Cross-domain recommendation without shared users or items
by sharing latent vector distributions

Tomoharu Iwata
Koh Takeuchi
NTT Communication Science Laboratories

1

Abstract

pected, but well-appreciated, items through serendipity [31].

We propose a cross-domain recommendation
method for predicting the ratings of items
in diﬀerent domains, where neither users nor
items are shared across domains. The proposed method is based on matrix factorization, which learns a latent vector for eachuser and each item. Matrix factorization
techniques for a single-domain fail in the
cross-domain recommendation task because
the learned latent vectors are not aligned overdiﬀerent domains. The proposed method
assumes that latent vectors in diﬀerent domains are generated from a common Gaussian distribution with a full covariance matrix. By inferring the mean and covarianceof the common Gaussian from given crossdomain rating matrices, the latent factors
are aligned, which enables us to predict ratings in diﬀerent domains. Experiments conducted on rating datasets from a wide variety
of domains, e.g., movie, books and electronics, demonstrate that the proposed method
achieves higher performance for predicting
cross-domain ratings than existing methods.

To create this functionality, a number of cross-domain
recommendation methods have been proposed [6, 8,
14, 23]. Most of the methods assume that there areshared users or items. However, shared users anditems might be unavailable if the diﬀerent domains
have totally diﬀerent items, diﬀerent stores use different databases, and the sharing of user information
might be restricted to preserve privacy.

Introduction

Recommender systems are widely used in online stores
because they help users to find the users’ favorite items
from the huge number of items available. They usually suggest items to users in a single domain, e.g.
the items available in that store. However, if a store
recommends items from diﬀerent stores, the store can
improve its profit. Moreover, presenting items thatare not those of the store can allow users to find unexAppearing in Proceedings of the 18th International Conference on Artificial Intelligence and Statistics (AISTATS)
2015, San Diego, CA, USA. JMLR: W&CP volume 38.
Copyright 2015 by the authors.

Our task in this paper is to recommend items indiﬀerent domains, where neither users nor items areshared across the domains, as shown in Figure 1. For
this task, we propose a cross-domain recommendation
method based on matrix factorization [18, 25], which
has been successfully used for single-domain recommendation. Matrix factorization learns a latent vector
for each user and each item given a user-item rating
matrix, and predicts missing ratings using the latentvectors. However, standard matrix factorization techniques fail to predict cross-domain ratings in the absence of shared users and items, because the latentvectors are inferred individually in each domain, andthe latent factors are not aligned across diﬀerent domains.
The proposed method assumes that latent vectors in
diﬀerent domains are generated from a common Gaussian distribution with an unknown mean vector andfull covariance matrix as shown in Figure 2. By inferring the shared mean and covariance of the Gaussian
from given cross-domain rating matrices, the latent
factors are aligned across diﬀerent domains, which enables us to predict ratings in diﬀerent domains.
When two domains are related, e.g., movie and bookratings, their latent factors would be similar. Supposethat each latent factor represents a genre; for example
a high value of a factor in an item latent vector indicates the item is SF movie, and a high value of thefactor in a user latent vector indicates the user likes SF
movie. We can expect that users who like SF movie
also like SF books. By aligning latent factors, e.g.,Cross-domain recommendation without shared users or items by sharing latent vector distributionsusers

the SF movie factor and SF book factor, across diﬀerent domains, SF movies can be recommended to users
who like SF books. The mean and covariance values
help to align the latent factors. The mean value represents popularity of the latent factor. The popularityof romance movies would be similar to the popularity of romance books. The covariance value represents
correlation between two latent factors. The correlation between genres would be similar across diﬀerent
but related domains; users who like fantasy movies
like SF movies, and also users who like fantasy books
like SF books. The proposed method finds the correspondence between latent factors in diﬀerent domains
by using the mean and covariance information withshared Gaussian distributions.
The remainder of this paper is organized as follows.
Section 2 reviews related work. In Section 3, we propose a cross-domain recommendation method based on
matrix factorization with shared latent vector distributions, and present its inference procedures based on
Gibbs sampling. Section 4 demonstrates the eﬀectiveness of the proposed method with experiments on rating datasets in a wide variety of domains, e.g., movie,
books and electronics. Finally, we present concluding
remarks and a discussion of future work in Section 5.

2

Related work

Collaborative filtering is a technique to predict missinguser preferences by using observed user-item preference data. Single-domain collaborative filtering methods, such as matrix factorization [18, 25] and nearest
neighbor based methods [27], can be used for crossdomain recommendations by aggregating datasetsfrom multiple domains into a single user-item rating
matrix if there are shared users or items. However, theoverlap of users and items across diﬀerent domains issmall and single-domain collaborative filtering meth-user 1 4
user 2 5 1
2user 3

covarianceuser A
user B
user C

movie 3
movie 2
movie 1Figure 1: Two domain rating matrices without sharedusers or items. Ratings in unshaded parts can be observed, and those in shaded parts are missing. The
task in our paper is, given rating matrices without
shared users or items, to predict missing ratings inoﬀ-diagonal blocks.

meanuser 1user 2user 3

movie 3
movie 2
movie 1domain 2 domain 1

itemsbook 3
book 2book 1book 3
book 2book 1domain 1 domain 2user A3
user B
2 5user C 3

mean covariance

Figure 2: The proposed method is based on matrix
factorization, where mean and covariance structuresof latent vectors are shared across diﬀerent domains.
The first domain contains items (book1, book2, · · · )and users (user1, user2, · · · ). The second domain contains items (movie1, movie2, · · · ) and users (userA,
userB, · · · ). The task is to predict ratings in diﬀerent
domains, e.g., a rating of userA to book1, and that ofuser2 to movie1.

ods are biased [6]. To alleviate the bias, a numberof cross-domain collaborative filtering methods havebeen proposed [6, 14, 23]; these methods also assume
shared users or items.
A limited number of cross-domain methods that do
not require shared users and items have been proposed [9, 20, 30]. Multi-domain collaborative filtering [30] is based on matrix factorization, in which therelationships among user latent factors in diﬀerent domains are modeled by a matrix-variate Gaussian distribution, and latent vectors are generated from Gaussian
distributions with zero mean and diagonal covariance.
Although this method is close to the proposed method,
the proposed method can find relationships across domains by utilizing both the mean vectors and full covariance matrices for both the user and item latent vectors. A rating-matrix generative model [20] transfers
information across domains by sharing implicit clusterlevel rating patterns. A cluster-level latent factor
model [9] assumes common latent and domain-specificrating patterns in matrix tri-factorization. These existing methods try to improve the performance of predicting ratings in a target domain by using data inother domains to alleviate the data sparsity problem
in the target domain. On the other hand, our task
is to predict ratings of user-item pairs in diﬀerent do-Tomoharu Iwata, Koh Takeuchi

mains. To our knowledge, this paper is the first study
to demonstrate the prediction of ratings in diﬀerent
domains without shared users or items. In [6], it isnoted that when there is no domain overlap, the only
approach available is non-personalized collaborative
filtering, e.g. recommending the most popular items,
or predicting item ratings by the average ratings overusers.
Content-based cross-domain recommendation methods have also been proposed [3, 22]. They require auxiliary information, such as user demographics and item
content information, to calculate similarities between
users/items in diﬀerent domains. In contrast, the proposed method does not require any information otherthan the rating matrices from the domains.
The proposed method is based on Bayesian probabilistic matrix factorization [25], which has achieved highpredictive performance for single-domain recommendation. In particular, the propose method applies, ineﬀect, Bayesian probabilistic matrix factorization toa single large rating matrix constructed by combining ratings in all of the domains, where ratings among
diﬀerent domains are assumed to be missing values.
The proposed method is related to unsupervised object matching methods, which find correspondence between objects, or users and items, in diﬀerent domains
without alignment information [11, 24, 29, 17, 15]. The
proposed method does not directly find the correspondence between user and item, but finds the correspondence between latent factors by sharing mean and covariance structures.
Latent semantic matching [13] learns latent vectors individually in each domain, and then uses unsupervised
object matching methods to find the alignment of latent factors across diﬀerent domains. Because of its
two stage procedure, the learned latent vectors might
not be suitable when aligned, and the accumulationof errors in the latent vector learning stage cannot becorrected in the factor alignment stage. On the other
hand, since the proposed method simultaneously infers
latent vectors and their factor alignments in a single
framework, latent vectors are inferred so as to be optimal when their factors are aligned across multiple
domains.

3

Proposed method

3.1

Model

We assume that we are given user-item rating matrices in D domains R = (R1 , · · · , RD ), where Rd ∈RNd ×Md is the rating matrix of domain d, Nd is thenumber of users in domain d, and Md is the number of

items in domain d. The (n, m) element of Rd , rdnm ,represents the rating of user n to item m in domain d;
Rd can contain missing values. The task is to predictratings of user-item pairs from diﬀerent domains.
We assume that each user has latent vector udn ∈RK×1 , and each item has latent vector vdm ∈ RK×1 ,where K is the number of latent factors. Rating rdnm
is generated from a Gaussian distribution with mean
−1−1u⊤
, rdnm ∼ N (u⊤).
dn vdm and variance α
dn vdm , αInferring the latent vectors individually in each domain
fails to predict cross-domain ratings because latent factors are randomly aligned if the overlap is nil. We try
to align latent factors by assuming that user latent vectors in all domains are generated from a common fullcovariance Gaussian distribution, udn ∼ N (µu , Λ−1u ),and item latent vectors in all domains are as well,
vdm ∼ N (µv , Λ−1This means that latent facv ).
tors have the same mean vector and covariance matrix across diﬀerent domains. By inferring the meanand covariance from given cross-domain rating matrices, the latent factors are aligned, which enables us topredict ratings in diﬀerent domains. For the parameters of the shared Gaussian distributions, (µu , Λu ) and(µv , Λv ), conjugate Gaussian-Wishart priors are used.
In summary, the proposed method generates rating
matrices from multiple domains R according to thefollowing process,1. Draw user latent vector precision matrix
ΛU ∼ W(W0 , ν0 )2. Draw user latent vector meanµU ∼ N (µ0 , (β0 ΛU )−1 )3. Draw item latent vector precision matrixΛV ∼ W(W0 , ν0 )4. Draw item latent vector meanµV ∼ N (µ0 , (β0 ΛV )−1 )5. For each domain d = 1, · · · , D(a) For each user n = 1, · · · , Nd
i. Draw user latent vector
udn ∼ N (µU , Λ−1U )(b) For each item m = 1, · · · , Md
i. Draw item latent vector
vdn ∼ N (µV , Λ−1V )(c) For each user n = 1, · · · , Ndi. For each item m ∈ Mdn
A. Draw rating−1).
rdnm ∼ N (u⊤dn vdm , αHere, Mdn is a set of items rated by user n in domain d, and W(Λ|W0 , ν0 ) represents a Wishart distribution, where W is a K × K symmetric positive

Cross-domain recommendation without shared users or items by sharing latent vector distributions

W0

ν0

ν0
α

ΛU
µU

N

u

where

W0

ΛV

r

M D

β0

)

∑

α

udn rnm + Λv µv∑

Λdm = Λv + α

(5)

un u⊤
n,

(6)

n∈Ndm

β0

Figure 3: Graphic model representation of the proposed method.
definite matrix, and ν0 > K − 1 is the number of degrees of freedom. The mean of this distribution is νW.
Figure 3 shows the graphical model representation of
the proposed method, where the shaded and unshaded
nodes indicate observed and latent variables, respectively. The scales of ratings are assumed to be thesame over all domains. If the scales are diﬀerent, they
can be unified by normalizing ratings for each domain
in a preprocessing step.

are the mean and precision matrix of the posterior
Gaussian distribution of item latent vector vdm , respectively. Here, Ndm is the set of users who give
ratings to item m in domain d.
The shared mean and covariance matrix for user latent vectors are sampled from the following posterior
Gaussian-Wishart distribution
p(µu , Λu |U, W0 , ν0 , β0 , µ0 )= N (µu |µ0u , (β0u Λu )−1 )W(Λu |W0u , ν0u ),

(7)

where
N=

D
∑

Nd ,

µ0u =

d=1

3.2

,

n∈Ndm

µV

v

(
µdm = Λ−1dm

β0 µ0 + N ū,β0 + N(8)

Inference

The unknown parameters in the proposed method areDduser latent vectors U = ((udn )Nn=1 )d=1 , item latentDdvectors V = ((vdm )Mm=1 )d=1 , and shared Gaussian parameters (µu , Λu ) and (µv , Λv ). They are inferredby Gibbs sampling, where each parameter is sampled from its conditional distribution given the current state of all the other parameters. Because theproposed method uses conjugate priors for the parameters, the conditional distributions can be analyticallyobtained as described in [25]. The conditional probability for a user latent vector is given byp(udn |R, V, µu , Λu , α) = N (µdn , Λdn ),β0u = β0 + N,
−1
W0u
= W0−1 + N S̄u +

(9)

β0 N
(µ0 − ū)(µ0 − ū)⊤ ,β0 + N
(10)

d
1 ∑∑
udn ,
N
n=1

D

ū =

ν0u = ν0 + N,

N

(11)

d=1

d
1 ∑∑
S̄u =
(udn − ū)(udn − ū)⊤ .
N
n=1

D

(1)

N

(12)

d=1

where
(
µdn = Λ−1
dn

α

)

∑

vdm rnm + Λu µu

,

(2)

m∈Mdn

Λdn = Λu + α∑

In the same way, the shared mean and covariance matrix for item latent vectors are sampled from the posterior distribution
p(µv , Λv |V, W0 , ν0 , β0 , µ0 )

⊤
vm vm
,

(3)

m∈Mdn

(13)

where

are the mean and precision matrix of the posterior
Gaussian distribution of user latent vector udn , respectively. Similarly, the conditional probability for
an item latent vector is given byp(vdn |R, U, µv , Λv , α) = N (µdm , Λdm ),= N (µv |µ0v , (β0v Λv )−1 )W(Λv |W0v , ν0v ),

(4)

M=

D
∑

Md ,

µ0v =

d=1

β0v = β0 + M,

β0 µ0 + M v̄,β0 + Mν0v = ν0 + M,

(14)

(15)

Tomoharu Iwata, Koh Takeuchi−1W0v= W0−1 + M S̄v +β0 M(µ0 − v̄)(µ0 − v̄)⊤ ,β0 + M
(16)

d
1 ∑∑
v̄ =
vdm ,
M
m=1

D

M

(17)

d=1

d
1 ∑∑
(vdm − v̄)(vdm − v̄)⊤ .
M
m=1

D

S̄v =

M

(18)

d=1

To initialize the latent vectors we use the result of
the maximum a posteriori (MAP) estimation basedon the probabilistic matrix factorization model [21].
Then, we iterate the sampling of shared means andcovariances µu , Λu , µv , Λv by (7) and (13), user latentvectors U by (1), and item latent vectors V by (4).
The rating of user n in domain d to item m in domaind′ is estimated by averaging over the samples after aburn-in periodS
1 ∑ (s)′r̂dnd m =r ′ ,S s=1 dnd m

(19)
(s)

where S is the number of samples, and rdnd′ m is thesth sample that is sampled by the following Gaussian
distribution−1).
rdnd′ m ∼ N (u⊤dn vd′ m , α(20)

The averaging means that ratings are predicted by
the posterior distribution of the latent vectors in a
Bayesian framework, but not by the point estimates.
For all of the experiments, we used the following hyperparameters: W0 = I, ν0 = K, β0 = 1, µ0 = 0, α = 2.

4

Experiments

4.1

Datasets

We evaluated the proposed method on the following
three movie rating datasets: Movielens [12], EachMovie and Netflix [2], and Amazon review rating
datasets from the following six categories: Book, DVD,
Electronics, Kitchen, Music and Video [5]. The numbers of users, items and ratings for each dataset are
shown in Table 1.
For the movie datasets, we generated two rating matrices for each dataset with no overlap. In detail, users
and items are randomly split into two groups of equal
size, and the ratings between users and items assigned
into the same group are used for training, and those
assigned to diﬀerent groups are used for testing, as in
Figure 1.

Table 1: Statistics of datasets used in our experiments.
#users #items#ratings
Movielens
943
1,682
100,000
Eachmovie
61,265
1,623
2,811,718
Netflix
480,189
17,770 100,480,507
Book
21,23765,044
248,829
DVD
3,915
10,86835,350
Elec.
372
2,148
797
Kitchen
317
1,563
727
Music
4,157
14,73839,350
Video
1,816
3,339
12,070For the Amazon datasets, we generated two rating matrices using each pair of categories with no overlap. For
each category pair, users are randomly split into twogroups, and the ratings of users in the first (second)group to items in the first (second) category are used
for training, and the rest are used for testing.
4.2

Comparing the methods

We compared the proposed method with seven methods: CLFM, MF-MM, LSM, MF, Mean-U, MeanI and Mean, which are explained below. CLFM
(cluster-level latent factor model) [9] is a cross-domain
recommendation method that assumes common latent and domain-specific rating patterns in matrix
tri-factorization. With MF-MM (matrix factorization with mean matching), latent vectors for eachdomain are obtained individually by MAP estimation based on the probabilistic matrix factorization
model [21]. Then, the latent factors in diﬀerent domains are aligned by sorting them by their mean values. LSM (latent semantic matching) [13] finds alignment of the latent factors across diﬀerent domains by
convex kernelized sorting [7]. Convex kernelized sorting is a kernel based unsupervised object matching
method that finds correspondence by maximizing thedependency measured in terms of Hilbert-Schmidt information criterion [10]. The MF (probabilistic matrix
factorization) does not align latent factors. We included MF to show how well a single-domain method
works for our task. The Mean-U (mean value for eachuser) method predicts ratings by the mean value forthe user. It cannot be used for recommendation because it predicts the same rating for all the items foreach user. The Mean-I (mean value for each item)
method predicts ratings by the mean value for theitem. It cannot personalize recommendations, and it
suggests popular items. The Mean method predicts
ratings by the mean value of all ratings. For all of
the matrix factorization based methods (the proposed
method, CLFM, MF-MM and LSM), the number oflatent factors was set at K = 30.

Cross-domain recommendation without shared users or items by sharing latent vector distributions

4.3

Results
Proposed
CLFM
MF−MM
Mean−U
Mean−I
Mean

1.3

RMSE

1.2
1.1
1
0.9
0.8
0.7

10

20

30

40
50
60
#latent factors

70

80

90

100

Figure 4: Average test RMSEs with diﬀerent numbersof latent factors with the Movielens dataset.
2.6
2.4
2.2
2
RMSE

The task is to predict ratings in diﬀerent domains.
Table 2 shows the root mean squared error (RMSE)
for test data, as averaged over 30 experiments. The
proposed method achieved the lowest RMSE in 15of the 18 datasets. CLFM achieved low error in themovie datasets because the statistical properties of two
domains in each movie dataset are the same, which
are generated from a single rating matrix. However,
CLFM failed to predict ratings in diﬀerent categories
in the Amazon datasets, because diﬀerent categories
have diﬀerent properties. On the other hand, theRMSE values of the proposed method were consistently low for all datasets examined, which indicatesthat the proposed method can well handle domains
with diﬀerent properties. The RMSE of MF-MM wasalso relatively low in the movie datasets, but high in
the Amazon datasets. LSM yielded worse RMSE than
MF-MM, which indicates that the mean values of latent factors are informative for aligning factors. The
proposed method improves the predictive performanceby using the mean values as well as the covariance information for aligning factors. MF had high RMSE for
all datasets, which shows that single-domain methods
fail at cross-domain rating prediction given the lack ofshared users and items.

1.8
1.6
1.4
1.2
1

Table 3 shows the average test RMSE of the proposed
method for each pair of categories. The diﬃculty ofrating prediction largely depends on the item category. For example, the RMSEs for the Electronics
items were generally higher than those for the Music
items. The RMSE for the Video (DVD) users to theDVD (Video) items were lower than the average of theDVD (Video) items. This is reasonable because DVD
and Video are closely related categories. The RMSEs
for the Book items did not vary when user category
was changed, and the RMSEs of the Book users werehigh for all item categories. This is because there weremany users and items in the Book category, and latent
factors were learned mainly for the Book category but
not for the other category.
Figure 4 shows the average test RMSEs with diﬀerent numbers of latent factors in the Movielens dataset.
When there were more than a few latent factors, theproposed method achieved the best performance. The
proposed method did not overfit even when there weremany latent factors.
Figure 5 shows the test RMSEs over inference iteration with the Movielens dataset. The burn-in period was 1,000. In the burn-in period, the test RMSEs were calculated by a sample; after burn-in, they
were calculated by averaging over samples. At thefirst iteration, the RMSE was very high, where the

0.8
0

1000

2000
3000
#iterations

4000

5000

Figure 5: Test RMSEs over inference iteration number
with the Movielens dataset. The burn-in period was
1,000.
MAP estimate by the probabilistic matrix factorization was used. With a small number of iterations, theRMSE decreased drastically due to the use of sharedmean and covariance values. After the burn-in period,
the RMSE decreased again by averaging over multiple
samples with the Bayesian prediction framework.
Figure 6 shows the empirical mean vectors and covariance matrices of latent vectors for each domain estimated by (a) MF and (b) the proposed method. WithMF, the latent factors are not aligned; the mean andcovariance values of diﬀerent domains are diﬀerent. On
the other hand, with the proposed method, the latent
factors are aligned, which enables the latent vectors to
be used in predicting cross-domain ratings.

5

Conclusion

We proposed a cross-domain recommendation methodthat can predict ratings in diﬀerent domains even if

Tomoharu Iwata, Koh Takeuchi

Table 2: Average test RMSE. Values in bold typeface are statistically better (at the 5% level) from those innormal typeface as indicated by a paired t-test.
Proposed CLFM MF-MM LSM MF Mean-U Mean-I Mean
Movielens
0.925 0.963
1.090 1.948 2.696
1.063 1.093 1.266
Eachmovie
0.961 1.036
1.188 2.217 2.696
1.324 1.230 1.564
1.043 0.927
1.086 2.081 2.819
1.000 1.022 1.176
Netflix
Book-DVD
1.178 3.158
3.134 3.453 3.406
1.507 1.616 1.338
Book-Elec.
1.180 3.558
3.487 3.488 3.488
1.451 1.618 1.341
Book-Kitchen
1.181 3.557
3.485 3.487 3.487
1.442 1.617 1.343
1.138 3.284
3.225 3.476 3.507
1.388 1.544 1.278
Book-Music
Book-Video
1.184 3.385
3.332 3.471 3.477
1.479 1.629 1.353
DVD-Elec.
1.135 3.452
3.415 3.420 3.419
1.416 1.605 1.315
DVD-Kitchen
1.128 3.473
3.427 3.430 3.430
1.404 1.585 1.295
DVD-Music
1.041 2.945
2.951 3.412 3.514
1.328 1.344 1.110
DVD-Video
1.053 2.118
2.502 3.117 3.391
1.125 1.664 1.392
1.369 3.600
3.436 3.437 3.441
2.024 1.930 1.787
Elec.-Kitchen
Elec.-Music
0.953 4.465
3.596 3.599 3.597
1.056 1.139 0.914
Elec.-Video
1.317 3.470
3.326 3.329 3.328
1.827 2.061 1.764
Kitchen-Music
0.948 4.503
3.601 3.603 3.604
1.035 1.123 0.900
Kitchen-Video
1.300 3.490
3.340 3.346 3.347
1.796 2.080 1.711
Music-Video
1.045 3.273
3.245 3.527 3.540
1.333 1.334 1.106
average
1.112 2.940
2.937 3.214 3.344
1.389 1.513 1.331
Table 3: Average test RMSE of the proposed method for each pair of categories in Amazon datasets. For
example, the value in the Book-row of the DVD-column shows the RMSE for the rating prediction of users in
the Book category to items in the DVD category. Values in bold typeface indicate the lowest values in each item
category (column).
user \ item
Book
DVD
Elec. Kitchen Music Video average
Book
1.201
1.551
1.402
1.038
1.310
1.300
DVD
1.175
1.489
1.376
0.962 1.094
1.219
Elec.
1.178
1.126
1.295 0.938
1.312
1.170
Kitchen
1.180
1.121
1.437
- 0.938
1.300
1.195
Music
1.155
1.125
1.493
1.381
1.303
1.291
Video
1.177 1.033 1.405
1.282
0.955
1.170
average
1.173
1.121
1.475
1.347
0.966
1.264
1.047

neither shared users nor items are available. The proposed method aligns latent factors in diﬀerent domains
by assuming the latent factors have common mean andcovariance structures that are shared by all domains.
Experiments on real-world data sets confirmed thatthe proposed method achieved better predictive performance for ratings in diﬀerent domains.
Although our results are encouraging, we must extend
our approach in a number of directions. First, theinterpretability of the inferred latent factors must be
improved. The inferred latent factors in our experiments were not interpretable as genres or topics. It
is because the proposed method allows latent factors
to have arbitrary sign, and involves complex cancellations between positive and negative latent factors [19].
The interpretability might be improved by using nonnegativity constraints. Second, we would like to ex-

tend the proposed method to handle domain heterogeneity. The proposed method described here assumes
that all domains share a common distribution for generating latent vectors. However, some latent factors
are used only for a certain domain, and relationships
between domains would be diﬀerent. By introducing
domain-specific latent factors as well as shared, domain heterogeneity could be modeled as introduced in
canonical correlation analysis [1, 16, 26, 28]. Third, theproposed framework can be applied to diﬀerent typesof datasets by using other distributions for the observation. The proposed method assumes Gaussian observation noise for rating matrices. In the case of bagof-words data, multiple topic models with shared meanand covariance structures can be an option, where acorrelated topic model [4] is used as a component. Finally, the performance of the proposed method can beCross-domain recommendation without shared users or items by sharing latent vector distributions
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Figure 6: Empirical mean vectors and covariance matrices of latent vectors for each domain estimated by (a)single-domain matrix factorization and (b) the proposed cross-domain method. The empirical mean vector of∑Nduser latent vectors for domain d is calculated by ūd = N1d n=1
udn , and the covariance matrix for domain∑Nd1⊤d is calculated by S̄ud = Nd n=1 (udn − ūd )(udn − ūd ) . The empirical mean vector of item latent vectors∑Md
for domain d is calculated by v̄d = M1d m=1
vdm , and the covariance matrix for domain d is calculated by∑Md
S̄vd = M1d m=1(vdm − v̄d )(vdm − v̄d )⊤ . With regard to the bar graph for the mean, the horizontal axis
represents latent factor indices, and the vertical axis represents their mean values. For covariance, the horizontal
and vertical axes represent latent factor indices, and the color shows their covariance values. They were calculatedby the last sample in the inference with the proposed method.
improved in a semi-supervised setting by using a small
number of shared users or items, and auxiliary information, such as user demographics and item content
information.

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