG. 4 ; 
           [0016]      FIG. 6  is a block diagram of an example method of operating a recommender system using parallel processing; 
           [0017]      FIG. 7  is a block diagram of another example method of operating a recommender system; 
           [0018]      FIG. 8  is an example factor graph; 
           [0019]      FIG. 9  is an example factor graph for use in predicting click events; 
           [0020]      FIG. 10  shows two example factor graphs; 
           [0021]      FIGS. 11 and 12  are tables of message update equations for exact factor nodes; 
           [0022]      FIG. 13  is a table of message update equations for order factor nodes; 
           [0023]      FIG. 14  illustrates an exemplary computing-based device in which embodiments of a recommender system may be implemented. 
       
    
    
       [0024]    Like reference numerals are used to designate like parts in the accompanying drawings. 
       DETAILED DESCRIPTION 
       [0025]    The detailed description provided below in connection with the appended drawings is intended as a description of the present examples and is not intended to represent the only forms in which the present example may be constructed or utilized. The description sets forth the functions of the example and the sequence of steps for constructing and operating the example. However, the same or equivalent functions and sequences may be accomplished by different examples. 
         [0026]    Although the present examples are described and illustrated herein as being implemented in a recommender system for recommending items to users based on previous explicit ratings each user has given to certain items, the system described is provided as an example and not a limitation. As those skilled in the art will appreciate, the present examples are suitable for application in a variety of different types of recommender systems where the ratings may be implicit, or where any information indicating how useful a user found a certain item may be used. 
         [0027]    Although the present examples are described and illustrated herein as being for predicting a rating that a user gives to an item the examples are also suitable for predicting any user behavior made by a user in response to an item. For example, a click event, a product purchase, a user interface input event, or other action made by a user. 
         [0028]      FIG. 1  is a schematic diagram of an example recommender system  101 . The recommender system may be implemented using a general purpose computer or any suitable processor. It is arranged to receive a user description  100  which may comprise only a user identifier (such as a customer number or code), or, in some embodiments, a user identifier as well as one or more features describing the user (such as the user&#39;s age, sex, native language, or other user data). The recommender system may optionally have access to a user database  104  which stores user descriptions. 
         [0029]    The recommender system has access to an item database  105  which stores item descriptions. In some embodiments an item description is only an item identifier such as a product code or other number. In other embodiments an item description comprises an item identifier as well as one or more features describing the item (such as price, manufacturer, or other item data). 
         [0030]    The recommender system comprises a model  108  stored in memory. Examples of this model are described below. The model is a probabilistic model which enables a latent rating to be predicted given a user description and an item description. A latent rating is an unobserved variable related to a rating that a user is predicted to give an item. Given a particular user description  100  the recommender system  101  is able to use the model  108  to generate predicted latent ratings  102  for each of the item descriptions in the item database  105 . Using these predicted latent ratings  102  the recommender system is arranged to select one or more of the item descriptions and recommend the corresponding items  103  to the user. The selection is made on the basis of the predicted latent ratings  102  in any suitable manner. For example, the items with the top five predicted latent ratings are selected and presented as a ranked list. 
         [0031]    User behavior is observed relating to the recommended items. For example, if the user makes a purchase of one of the recommended items or if a user clicks on a link to one of the recommended items. The observed user behavior  106  may be input to the recommender system  101  as historical data  107 . Other historical data  107  may also be input comprising observed user description, item description and rating triples. The recommender system  101  comprises a machine learning system which trains the model  108  using the historical data  107 . 
         [0032]    In some embodiments where the user descriptions and the item descriptions comprise only identifiers, the model is a collaborative filtering model. In other embodiments where the user descriptions and the item descriptions comprise identifiers and features, the model is a combined collaborative filtering model and content-based approach model. 
         [0033]    In embodiments where the model is a combined collaborative filtering model and content-based approach model it is possible to produce improved predictions for users new to the system (due to their description in terms of features, such as sex, age, job) as compared with purely collaborative filtering based approaches. In addition, it is possible to make accurate, personalized, predictions for longer term users because user IDs are taken into account by the model. 
         [0034]    As illustrated in  FIG. 2  the recommender system may be arranged to produce predicted ranks  200 . The predicted ranks may be obtained from the predicted latent ratings by using user-specific thresholds. If a predicted latent rating falls between two specified thresholds it becomes a particular rank. The specified thresholds may themselves be learnt by the recommender system. Thus in some embodiments, the recommender system is able to learn a user-specific rating scale and this is described in more detail below. 
         [0035]    As illustrated in  FIG. 3  the recommender system may be arranged to produce predicted click events  300 . In this case, the recommender system, given a user description and an item description, predicts whether the user will click on the item if it is presented. These predicted click events  300  may be used to select items to recommend  103  or may be used for any other suitable purpose. 
         [0036]    The model  108  stored at the recommender system may be a probabilistic, bi-linear model. For example, the model combines two linear models, one of an item and one of a user. For example, each user is represented by a set of true/false features such as “age=20”, “occupation=programmer”, “sex=male” and a unique identifier (ID). For example, each item is represented by a set of features such as “genre=War”, “date=1 st  Dec” and “ID=23987” in the case that the items are movies. In the following description, we will denote an index over features by j. Each feature is mapped to a vector of floating-point numbers referred to herein as a “trait vector” as each element of the trait vector corresponds to a particular trait of the feature. The ith trait of feature j is denoted v ij  in the following. Any suitable data structure may be used to store these vectors such as an array, list, or a hash table. 
         [0037]    For each user it is possible to calculate a total trait vector s as a weighted sum of all user feature trait vectors. The weights, x, depend on the user in question. Similarly, for each item it is possible to calculate a single trait vector t as a weighted sum of all item feature trait vectors with the weights, y, depending on the item in question. In some embodiments sparsely active binary features are used (such as user “ID=1234” or “age&gt;18” and the total user trait vector may then be calculated by summing the user trait vectors for each of the features active for the particular user (corresponding to the non-zero elements of x). 
         [0038]    In other words, there is a linear model for each item trait vector given by:: 
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         [0000]    and a linear model for each user trait vector given by: 
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         [0000]    where u ij  and v ij  are the latent feature trait values to be learnt as described below and x is a user feature vector and y is an item feature vector. 
         [0039]    The combination rule for combining the two linear models may comprise an inner product of the latent item traits and latent user traits:
       latent rating, r, is proportional to S T t . The inner product is comprised of the sum over the pairwise products of the components.       
 
         [0041]    The latent rating r is also an unobserved variable. The latent rating a particular user gives to a particular item may be modeled as the inner product of the corresponding user and item latent trait vectors. 
         [0042]    Thus the model at the recommender system may be of a latent rating.  FIG. 4  is a block diagram of a method at the recommender system of training the model. It comprises accessing  400  the model  108 , which may be a model of a latent rating, the model having variables which may be latent user traits and latent item traits. The recommender system is arranged to maintain probability distributions describing belief about the variables of the model (block  401 ). Any suitable probability distribution type may be used such as a Gaussian distribution. When information is received about an observed user, item and associated rank (block  402 ) this information is used to update the probability distributions (block  403 ). This update process may be repeated for all available historical data. 
         [0043]    Learning a probability distribution for each model variable is a complex and technically difficult task requiring significant memory resources. In an embodiment, assumed-density filtering is used to achieve this learning process and particular data structures, referred to as factor graphs, are formed in memory. Ratings may be observed one at a time and the observed information is incorporated into the model before moving on to the next rating (as explained above with reference to  FIG. 4 ). In this way memory overhead may be reduced and the recommender system is able to update immediately after each rating without the need to revisit old data. 
         [0044]    In an embodiment the update process is carried out by arranging the recommender system to build a factor graph in memory (block  500 ) for each observed rating. Some of the nodes of the factor graph are instantiated with the observed user, item and rank information and message passing is carried out along edges of the factor graph using an approximate sum-product algorithm (block  501 ). Approximations may be made using Expectation Propagation (EP) and Variational Message Passing (VMP). In this way, statistics describing the probability distributions are updated (block  403  of  FIG. 4 ). 
         [0045]    Once the model is trained it is possible to carry out prediction for a new pair of user features x and item features y. In this way a predicted latent rating is obtained for a specified user and item pair. This may be repeated to obtain predicted latent ratings for many items given a particular user. Using the predicted latent ratings items may then be selected for recommending to the user. 
         [0046]    The method described with respect to  FIG. 4  may also be used to update the model on the fly as new ratings are made, without having to revisit previous data (in contrast with some previous recommender systems). Because it is not necessary to revisit previous data the recommender system is very efficient. As a result, user experience may be improved as the system is able to adapt to a users taste immediately even within a single user session. In addition, in some embodiments, the recommender system is arranged to take into account the effect of a user&#39;s preferences and an item&#39;s popularity changing with time. In some embodiments, the data structures used to store and enable the model to be updated are specially designed in order to enable parallel hardware to be used. This improves scalability of the recommender system. 
         [0047]    In some embodiments parallel hardware is used by the recommender system in order to speed up operation. Any suitable parallel hardware may be used. In order to enable the parallel hardware to be used effectively the recommender system incorporates a monitor as now described with reference to  FIG. 6 . Using parallel hardware is typically not straightforward as concurrently executed instructions may conflict with one another producing unwanted or unexpected results. By carefully arranging the recommender system to form the factor graph data structures in a particular manner and by using the monitor described below it is possible to use parallel hardware. 
         [0048]    The recommender system accesses the model (block  400 ) and maintains probability distributions (block  401 ) describing belief about the variables of the model. The recommender system receives two or more parallel streams of ratings from different subsets of the items at the same time. This is illustrated in  FIG. 6  as two parallel streams of ratings  600  and  601  from two subsets of items. The received observed rating, item and user data is used to update the probability distributions (block  602 ) as described above with reference to  FIG. 4 . However each update is treated as an atomic transaction and a monitor  603  is used at the recommender system to ensure that if a single user variable is to be updated based on two different item ratings (because, for example, this user rated both items) then these are processed in series. In this way the monitor is arranged to prevent conflict in the concurrent updates to the model. In other embodiments, the parallel streams of ratings are from different subsets of users. In this case the monitor is used to ensure that if a single item variable is to be updated based on two different user ratings (because, for example, this item was rated by both users) then these are processed in series. 
         [0049]    In some embodiments the recommender system is arranged to take into account the possibility of a user&#39;s taste changing with time and/or an item&#39;s popularity changing with time. This is achieved by increasing the variance of the probability distribution associated with each latent trait at specified time intervals. For example, the time interval may be a day so that the variance is increased once per day. Any other suitable time step may be used depending on the particular application that the recommender system is being used for. For example, the variance may be increased by adding Gaussian noise to the probability distribution (which may be a Gaussian distribution).  FIG. 7  is the same as  FIG. 4  and in addition shows the step  700  at which the variance of the probability distributions may be increased. 
         [0050]    In some embodiments the recommender system is arranged to learn, for each user, a particular scale used by that user to rate items. For example, users may make ratings in the form of discrete star values, with 5 stars for excellent, 4 stars for good, 3 stars for satisfactory and so on. The rating predicted by the recommender system is a latent variable and this is converted into a star value for an item by comparing the latent variable value to some specified thresholds. The specified thresholds may be learnt for each user. For example, the specified thresholds are learnt using Expectation Propagation as described in more detail below. 
         [0051]    A particular example is now described in detail. 
         [0052]    In an example, the recommender system receives triples (x,y,l) of user descriptions x ∈           n , item descriptions y ∈           m  and ranks l ∈ {1, . . . L}. The ranks form an ordinal scale and can only be compared where i          j         l i &gt;l j . The model assumes that there exists a latent rating r ∈         which is stochastically generated as follows: 
         [0000]        p ( r|x,y,U,V,u,v ):= N ( r;  r   ( x,y,U,V,u,v ), β 2 ),   (1) 
         [0000]    where N(t; μ,σ 2 ) denotes the density at t of a Gaussian distribution with mean μ and variance σ 2 . The expected rating is given by 
         [0000]          r   ( x,y,U,V,u,v ):= x   T   U   T   Vy+x   T   u+y   T   v.    (2) 
         [0053]    The matrices U ∈           Kxn  and V ∈           Kxn  and the vectors u ∈           n  and V ∈           m  are the variables of the model which map both users and items into the latent K-dimensional trait space via s:=Ux and via t:=Vy. The expected rating is then determined as the inner product between the low dimensional user and item representations shifted by both a user specific bias x T u and an item specific bias y T v. Hence, the more aligned the vectors s and t are, the higher the expected rating r. Also, the expected rating is proportional to the length ∥s∥ of the latent user traits and to the length ∥t∥ of the latent item traits. The model may use low dimensional representations of both users and items. This enables recommender system to generalise across users and items, i.e., to learn from one user about another and from one item about another. Working with a small number K of latent traits, K&lt;&lt;m and K&lt;&lt;n, also has benefits in that the number of model variables to be stored is small. Both users and items can be described by arbitrary features. 
       Ordinal Regression Model 
       [0054]    As described, the recommender system may predict a real-valued latent rating r but observations may be ranks, l. The main difference between ratings and ranks is that the latter can only be compared but not subtracted from each other. In order to address this, latent ratings may be related to ranks via a cumulative threshold model. For example, user-specific thresholds b ∈           L−1  are used as follows: 
         [0000]      Rating level  l               {tilde over (b)}   l−1   &lt;r&lt;{tilde over (b)}   l ,   (3) 
         [0000]    where {tilde over (b)} 0 :=−∞, {tilde over (b)} L :=+∝ and p({tilde over (b)} l |b l )=N({tilde over (b)} l :b l ,γ 2 ). In other words, the latent rating axis is divided into L consecutive intervals (b l−1 , b l ) of varying length each representing the region in which the user gives the same rank to an item. Though there is an over-parameterisation in scale if x=e i —either the vector u i  or b is scaled—the varying lengths of each interval call for a threshold model. 
       Dynamics Model 
       [0055]    In order to address the issue of adapting to time-varying user preferences, item popularity and user rank models, the recommender system may arrange the latent variables U,V,u,v and b to vary with time. For example, for the threshold b a Gaussian dynamics model may be used, where p(b l   (t+1) |b l   (t))=N(b   l   (t+1) ;b l   (t) ,τ b   2 ). Note that this dynamics model is anchored at (t 0 ) where bland b l  and τ b   2  are replaced by a prior mean μ b  and variances of σ b   2 . An analogous model is used for all other latent variables. Here, superscripts (t) are used for time series indices; this should not be confused with the (t)th power. 
       Factorisation 
       [0056]    The model parameters to be learned are the variables U,V,u,v and b which determine how users and items are mapped to the K-dimensional trait space and how similarity in the trait space is mapped to a rank. Since the amount of data per user and/or per item is scarce, the recommender system maintains knowledge of the uncertainty about the unknown quantities. In some embodiments the knowledge about these parameters is stored at the recommender system in terms of factorising Gaussian probability distributions. Complete factorisation of all these parameters may then be assumed: 
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         [0057]    For each of the components of the matrices U and V and each of the components of the vectors u, v and b, the recommender system maintains a Gaussian belief. Given a stream of ranking triples (x,y,l) approximate posterior distributions for the parameters are learned using an example inference method discussed below. 
         [0058]    As mentioned above, for each observed rating, a small factor graph is formed in memory by the recommender system. More detail about the process of forming the factor graph is now given with reference to  FIG. 8 . The factor graph comprises nodes  802 ,  803 ,  804 ,  805  connected by links  801 . The nodes are either variable nodes (circles) or factor nodes (rectangles). Variable nodes represent storage locations and factor nodes represent computational units. The factor nodes read and write information to their neighboring variable nodes according to calculation rules described later. 
         [0059]    The factor graph of  FIG. 8  is for a single rating triple (x, y, l). In this example, all the variables are one dimensional Gaussian variables. The factor nodes  802 ,  803  at the top of the diagram are functions which access a database or other store to obtain belief distributions for the user traits  802  and item traits  803 . These computational units feed parameters describing the trait belief distributions into the corresponding variable nodes  804 ,  805 . 
         [0060]    A weighted sum is carried out represented by factor nodes  806  and  807  to obtain the latent user trait  808  and latent item trait  809  belief distributions. 
         [0061]    A product factor labeled * in  FIG. 8  (reference numeral  810 ) is applied to take the inner product of the latent user trait and latent item trait and obtain the latent rating  811 . Z_k is the result of the product for trait k. The variance of the belief distribution for the latent rating may be increased using the factor  813  (≈β) which can be thought of as adding noise to the distribution. Similarly, factors  812  (≈γ) add noise. 
         [0062]    The shaded box  815  of  FIG. 8  indicates the part of the factor graph which may be considered as a factor-node itself which performs its message update equations by iterating message passing within. This structure of the factor graph is particularly useful for enabling the recommender system to use parallel processing. 
         [0063]    The factor graph of  FIG. 8  is a data structure which may be physically represented in memory in a computer system arranged to provide the recommender system. It is formed by the recommender system in order to specify the model of equations (1), (2), (3) and (4) above. In  FIG. 8 , plate notation is used as a short-hand for replicating parts of the factor graph over the user variables i ∈ {1, . . . , n}, item variables j ∈ {1, . . . , m} and latent trait space dimensions k ∈ {1, . . . , K}. Inference in this model is performed using message passing along the edges of the graph using the sum-product algorithm as described in F. R. Kschischang, B. Frey and H. -A. Loeliger “Factor graphs and the sum-product algorithm”, IEEE Trans. Inform. Theory, 47(2):498-519, 2001, which is incorporated herein by reference in its entirety. 
         [0064]    The process of message passing comprises carrying out a calculation associated with a factor node (square node in  FIG. 8 ) using distribution parameters from neighbouring variable nodes and passing the results to one of the neighbouring variable nodes (circular nodes in  FIG. 8 ). The direction of passing the results (also referred to as a processing schedule) is explained in more detail now. 
         [0065]    The processing schedule is preferably divided into three phases: pre-processing, chain processing, and post-processing. An example pre-processing schedule starts at the top factor nodes ( 802 ,  803 ). Computation proceeds downward along each column until the s and t variables are reached (nodes  808  and  809 ). The post processing schedule is the reverse of the pre-processing schedule but stopping at the trait nodes  804 ,  805 . The chain processing schedule involves iterating the calculations of the messages within region  815  of  FIG. 8  until their values no longer change substantially. 
         [0066]    Each message that is passed in the processing schedules represents a non-trivial calculation and details of those calculations are given below. General update equations for use in carrying out the computations along the arrows in the message passing process are now given. Those general update equations are tailored for use with Gaussian distributions as shown. 
         [0000]    Factor Node Update with Gaussian Messages 
         [0067]    Consider the factor graph of  FIG. 10 . 
         [0068]    Suppose it is required to update the message m f→x  and the marginal p x . Then, the general update equations are as follows: 
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                       z 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   p 
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               = 
               
                 MM 
                  
                 
                   [ 
                   
                     
                       
                         m 
                         
                           f 
                           → 
                           x 
                         
                         true 
                       
                        
                       
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                         x 
                         ) 
                       
                     
                      
                     
                       
                         m 
                         
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                         ) 
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   m 
                   
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                     → 
                     x 
                   
                 
                  
                 
                   ( 
                   x 
                   ) 
                 
               
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                
               
                 
                   
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                       → 
                       f 
                     
                   
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         [0000]    where MM[.] returns the distribution in the Gaussian family with the same moments as the argument and all quantities on the right are normalized to be distributions. In the following the exponential representation of the Gaussian is used, that is, 
         [0000]        G ( x;τ,π )∝ exp(π x   2 −2 τx ) 
         [0069]    This density has the following relation to the standard density 
         [0000]    
       
         
           
             
               
                 G 
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                   ( 
                   
                     
                       x 
                       ; 
                       τ 
                     
                     , 
                     π 
                   
                   ) 
                 
               
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                       x 
                       ; 
                       
                         τ 
                         π 
                       
                     
                     , 
                     
                       1 
                       π 
                     
                   
                   ) 
                 
               
             
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              
             or 
           
         
       
       
         
           
             
               N 
                
               
                 ( 
                 
                   
                     x 
                     ; 
                     μ 
                   
                   , 
                   
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                     2 
                   
                 
                 ) 
               
             
             = 
             
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                     ; 
                     
                       μ 
                       
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                         2 
                       
                     
                   
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                 ) 
               
             
           
         
       
     
         [0070]    In the case of exact factor nodes the update equations are given in  FIGS. 11 and 12 . In the case of order factor nodes such as  813  and  814  in  FIG. 8 , the update equations are given in  FIG. 13 . 
         [0071]    In these update equations the symbol a represents weightings which in a preferred example are set to 1. Also, in the update equations v and w correspond to: 
         [0000]    
       
         
           
             
               V 
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
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                   t 
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                 Φ 
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                   t 
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                 ( 
                 
                   
                     V 
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                       ( 
                       t 
                       ) 
                     
                   
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                 ) 
               
             
           
         
       
     
         [0072]    The following approximate message equations may be used for the product factor  810 . f(s k ,t k ,z k )=identity function(z k =s k ·t k ). For the rest of this paragraph the index of the latent dimension, k, is dropped and the equations below correspond to a single latent dimension. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       m 
                       
                         * 
                         
                           → 
                           z 
                         
                       
                     
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                             z 
                             ; 
                             
                               
                                 〈 
                                 s 
                                 〉 
                               
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                                   s 
                                   2 
                                 
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                                   2 
                                 
                                 〉 
                               
                             
                             - 
                             
                               
                                 〈 
                                 
                                   s 
                                   2 
                                 
                                 〉 
                               
                                
                               
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                                   t 
                                   2 
                                 
                                 〉 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       m 
                       
                         * 
                         
                           → 
                           s 
                         
                       
                     
                     = 
                     
                       N 
                       ( 
                       
                         
                           s 
                           ; 
                           
                             
                               
                                 〈 
                                 z 
                                 〉 
                               
                                
                               
                                 〈 
                                 t 
                                 〉 
                               
                             
                             
                               t 
                               2 
                             
                           
                         
                         , 
                         
                           
                             
                               〈 
                               
                                 z 
                                 2 
                               
                               〉 
                             
                             - 
                             
                               
                                 〈 
                                 z 
                                 〉 
                               
                               2 
                             
                           
                           
                             〈 
                             
                               t 
                               2 
                             
                             〉 
                           
                         
                       
                       ) 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Here,                     denotes the mean of the marginal p(t) and          t 2            denotes the non-centred second moment of the message marginal p(t). Marginals are used for the inputs for the product factor as this update is a Variational approximation instead of an EP approximation (unlike the other factors in the model). These marginals may be obtained by multiplying all messages into the s and t variables (including the upward messages), hence the process of iterating the computation until convergence. The upward messages into the s and t variables are not initially available so in the first iteration these may be set to uniform distributions. The message for m* →t  is obtained from m* →s  by swapping the roles of s and t.          z          denotes the mean of the message from the sum factor upwards to the zk variable. 
         [0073]    In some embodiments assumed density filtering is used whereby the processing schedule is arranged such that the inference algorithm only passes messages forward in time. 
         [0074]    In the example of  FIG. 8 , in order to carry out prediction on a new pair of user x and item y, the recommender system is arranged to loop through all L possible values for the rank and compute the evidence of the full factor graph. Due to the approximate nature of the factors &gt;, &lt; and *, the sum of all evidences is not one and evidences may be re-normalised to arrive at a predictive probability, π l =p(l|,x,y,D). D represents the whole sequence of ranking triples in a training set. 
         [0075]    For a single point prediction θ the numbers π l  may be used to minimise the expectation of a given cost function. For example, for the mean-squared error the following expected loss L MSE (θ, π)=Σπ i (θ−i) 2  may be minimised for θ* MSE (π)=Σiπ i . 
       Parallel Inference 
       [0076]    The message passing process may be parallelised by exploiting that the incoming messages from a variable to a factor, m i→f , are ideally computed by dividing a cache of p(u i ) by the message m f→i . Hence, as long as both the cache p(u i ) and the incoming message m f→i  are updated in one atomic step, computations based on various messages from variables u i  to m i→f  can be parallelised. Thus, all the message exchanges within box  815  in  FIG. 8  may be performed in parallel over all triples provided that the updates of the messages and cached marginals over all elements in U,V,u,v and b are performed with a read-write lock. This can lead to a nearly linear performance increase over the number of cores in modern computers. 
         [0077]      FIG. 9  shows part of another factor graph which may be formed by the recommender system in order to predict click events. The factor graph of  FIG. 8  is used up to node  811  which is the same as node  900  of  FIG. 9 . Noise is added to the belief distribution for the latent rating as indicated by factor node  901 . The results are stored at variable node  902 . A factor node  903  is used to introduce a test as to whether the latent rating is positive. If so, a click event is predicted and the result stored at variable node  904 . 
         [0078]      FIG. 14  illustrates various components of an exemplary computing-based device  1400  which may be implemented as any form of a computing and/or electronic device, and in which embodiments of a recommender system may be implemented. 
         [0079]    The computing-based device  1400  comprises one or more inputs  1406  which are of any suitable type for receiving media content, Internet Protocol (IP) input, and including observed ratings, information about users and information about items. The device also comprises communication interface  1407  to enable the recommender system to access and communicate with other entities such as databases, search engines, web servers and the like. 
         [0080]    Computing-based device  1400  also comprises one or more processors  1401  which may be microprocessors, controllers or any other suitable type of processors for processing computing executable instructions to control the operation of the device in order to recommend items to users. Platform software comprising an operating system  1404  or any other suitable platform software may be provided at the computing-based device to enable application software  1403  to be executed on the device. 
         [0081]    The computer executable instructions may be provided using any computer-readable media, such as memory  1402 . The memory is of any suitable type such as random access memory (RAM), a disk storage device of any type such as a magnetic or optical storage device, a hard disk drive, or a CD, DVD or other disc drive. Flash memory, EPROM or EEPROM may also be used. 
         [0082]    An output is also provided such as an audio and/or video output to a display system integral with or in communication with the computing-based device. The display system may provide a graphical user interface, or other user interface of any suitable type although this is not essential. 
         [0083]    The term ‘computer’ is used herein to refer to any device with processing capability such that it can execute instructions. Those skilled in the art will realize that such processing capabilities are incorporated into many different devices and therefore the term ‘computer’ includes PCs, servers, mobile telephones, personal digital assistants and many other devices. 
         [0084]    The methods described herein may be performed by software in machine readable form on a tangible storage medium. The software can be suitable for execution on a parallel processor or a serial processor such that the method steps may be carried out in any suitable order, or simultaneously. 
         [0085]    This acknowledges that software can be a valuable, separately tradable commodity. It is intended to encompass software, which runs on or controls “dumb” or standard hardware, to carry out the desired functions. It is also intended to encompass software which “describes” or defines the configuration of hardware, such as HDL (hardware description language) software, as is used for designing silicon chips, or for configuring universal programmable chips, to carry out desired functions. 
         [0086]    Those skilled in the art will realize that storage devices utilized to store program instructions can be distributed across a network. For example, a remote computer may store an example of the process described as software. A local or terminal computer may access the remote computer and download a part or all of the software to run the program. Alternatively, the local computer may download pieces of the software as needed, or execute some software instructions at the local terminal and some at the remote computer (or computer network). Those skilled in the art will also realize that by utilizing conventional techniques known to those skilled in the art that all, or a portion of the software instructions may be carried out by a dedicated circuit, such as a DSP, programmable logic array, or the like. 
         [0087]    Any range or device value given herein may be extended or altered without losing the effect sought, as will be apparent to the skilled person. 
         [0088]    It will be understood that the benefits and advantages described above may relate to one embodiment or may relate to several embodiments. The embodiments are not limited to those that solve any or all of the stated problems or those that have any or all of the stated benefits and advantages. It will further be understood that reference to ‘an’ item refers to one or more of those items. 
         [0089]    The steps of the methods described herein may be carried out in any suitable order, or simultaneously where appropriate. Additionally, individual blocks may be deleted from any of the methods without departing from the spirit and scope of the subject matter described herein. Aspects of any of the examples described above may be combined with aspects of any of the other examples described to form further examples without losing the effect sought. 
         [0090]    The term ‘comprising’ is used herein to mean including the method blocks or elements identified, but that such blocks or elements do not comprise an exclusive list and a method or apparatus may contain additional blocks or elements. 
         [0091]    It will be understood that the above description of a preferred embodiment is given by way of example only and that various modifications may be made by those skilled in the art. The above specification, examples and data provide a complete description of the structure and use of exemplary embodiments of the invention. Although various embodiments of the invention have been described above with a certain degree of particularity, or with reference to one or more individual embodiments, those skilled in the art could make numerous alterations to the disclosed embodiments without departing from the spirit or scope of this invention.