DEIM Forum 2017 G5-3

A Gradient Descent Method for Matrix Factorization Using Latent
Dirichlet Allocation
Guan-shen FANG† , Kamei SAYAKA†† , and Satoshi FUJITA††† Information Engineering, Graduate School of Engineering, Hiroshima University,
1-3-2 Kagamiyama, Higashi-Hiroshima City Hiroshima, Japan†† Department of Information Engineering, Faculty of Engineering, Hiroshima University
1-3-2 Kagamiyama, Higashi-Hiroshima City, Hiroshima, Japan
E-mail: †{bou,s-kamei,fujita}@se.hiroshima-u.ac.jp
Abstract Recommendation of products plays a significant role in online service and social network nowadays. In
order to discover unpurchased ones that a user may like, a consideration is to predict how he evaluates them after hispurchasing according to his history feedbacks. Such feedback often includes a couple of numeric rating and textual
review for a specific purchased product. A traditional technique to model user’s rating is matrix factorization, whichuses latent factors to characterize both user’s interests and product’s properties. However, its ignorance of review
brings some drawbacks, including incorrect latent factors learnt and mediocre performance. In this paper, we aim
to combine the analysis of review into traditional matrix factorization and improve the learning of latent factors.
The experimental results show that by using proposed method in matrix factorization model, latent factors are
more accurately learnt and a more accurate prediction of rating for unpurchased products is realized. In the sceneof prediction for unpurchased item’s rating, the improvement in term of Mean Absolute Error is up to 3.756%.
Key words gradient descent, recommender system, topic model, Matrix Factorization

1. Introduction
As an eﬃcient way to communicate with their users, recommender system plays a significant role in online service
and social network nowadays. In order to discover and provide users the products that they potentially be interested
and buy in future, a consideration is to predict how they assess unpurchased products. The most well-known approachis Collaborative Filtering (CF) [1]. It assumes that users
sharing similar opinions to their common products in thepast, are also likely to have similar evaluation for a certain
product in the future. Among all the CF algorithms, latent
factor models [2], [5] are the most successful ones. It characterizes both products and users by vectors of latent factors,

Figure 1 A graph characterized the actual topics in reviews of

which comprise computerized alternatives to the human cre-two users to three products on Amazon. For a review,

ated genres. In order to infer such factors by machine learn-

the size of characters illustrates the proportion of theing algorithms, it uses the ratings in users’ feedbacks, which

topic; for a user, it represents how much the user men-

typically include couples of rating and review. High corre-tions it in all reviews.

spondence between user and new product leads to a highpredicted rating ― a recommendation.

the product, but cannot further explain the reasons in which

However, recent researches pointed out that the ignorance

he rates. This is the meanning of “latent”: only using his-of textual reviews in the feedbacks is the major shortcoming

tory ratings we cannnot establish the definit correspondanceof traditional latent factor models [6], [7], [9]. A rating in a

between latent factors and actual topics which represent ob-

user’s feedback only represents whether he likes or dislikes

vious genres, or properties of products. Therefore, it forcestraditional models to treat each of latent factors equally in

Comparing with existing approach [6] which both model thetheir learning phase. In contrast, there is a gap with therating and topics in the same time, proposed method makeactual case that users give their ratings. Users often has

MF outperform it with an improvement up to 28.442%.

their own overall opinion (i.e. like or dislike) to each factor,

The remainder of this paper is organized as follows. Sec-but for a specifc product, they may only consider a part oftion 2. overviews related works of latent factor models. Sec-

them when they rate and write the reviews. Examples aretion 3. describes the problem that we focus and briefly re-shown in Figure 1, which represents two users and the top-

views LDA and MF. Section 4. describes the detail of pro-

ics in their reviews to three products. For product B, user

posed method. Section 5. represents the method of evalua-

A made a low rating and pointed out the bad qualities oftion and shows its results. Finally, Section 6. concludes themusic’s playing and camera; conversely, user B rated highly

paper with future work.

and his important reason is that he is a fan of the product’s maker. It seems like there is a degree of importance

2. Related Workfor each of the topics, which is often adjusted case by case.

In recent years, latent factor-based approaches are popular

Due to such gap, the latent factors learnt only by ratings

for their eﬃciencies in dealing large scale datasource. In or-

are not correct. More seriously, such incorrect latent factors

der to model users’ ratings for further prediction, they trans-

lead to mediocre performance in missing rating’s prediction

fer the approximation of user-item rating matrix into mini-

for user. Therefore, it is necessary to understand the latent

mizing the sum-squared errors between actual and predictedfactors and realize such adjustments in their learning.

ratings, and solve it using Singular Value Decomposition [4].

In this paper, in order to solve the issue mentioned above,

Salakhutdinov et al. [5] proposed Probabilistic Matrix Fac-

we aim to use analysis of reviews to enhance traditional ma-

torization model (PMF) and introduces Gaussian priors astrix factorization (MF). Our objective is to relax the con-hyperparameters to present latent factor vectors. They alsostraint that equally treats latent factors, and further improve

pointed out that maximizing the log-posterior over items

the performance of MF. We use Latent Dirichlet Allocation

and user’s latent factors with hyperparameters kept fixed(LDA) to model the topics that users summarized in theiris equivalent to minimizing the sum-of-squared-errors objec-

reviews. Such topics always represent the reasons why theytive function. Both of these approaches ignore the reviewsgave the corresponding rating. Therefore, the topic propor-

in users feedback, which is pointed out as a significant draw-

tion in a review represents its importance of degree afore-back.

mentioned in user’s consideration of rating. Rather thanIn order to improve the interpretive ability of recommen-

integrating LDA and transforming such topics into latent

dation result, eﬀorts are made in combining semantic anal-factors in existing research [9], we propose a new gradientysis of reviews or introductory essays of products. Wang

descent method via LDA. Our idea is that the error of pre-et al. [6] proposed Collaborative Topic Regression (CTR), a

diction with actual rating is composited by portions of topics,

latent factor-based model combining PMF and LDA for sci-the more important the user considers a topic and mentioned

entific articles recommendation. For each article, its topic

in review, the more error the topic “contributed”. There-

proportion is infered as a stochastic vector using its title andfore, we relate each latent factor with a topic, and individu-

abstract. Each of the elements in such vector is added aally assign the learning rate in its learning. In other words,

latent variable, to form the article’s latent vector. In their

we simply let each latent factor characterize the interest oflearning algorithm, the latent vectors for user and article

user/quality of item for one topic, and accurately update its

and its topic proportion are learnt dependently. To solve thevalue in learning process. Such method has various advan-

problem of many users have few feedbacks, namely data spar-tages: 1) one topic corresponds with one factor and interpretsity of users, Purushotham et al. [7] improved CTR with inte-it; 2) with such relation each factor is accurately learnt, andgration social network structure, i.e. follower relationship offurther provide accurate prediction of rating, and 3) compar-

users. They transform the corelation of users into social net-

ing with the approach which integrate topic model into MF,

work matrix. For such matrix is approximated by the inner

it is easy to implement and time saving in learning process.

product of user latent matrix and social factor feature matrix

In evaluation, we conducte a series of experiments using

by matrix factorization. In other words, user latent matrix

The results show that in theis shared in decomposition of social network matrix and rat-

problem of missing rating’s prediction, when using proposed

ing matrix. On the other hand, Wang et al. [8] proposed

method in optimization of objective function and update theCollaborative Deep Learning based on CTR, to improve its

latent factor vectors both for user and item, MF gains an im-

performance. Instead of LDA, they use Stacked Denoisingprovement up to 3.756% in terms of Mean Absolute Error.

Autoencoder to infer the topic proportion of scientific article

real-world actual datasets.

or semantic introduction of item. These CTR-based method

Vj ∈ RK . The elements in Ui measure the extent of interest

directly use topic proportion of item as stochastic vector in-of ui of such factors, on the other hand, the elements of Vj

stead of latent factor vector, or to combine with latent vari-measure the extent to which the item possesses those factors

ables to form such vector in rating matrix’s approximation.

with positive or negative. Therefore, the product of Ui andAnother consideration in combining latent factor model

Vj represents the interaction of ui and vj , further approxi-

and topic model is to transform the topic proportion of item

mates the corresponding rating ri,j , leading to the estimate:

into its latent factor vector. McAuley et al. [9] developed astatistical model named ‘Hidden Factors of Topics’, trying

r̂i,j = UiT Vj + µ + bi + bj

(1)

to use topics to interpret latent factors in MF. For a specificwhere bi and bj denote the observed bias of rating of user ui

item, its related reviews are defined as a single document,

and item vj respectively, and µ is the average overall rating.

which is represented by topics having the same size to latent

The objective is to learn Ui and Vj by given training set

factors. Therefore, the latent factors having been inferedincluding observed ratings, by minimizing the regularized

are normalized by introducing a parameter as the stochastic

squared error loss:∑
L = min[ci,j (ri,j − r̂i,j )2

topic proportion directly. Zhang et al. [10] further analyzed
sentiment contained in reviews in their explicit factor model.

U,V

Addition to implicit (latent) factors, they extract the nouns

i,j

(2)

+λ(∥Ui ∥ + ∥Vj ∥ + bi + bj )]2

represent properties of item, as explicit feature. After that,

2

2

2

the polarities of such explicit features are extracted from re-where λ is the parameter to controll the regularization toviews to form two explicit feature matrics for both size of

avoid over-fitting in learning, and ∥ · ∥2 denotes the L2 norm.

user and item. In their model’s learning, the user and item

ci,j is a confidence parameter of rating ri,j , which indicatedlatent factors are not only to fit the rating matrix, but also

how much we trust it. In other words, a large ci,j means thathave to fit such sentiment explicit feature matrics.

ri,j is a deliberate rating so that we trust it more; a small

Although in state-of-the-art approaches semantic analysisof review are integrated to gain better performance and rec-ci,j should be assigned to some observed rating that do notdeserve seriously treatment such as advertisings.

ommendation’s interpretive ability than traditional MF, they

In order to optimize Equation (2), gradient descent algo-focus on the “document” formed by reviews published by in-rithm [11] is used to loop through all ratings in training set.

dividual user or related with single item. In other words,

Ui and Vj are updated by modifing them to the opposite

they can not cope with the case that individual user may

direction of the gradient of error:

adjusts the importances of factors when he faces to diﬀerent
items.

Ui ← Ui + γ[ci,j (ri,j − r̂i,j )Vj − λUi ]Vj ← Vj + γ[ci,j (ri,j − r̂i,j )Ui − λVj ]3. Preliminaries(3)

where γ is the learning rate. With Ui and Vj updated alter-3. 1 Problem Definition

nately, a missing rating can be predicted following Equation

The problem we study is to accurately learn latent factor(1).

vector for user and item, to provide the prediction to missing

3. 3 Latent Dirichlet Allocation (LDA)

ratings for users by their history feedbacks. Such rating is

Topic model is the algorithm used to discover topics in aoften in scale of [a, b] (e.g. ratings of one to five stars onlarge set of documents, i.e., corpus. Such topic is a proba-

Amazon), and couples with a semantic review. Suppose webility distribution over all words. The words associate withhave I users and J items, and the rating made by user ui

a single theme make their corresonding elements bias in theto item vj is denoted as ri,j , its corresponding review is de-distribution of a topic. Therefore, topic model provides annoted by di,j respectively. We assume that ri,j must have ainterpretable dimension reduction of corpus.

di,j having been written. Therefore, a feedback is a 4-tuple

LDA [3] is a generative probabilistic topic model of a set(ui , vj , ri,j , di,j ). For user ui , Ui denotes its latent factor

of semantic documents. Its basic idea is that documents arevector, and Vj denotes the one for item vj .

represented as random mixtures over latent topics, where

3. 2 Matrix Factorization for Recommendationeach topic is characterized by a distribution over words.

Matrix Factorization [2] is a basic but successful method

Assume there are K topics in corpus, and one topic tk

to predict the missing rating in recommendation. It mapps(k ∈ [1, . . . , K]) is a probability distribution of all words.

users and items into a joint latent factor space with K di-

For each document wj in corpus, the generative process is

mensions. Accordingly, each user ui is associated with aas follows:

vector Ui ∈ R , and each item vj is associated with a vector
K•

Choose topic proportion θj of wj ∼ Dir(α)

•Table 1 The description of dataset used in experiments.

For each of the words wn :– Choose a topic assignment zj,n ∼ Multinomial(θj )datasetYelp– Choose a word wn ∼ Multinomial(βzj,n )automotive digital music♯users1114

369

The process represents the assumption how a document is♯items

5781

369

4202
3279

generated. Overall topic size K is assumed to be known and♯ratings

16014

2549

52500

tk is shared by corpus. For each document wj , a specificaverage ratings

3.81

4.33

4.216

134.8

188

234

average

topic proportion θj is provide as its representation.
The objective is to estimate the maximun likelihood of βk♯words ofand α to generate the documents of corpus. We approximate

reviewthem by using an variational EM algorithm [3] which maximizes their lower bound. Further, the parameters θj and zj,ndetermine their learning rates. Note that r̂i,j is calculated

can be updated via Gibbs sampling [12] iteratively.

following Equation (1). It follows the idea that 1) since latent factors has no definit correspondance with actual topics,

4. Proposed Method

we force each of its elements to relate with a topic; 2) theIn our method, we use LDA to uncover the latent topics

rating error ri,j − r̂i,j is composited by the portions of top-

in user’s review text. Diﬀerent with existing method [8], [9],k
ics with importances θi,j
. In other words, the important the

we propose a new gradient descent method to correlate thetopic considered in ri,j , the more its corresponding factors

topics to latent factors. Our idea is to adjust the learning

are updated. For the omitted topics in di,j since they do notrate with topics’ proportions, to unequally treat each latentproduce error, we do not update their corresponding factors

factor for a specific rating.

respectively. Another issues for us is to determine a properFirstly, each review in users’ feedbacks is taken as a sin-value for constant α. Comparing with ordinary MF whichgle document and all reviews form the corpus. With suchkonly uses a constant learning rate, since the average of θi,j
is

corpus, we independently train LDA to infer the topic dis-

1/K, in proposed method the step of updating in each loop

tribution θi,j for review di,j , which is published by user ui

reduces to 1/K overall. Thus, we assign Kγ for α, where γkto item vj . Suppose we have K topics overall, and let θi,jis the learning rate used in MF. For example, if the learningkdenote the proportion of k-th topic in θi,j . Thus θi,j
∈ [0, 1]rate γ was 0.01, for proposed method when K = 5, we set

and k ∈ 1, . . . , K. Since di,j summarizes the reasons that ui

α = 0.05.

kgives the corresponding rating ri,j , θi,jrepresents the impor-

After the optimization of each Ui and Vj , we can use themtance of topic tk in ri,j . In other words, for specific vj , theto predict the missing ratings of users following Equation (1)topics that ui does not care or considers it normal are notof MF.

mentioned in di,j . On the other hand, if ui considers tk very
important for vj , he will mention it more in di,j and adjust

5. Evaluation
In order to evaluate the performance of proposed method,ri,j intenser from his base line than other topics.
Although we can simply introduce the importance of topics

we conduct several experiments to compare with existing
methods by using real-world datasets. We aim to answer

into rating’s modeling following the estimation

the questions of 1) whether the loss (i.e. square error inr̂i,j = UiT Hi,j Vj + µ + bi + bj

Equation (2)) rapidly converge in learning process by using

where Hi,j is a K × K diagonal matrix with θi,j as its diago-

proposed method; 2) whether proposed method can improve

nal elements, for unpurchased item vj′ , Hi,j ′ is unknown and

MF to gain a better performance in the scene of rating’s

hard to predict. As the solution, we re-define the learning

prediction.
5. 1 Datasets

rate γ in Equation (3) asAs shown in Table 1, we use three datasets provided bykγ = αθi,jYelp and Amazon in our experiments. For Yelp dataset（注 1）,

where α is a constant. Therefore, for the K-dimensional la-

we focus on the feedbacks that relate with the POIs in Ari-tent factor vectors Ui and Vj , the update equation (3) of thezona, USA, and are published in 2014. For Amazon datasets

elements Uik and Vjk are re-writed asprovided by McAuley [13], we use the feedbacks of categorieskUik ← Uik + αθi,j[ci,j (ri,j − r̂i,j )Vjk − λUik ]kVjk ← Vjk + αθi,j[ci,j (ri,j − r̂i,j )Uik − λVjk ]of “automotive” and “digital music”. All datasets are filtered(4)

where for each pair of elements Uik and Vjk we individuallyby constrains of 1) users who at least have 5 feedbacks; 2)（注 1）：http://www.yelp.com/dataset_challenge

5. 3 Impact to Recommendation
In this section, we evaluate the performance of proposed
method when it is used as optimization method for latent
factor model.
5. 3. 1 Comparisons and Implementation
We select traditional MF [4] and Collaborative Topic Regression (CTR) [6] as baselines. For both of them, latent factors are optimized by the gradient descent methods following
their researches. Additionally, in order to evaluate the performance of proposed method, we use proposed method tolearn latent factors of MF instead of [11], as the improvementof traditional MF. Especially, the initial values for elementsin Ui and Vj are fixed to 0.001. For traditional MF we tried
several values for λ and finally fix it to 0.001. On the otherFigure 2 The square error in learning process on dataset.

hand, we set ci,j = 1 if ui have reviewed and rated vj , anditems that is reviewed by at least 3 users; 3) the feedbacks

ci,j = 0 otherwise. For CTR, since it analysis the topics inincluding a review that has at least 10 nouns. In the lasttextual introduction essay of item instead of the individualline of Table 1, we list the average size of words included

review, we take the reviews related to an item as a single doc-in reviews. It indicates that reviews are well summarizedument as its introduction essay. Following the parameters’

by users, and will not aﬀect the performance of inference ofsetting in their research, we set λu equals to 0.01. At last, we

topics. For each dataset, we randomly take 80% of its feed-

fix K to 5 and 10 overall to observe their performances. Also

backs as training set, and the rest as testing set. In ordernote that both for YELP and Amazon datasets the ratings

to independently train LDA, for each review in the training

is in scale of [1, 5], thus we limit the predicted rating in thisset, we extract the nouns as words, and conduct a stemming

range.
5. 3. 2 Evaluation Methodologies

to each words.

For each dataset, we form and use a common training set

5. 2 Convergence of Square Error
Firstly, we simply conduct learning processes both using

for both proposed method and existing models in various K

proposed method and the method [11] in traditional MF byvalues. In training phase, the parameters Ui and Vj are iter-using YELP dataset, with K = 5 and λ = 0.01. In each loop

atively learnt until the loss calculated by objective function

to update Ui and Vj we observe the square error. For tradi-

convergences. After that, we evaluate the performances bytional MF, we set Ui and Vj with randomly generated values

observing the accuracy of prediction of the ratings in test

following the zero mean gaussian distribution of N (0, λ ).

set. As quantification, we use mean absolute error (MAE)

For proposed method, we initially set a extremely small

and root mean square error (RMSE) which are calculated as

unique value of 0.001 for the elements in Ui and Vj . The

follows:2

reason is that such assignment makes the initially predictionof rating be the average of ratings, and minimize the eﬀect∑
M AE = N1(|r − r̂i,j |),√ i,j∑ i,j
1
RM SE = N i,j (ri,j − r̂i,j )2

of initialization. The learning rate for traditional MF is set
to 0.01, and 0.05 for proposed method since K = 5.
Figure 2 summarized the square error of traditional MF
and proposed method in learning process. It shows that comparing with existing gradient descent method [11], the con-

where N denotes the number of feedbacks in test set, and| · | denotes the absolute value. In general, RMSE is moresensitive for large prediction’s errors than MAE.
5. 4 Experimental Resultsvergence of square error of proposed method is more slower,

Table 2 shows the results of MAE and RMSE of traditional

but the step of updating is almost same. The reason is con-

MF, CTR and MF using proposed method with K equals to 5

sidered that the initial value is small and learning rate is set

and 10. The emphasised numbers represents the lowest MAE

conservatively. On the other hand, the minimum of squareand RMSE for each dataset. When K = 5, MF using pro-error is also higher than existing method. This is because

posed method shows the best performance for all datasets.

that for the topics having been seldom mentioned in reviews,

Comparing with traditional MF, our proposed method im-

their corresponding factors are also seldom updated so thatproves it from 0.743% to 3.756% in terms of MAE, 1.727% tothey remain the initial values and hard to reduce.

3.361% in RMSE. When K = 10, for dataset ‘digital music’traditional MF outperforms porposed method. It represents

Table 2 The performances of MF, CTR and proposed method in K = 10 and K = 50.
MF
dataset

MAE

RMSE

CTR
MAE

RMSE

proposed method improvement of MAE improvement of RMSE
MAE

RMSE

vs. MF

vs.CTRvs. MF

vs.CTR
21.815%K=5
YELP

1.02801.36491.3293

1.69100.99701.32213.015%

24.998%3.135%

digital music

0.88291.30270.93481.36730.84971.25893.756%

9.102%

3.361%

7.925%

automotive

0.7404

1.0712

1.027

1.37940.73491.05270.743%

28.442%

1.727%

23.684%-25.96%

K = 101.39201.73630.9634

1.29181.03%

30.79%

0.57%

digital music 0.7935 1.1435 0.8813

YELP

0.97341.22760.85281.2305-7.472%

3.228%-7.604%-0.236%

automotive

1.29300.71141.02993.382%

27.028%1.464%

20.348%0.7363

1.2992
1.0452

0.9749that for MF, our method are more aﬀected by parameter
K than existing gradient descent method [11]. Comparing
with CTR, which is a model that integrates LDA into MF to
improve the explanation of latent factor, by using our proposed method MF also shows better performances in mostof the experiments. It indicates that our proposed method
improves MF in its factor’s learning process, and enhances
it in the scene of rating’s prediction.
Combining with the experimental result in Section 5. 2,in optimization of objective function Equation (2), althoughproposed method cannot reaches the same level of loss toexisting gradient descent [11], it trains Ui and Vj more accurately.

6. Concluding Remarks
In this paper, we propose a gradient descent for the optimization of tradional latent factor model like MF. By using
LDA to model the topics in user’s review, we relate latent
factors with topics and individually determine the learning
rates via proportions of topics. The performance of optimization of proposed method is experimentally evaluated bya scene to use MF in prediction of unknown ratings of user.
The results show that proposed method provides more accurately learnt latent factors. Overall, it indicates that our
proposed method achieves the objectives to release the limination of equally treatment of latent factors, and improve
the performance of MF.
In future work, we want to further improve the speed of
optimization of proposed method. By considering the epoch
in learning process, we can further adjust the learning rate tofasten the convergence of loss. On the other hand, since therunning time of proposed method has not analyzed, we willcalculate it and make the comparison with existing method.

References[1] D. Goldberg, D. Nichols, B. M. Oki, D. Terry, “Using collaborative filtering to weave an information tapestry,” Com-

munications of the ACM, Vol. 35, No. 12, pp.61–70, 1992.
[2] Y. Koren, R. M.Bell and C.Volinsky, “Matrix factorization
techniques for recommender systems,” IEEE Computer,Vol. 42, No. 8, pp. 30–37. 2009.
[3] D. M. Blei, A. Y. Ng, M. I. Jordan, “Latent dirichlet allocation,” Journal of machine Learning research, Vol. 3, No.
Jan, pp. 993–1022, 2003.
[4] Y. Koren, “Factorization meets the neighborhood: a multifaceted collaborative filtering model,” Proceedings of the
14th ACM SIGKDD international conference on Knowledge
discovery and data mining, pp. 426–434, 2008.
[5] R. salakhutdinov, A. Mnih, “Probabilistic matrix factorization,” NIPS, Vol. 20, pp. 1–8, 2011.
[6] C. Wang, D. M. Blei, “Collaborative topic modeling forrecommending scientific articles,” Proceedings of the 17th
ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 448–456, 2011.
[7] S. Purushotham, Y. Liu, C. C. J. Kuo, “Collaborative Topic
Regression with Social Matrix Factorization for Recommendation Systems,” CoRR, Vol. abs/1206.4684, 2012.
[8] H. Wang, N. Wang, D. Y. Yeung, “Collaborative deep learning for recommender systems,” Proceedings of the 21th
ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1235–1244, 2015.
[9] J. McAuley, J. Leskovec, “Hidden factors and hidden topics:
understanding rating dimensions with review text,” Proceedings of the 7th ACM conference on Recommender systems, pp. 165–172, 2013[10] Y. Zhang, G. Lai, M. Zhang, Y. Zhang, Y. Liu, S. Ma, “Explicit factor models for explainable recommendation basedon phrase-level sentiment analysis,” Proceedings of the 37th
international ACM SIGIR conference on Research and development in information retrieval, pp. 83-92, 2014.
[11] S. Funk, “Netflix Update:
Try This at Home,”
http://sifter.org/ simon/journal/20061211.html, 2006[12] G. Heinrich, “Parameter estimation for text analysis,” University of Leipzig, Tech. Rep, 2008[13] J. McAuley, C. Targett, Q. Shi, A. van den Hengel, “Imagebased recommendations on styles and substitutes,” Proceedings of the 38th International ACM SIGIR Conference
on Research and Development in Information Retrieval, pp.
43–52, 2015