# Dual Metric Learning for Effective and Efficient Cross-Domain
Recommendations

Pan Li, and Alexander Tuzhilin Pan Li and Alexander Tuzhilin are with the
Department of Technology, Operation and Statistics, Stern School of Business,
New York University, New York, NY, 10012.
E-mail<EMAIL_ADDRESS>
###### Abstract

Cross domain recommender systems have been increasingly valuable for helping
consumers identify useful items in different applications. However, existing
cross-domain models typically require large number of overlap users, which can
be difficult to obtain in some applications. In addition, they did not
consider the duality structure of cross-domain recommendation tasks, thus
failing to take into account bidirectional latent relations between users and
items and achieve optimal recommendation performance. To address these issues,
in this paper we propose a novel cross-domain recommendation model based on
dual learning that transfers information between two related domains in an
iterative manner until the learning process stabilizes. We develop a novel
latent orthogonal mapping to extract user preferences over multiple domains
while preserving relations between users across different latent spaces.
Furthermore, we combine the dual learning method with the metric learning
approach, which allows us to significantly reduce the required common user
overlap across the two domains and leads to even better cross-domain
recommendation performance. We test the proposed model on two large-scale
industrial datasets and six domain pairs, demonstrating that it consistently
and significantly outperforms all the state-of-the-art baselines. We also show
that the proposed model works well with very few overlap users to obtain
satisfying recommendation performance comparable to the state-of-the-art
baselines that use many overlap users.

###### Index Terms:

Cross Domain Recommendation, Dual Learning, Metric Learning, Orthogonal
Mapping

##  1 Introduction

Cross domain recommendation [1, 2] constitutes an important method to tackle
sparsity and cold-start problems [3, 4], thus becoming the key component for
online marketplaces to achieve great success in understanding user
preferences. Researchers have proposed various mechanisms to provide cross-
domain recommendations through factorization approaches [5, 6, 7, 8, 9, 10]
and transfer learning methods [11, 12, 13, 14, 15] that transfer user
preferences from the source domain to the target domain [11]. For instance, if
a user watches a movie, we may recommend the novel on which that movie is
based to that user [6].

However, most of the existing methods only focus on the unidirectional
learning process for providing cross-domain recommendations, i.e., utilizing
information from the source domain to improve the recommendations in the
target domain. This is unfortunate since the duality nature of the cross-
domain recommendation task is not explored, and it would also be beneficial to
leverage user preference information from the target domain to predict user
behaviors in the source domain. For example, once we know the type of books
that the user would like to read, we can recommend movies on related topics to
form a loop for better recommendations in both domains.

Dual learning models [16], meanwhile, have been shown in the previous research
to be effective and efficient in multiple applications to address duality
nature in the learning tasks, including domain adaption [17], transfer
learning [18, 19] and machine translation [20]. Therefore in this paper, we
proposed to apply the dual learning mechanism to the problem of cross-domain
recommendations, with the assumption that improving the learning process in
one domain would also help the learning process in the other domain. We
address this assumption by proposing a unifying mechanism that extracts the
essence of preference information in each domain and bidirectionally transfer
user preferences between different domains to improve the recommendation
performance across different domains simultaneously.

Note that existing cross-domain recommendation models typically require a
large amount of overlap users between different domains as ’pivots’, in order
to learn the relations of user preferences and produce satisfying
recommendation performance [2, 21]. These overlap users have consumed items in
both categories, such as watching a movie and reading a book. However,
collecting sufficient amount of overlap users could be difficult and expensive
to accomplish in many applications. For example, there might be only a limited
number of users who purchased books and digital music altogether on Amazon.
Therefore, it is important to overcome this problem and minimize the required
common user overlap across the two domains in cross-domain recommendations.

To address this issue, one potential solution is to construct those ’pivots’
for cross-domain recommendations based on not only the overlap users, but also
those users with similar preferences, with the assumption that if two users
have similar preferences in a certain domain, their preferences would also be
similar across other domains. While the information of user pairs with similar
preferences are not explicitly recorded in the dataset, metric learning models
[22] are capable of learning the distance metric between users and items in
the latent space through shared representations, and capturing user pairs with
similar preferences accordingly [23]. In particular, the metric learning model
would pull latent embeddings of overlap user pairs close to each other in the
latent space, and also pull those similar user pairs close to each other due
to triangle inequality of the distance measure [24]. Besides that, as the
metric learning model is capable of learning a joint latent metric space to
encode not only users’ preferences but also the user-user and item-item
similarity information, it also significantly contributes to the improvements
of recommendation performance [23].

Therefore, in this paper we also propose to supplement the aforementioned dual
learning mechanism with the metric learning approach to effectively identify
the latent relations in user preferences across different domains using the
minimal amount of overlapping users. Moreover, we empirically demonstrate
that, by iteratively updating the dual metric learning model, we
simultaneously improve recommendation performance over both domains and
outperform all the state-of-the-art baseline models. And crucially, the
proposed Dual Metric Learning (DML) model provides these strong recommendation
performance results with only few overlapping uses.

In this paper, we make the following contributions. We

  * •

propose to apply the dual learning mechanism for providing cross-domain
recommendations to address the duality of learning tasks

  * •

propose to incorporate the metric learning model into the dual learning
mechanism to reduce the requirement of having large amounts of overlapping
users

  * •

present a novel cross-domain recommendation model DML and empirically
demonstrate that it significantly and consistently outperforms the state-of-
the-art approaches across all the experimental settings

  * •

provide theoretical foundations to (a) explain why DML reduces the need for
the user overlap information; (b) demonstrate the convergence condition for
the simplified case of our model; and (c) illustrate that the proposed model
can be easily extended to multiple-domain recommendation applications

  * •

conduct extensive ablation experiments to study (a) the cross-domain
recommendation performance of different approaches; (b) the impact of
different architectures of neural networks; (c) the influence of including and
excluding explicit feature information during the representation learning
stage; (d) the impact of different numbers of overlap users included in the
model; (e) the sensitivity of hyperparameters, particularly the dimension of
latent embeddings; (f) the scalability of the proposed approach; and (g) the
convergence behavior of the proposed approach.

##  2 Related Work

The work related to the proposed model comes from the following research areas
that we review in this section: cross domain recommendations, dual learning,
metric-learning for recommendations and deep-learning based recommendations.

###  2.1 Cross Domain Recommendations

Cross-domain recommendation approach [2] constitutes a powerful tool to deal
with the data sparsity problem. Typical cross domain recommendation models are
extended from single-domain recommendation models, including CMF [5], CDCF [6,
7], CDFM [8], HISF [25], NATR [15], DTCDR [26], Canonical Correlation Analysis
[27], Dual Regularization [28], Preference Propagation [29] and Deep Domain
Adaption [30]. These approaches assume that different patterns characterize
the way that users interact with items of a certain domain and allow
interaction information from one domain to inform recommendation in the other
domain.

The idea of information fusion also motivates the use of transfer learning
methods [12, 13, 15, 31, 32] that transfer extracted information from the
source domain to the target domain. The existing transfer learning methods for
cross-domain recommendation include Collaborative DualPLSA [33], JDA [34] and
RB-JTF [35].

However, these models do not fundamentally address the relationship between
different domains, for they do not improve recommendation performance of both
domains simultaneously, thus might not release the full potential of utilizing
the cross-domain user interaction information. Also, they do not explicitly
model user and item features during recommendation process, and usually
require large amount of overlap users. In this paper, we propose a novel dual
metric learning mechanism combining with autoencoders to overcome these issues
and significantly improve recommendation performance.

###  2.2 Dual Learning

Transfer learning [11] deals with the situation where the data obtained from
different resources are distributed differently. It assumes the existence of
common knowledge structure that defines the domain relatedness, and
incorporate this structure in the learning process by discovering a shared
latent feature space in which the data distributions across domains are close
to each other. The existing transfer learning methods for cross-domain
recommendation include Collaborative DualPLSA [33], Joint Subspace Nonnegative
Matrix Factorization [liu2013multi] and JDA [34] that learn the latent factors
and associations spanning a shared subspace where the marginal distributions
across domains are close. In recent years, researchers have utilized up-to-
date deep learning techniques to design the transfer learning models for
providing cross-domain recommendations, including CATN [36] and MMT-Net [37],
and they have all achieved satisfying recommendation performance.

In addition, to exploit the duality between these two distributions and to
enhance the transfer capability, researchers propose the dual transfer
learning mechanism [18, 17, 19] that simultaneously learns the marginal and
conditional distributions. Recently, researchers manage to achieve great
performance on machine translation with dual-learning mechanism [20, 38, 39].
All these successful applications address the importance of exploiting the
duality for mutual reinforcement. However, none of them apply the dual
transfer learning mechanism into cross-domain recommendation problems, where
the duality lies in the symmetrical correlation between source domain and
target domain user preferences. In this paper, we utilize a novel dual-
learning mechanism and significantly improve recommendation performance.

###  2.3 Metric Learning for Recommendations

Metric learning [22] learns a distance metric that preserves the distance
relation among the training data and assigns shorter distances to semantically
similar pairs. Classical metric learning approach attempts to learn a
Mahalanobis distance metric

$d_{A}(x_{i},x_{j})=\sqrt{(x_{i}-x_{j})^{T}A(x_{i}-x_{j})}$ (1)

where $A\in R^{m\times m}$ is a positive semi-definite matrix [40] that
projects each input $x$ to the Euclidean space $R^{m}$. The above global
optimization pulls all similar pairs together, and pushes dissimilar pairs
apart.

There are many different ways to learn the distance metric for providing
recommendations, including convex optimization, kernel methods and neural
networks [41, 42, 43, 44, 45]. Recently, researchers proposed Collaborative
Metric Learning (CML) model [23] that projects users and items into the shared
latent space. Therefore, it is capable of modeling the intensity and the
heterogeneity of the user–item relationships in the implicit feedback.
Researchers in [46, 47, 48] further improve the framework of CML and obtain
better recommendation performance.

However, none of the existing metric learning works focus on the application
of cross-domain recommendations to reduce the requirement of overlap user
information. In this paper, we use metric learning to map the users into the
shared latent space across different domains. By effectively utilizing
information of overlap users, we achieve satisfying cross-domain
recommendation performance using only very few users, thus significantly
improving the practicality of the cross-domain recommendation models.

###  2.4 Deep Learning-based Recommendations

Recently, deep learning has been revolutionizing the recommendation
architectures dramatically and brings more opportunities to improve the
performance of existing recommender systems. To capture the latent
relationship between users and items, researchers propose to use deep learning
based recommender systems [49, 50], especially embedding and autoencoding
methods [51, 52, 53, 54, 55, 56, 57, 58] to extract the latent essence of
user-item interactions for the better understanding of user preferences.

However, user preferences in different domains are learned separately without
exploiting the duality for mutual reinforcement, for researchers have not
addressed the combination of deep learning methods with dual learning
mechanism in recommender systems, which learns user preferences from different
domains simultaneously and further improves the recommendation performance. To
this end, we propose a deep dual metric learning model that captures latent
interactions across different domains. The effectiveness of dual learning over
the existing methods is empirically demonstrated by extensive experiments.

##  3 Model

In this section, we introduce the proposed Dual Metric Learning (DML) model
for improving cross-domain recommendation performance and reducing the need of
overlap information. We illustrate the proposed model in Figure 1 and present
the framework in Algorithm 1. We will now describe the details of DML model in
the following sections.

Figure 1: Dual Metric Learning for Cross-Domain Recommendations Algorithm 1
Dual Metric Learning (DML) Model

1:Input: Domain $A$ and $B$, user embeddings $W_{u_{A}}$ and $W_{u_{B}}$, item
embeddings $W_{i_{A}}$ and $W_{i_{B}}$, ratings $r_{A}$ and $r_{B}$, neural
recommendation models $RS_{A}$ and $RS_{B}$, orthogonal mapping $X$, overlap
users $ou_{A}=ou_{B}$

2:repeat

3: for ($W_{u_{A}}$, $W_{i_{A}}$, $r_{A}$) in $A$, ($W_{u_{B}}$, $W_{i_{B}}$,
$r_{B}$) in $B$ do

4: Estimate $r_{A}^{\prime}=RS_{A}(W_{u_{A}},W_{i_{A}})$ and
$r_{B}^{\prime}=RS_{B}(W_{u_{B}},W_{i_{B}})$

5: Backpropogate $L_{A}=||r_{A}-r_{A}^{\prime}||$ and
$L_{B}=||r_{B}-r_{B}^{\prime}||$

6: Update $RS_{A}$ and $RS_{B}$

7: end for

8: for $W_{ou_{A}}$, $W_{ou_{B}}$ in Overlap Users do

9: Backpropogate $L_{o_{A}}=||XW_{ou_{A}}-W_{ou_{B}}||$ and
$L_{o_{B}}=||X^{T}W_{ou_{B}}-W_{ou_{A}}||$

10: Update and Orthogonalize $X$

11: end for

12: for ($W_{u_{A}}$, $W_{i_{A}}$, $r_{A}$) in $A$, ($W_{u_{B}}$, $W_{i_{B}}$,
$r_{B}$) in $B$ do

13: Estimate $r^{*}_{A}=RS_{B}(XW_{u_{A}},W_{i_{A}})$ and
$r^{*}_{B}=RS_{A}(X^{T}W_{u_{B}},W_{i_{B}})$

14: Backpropogate $L_{A^{*}}=||r_{A}-r^{*}_{A}||$ and
$L_{B^{*}}=||r_{B}-r^{*}_{B}||$

15: Update $RS_{A}$ and $RS_{B}$

16: end for

17:until convergence

###  3.1 User and Item Embeddings

To effectively extract latent user preferences and efficiently model user and
item features, we utilize the autoencoder framework to learn the latent
representations of users and items, which transforms the heterogeneous and
discrete feature vectors into continuous feature embeddings. The goal is to
train two separate neural networks: the encoder network that maps feature
vectors into the latent space, and the decoder network that reconstructs
feature vectors from latent embeddings. Due to effectiveness and efficiency of
the training process, we formulate both the encoder and the decoder as multi-
layer perceptron (MLP).

Specifically, we denote features of user $a$ as
$u_{a}=\\{u_{a_{1}},u_{a_{2}},\cdots,u_{a_{m}}\\}$ and features of item $b$ as
$i_{b}=\\{i_{b_{1}},i_{b_{2}},\cdots,i_{b_{n}}\\}$, where $m$ and $n$ are the
respective dimensions of feature vectors. MLP learns the hidden
representations by optimization reconstruction loss $L$:

$L=||u_{a}-MLP_{dec}(MLP_{enc}(u_{a}))||$ (2)

where $MLP_{enc}$ and $MLP_{dec}$ represents the MLP network for the encoder
and the decoder. In this step, we train the autoencoder separately for users
and items in different domains to avoid the information leak between domains.
We subsequently obtain the user embeddings and item embeddings as
$W_{u_{a}}=MLP_{enc}(u_{a})$ and $W_{i_{b}}=MLP_{enc}(i_{b})$ for the
recommendation stage.

Note that regularization technique is typically crucial to the feasibility and
efficient optimization of the metric learning model. Especially when the data
points are distributed in a high-dimensional space, they might spread too
widely because of the curse of dimensionality and thus become ineffective to
be mapped into the same shared latent space. Therefore, we restrict all the
user and item embeddings $||W_{u}||$ and $||W{i}||$ within the unit ball,
i.e., $||W_{u}||^{2}\leq 1$ and $||W{i}||^{2}\leq 1$, to ensure the robustness
of the learned metric. As such, we do not need to impose additional
regularization during the subsequent metric learning process.

###  3.2 Metric Learning

In this section, we introduce the proposed DML recommendation model to
efficiently capture relative preferences between different domains through
overlap users. To start with, we identify and extract overlap users and their
embeddings in two domains as a set of embedding-embedding pairs $S$ that match
with each other. Note that in the previous stage, we separately train user
embeddings and item embeddings for each domain. Due to the heterogeneity of
user preferences in different domains, the distribution of user embeddings in
each domain should also be different. The underlying assumption we make is
that, if two users have similar interest in one domain, then those two users
would have similar interest in the other domain as well. We aim at addressing
this assumption to incorporate consumption behaviors in one domain for better
predictions of the other domain. Therefore, these overlap users can be used as
‘pivots’ to learn the relations of user preferences and behaviors in different
domains.

We aim to learn a transitional metric learning mapping to encode these
relationships in a shared latent space, and the learned mapping should pull
the overlap user embedding pairs closer and push the other pairs relatively
further apart. According to the triangle inequality, this learning process
will also cluster the items that are purchased by the same user across
different domains together, and the users who purchase similar items across
different domains together. Eventually, the nearest neighbor points for any
given user in the shared latent space will become its representation in the
other domains respectively. Through the metric learning mapping that
effectively captures the relations between overlap users, we propagate these
relations not only to overlap user-user pairs, but also to those user-item
pairs and non-overlap user-user pairs for which we did not directly observe
such relationships.

Distinguished from the classical metric learning settings, the proposed DML
model restricts this transitional mapping to be an orthogonal mapping. We
point out that orthogonality is important for cross-domain recommendations for
the following reasons:

  * •

Orthogonality preserves the similarities between embeddings of different users
during the metric learning process since orthogonal transformation preserves
inner product of latent vectors.

  * •

Orthogonality mapping $X$ will automatically give the inverse mapping as
$X^{T}$, because $Y=X^{T}(XY)$ holds for any given orthogonal mapping $X$,
thus making inverse mapping matrix equivalent to its transpose. This
simplifies the learning procedure and enables the dual learning process.

  * •

Orthogonality restricts the parameter space of the transitional mapping and
thus requires less overlap user information to determine this mapping
function, as we will discuss in the next section.

Most importantly, the proposed DML model utilizes dual learning mechanism to
update the recommender system for both domains simultaneously. We use the
metric learning output of user embeddings in the source domain as the input of
recommender system in the target domain, while in turn using the output of the
target domain as the input of the source domain. In this way, we could improve
the metric learning mapping as well as recommender systems for both domains
simultaneously in each iteration. The learning process can then be repeated
iteratively to obtain better metric mappings and recommender systems each time
until the convergence criterion is met. As a result, it pushes the metric
learning framework to better capture user preferences and thus provide even
better recommendation performance. We illustrate the dual orthogonal metric
learning process in Figure 2.

Figure 2: Dual Metric Learning

Specifically, we denote the user embeddings for domain $A$ and $B$ as
$W_{u_{A}}$, $W_{u_{B}}$ that we obtained in the previous section. We also
denote the overlap users between two domains as $ou_{A}=ou_{B}$, and their
user embeddings $W_{ou_{A}}$, $W_{ou_{B}}$ correspondingly. The goal of
proposed DML model is to find the optimal mapping matrix $X$ to minimize the
sum of squared Euclidean distances between the mapped user embeddings
$XW_{ou_{A}}$ and the target user embeddings $W_{ou_{B}}$ for the same overlap
users:

$L_{X}=argmin_{X}\sum_{W_{ou_{A}},W_{ou_{B}}\in
ou_{A},ou_{B}}||XW_{ou_{A}}-W_{ou_{B}}||^{2}$ (3)

This optimization is equivalent to its dual form:

$L_{X^{T}}=argmin_{X}\sum_{W_{ou_{A}},W_{ou_{B}}\in
ou_{A},ou_{B}}||W_{ou_{A}}-X^{T}W_{ou_{B}}||^{2}$ (4)

We constrain the metric learning mapping $X$ to be an orthogonal mapping (i.e.
$XX^{T}=X^{T}X=I$), which serves to enforce structural invariance of user
preferences in each domain, while preventing a degradation in mono-domain
recommendation performance for learning better transitional mappings. We
optimize Equation (3) and (4) simultaneously to learn the orthogonal metric
mapping.

###  3.3 Dual Learning

To further improve performance of recommendations, we propose to combine dual
learning mechanism with cross domain recommendations, where we transfer user
preferences between the source domain and the target domain simultaneously.
Consider two different domains $D_{A}$ and $D_{B}$ that contain user-item
interactions as well as user and item features. To obtain better understanding
of user preferences in $D_{A}$, we also utilize external user preference
information from $D_{B}$ and combine them together. Similarly, we could get
better recommendation performance in $D_{B}$ if we utilize user preference
information from $D_{A}$ at the same step. To leverage duality of the two
transfer learning based recommendation models and to improve effectiveness of
both tasks simultaneously, we conduct dual learning recommendations for the
two models together and update the recommendation models for both domains
accordingly.

Specifically, we propose to model cross-domain user preferences utilizing the
metric mapping $X$ learned in the previous stage and taking into account
heterogeneous user behavior in both domains. In that sense, we estimate user
ratings in domain pairs $(A,B)$ as follows:

$r^{*}_{A}=RS_{A}(XW_{u_{A}},W_{i_{A}})$ (5)
$r^{*}_{B}=RS_{B}(X^{T}W_{u_{B}},W_{i_{B}})$ (6)

where $W_{u_{A}},W_{i_{A}},W_{u_{B}},W_{i_{B}}$ represents user and item
embeddings and $RS_{A},RS_{B}$ stand for the domain-specific neural
recommendation model for domain A and B respectively. Here, we intentionally
transfer the user representations from one domain to the other and force the
recommendation to learn the user preferences from the other domain in order to
provide better recommendations. As shown in Figure 1, dual learning entails
the transfer loop across two domains and the learning process goes through the
loop iteratively. It is also important to study the convergence property of
the model, which we discuss in the next section.

###  3.4 Summary of DML Model

To conclude, we propose the DML method to model the overlap user information
for providing better cross-domain recommendations. Specifically, we impose the
orthogonal constraint to the metric learning mapping and perform dual learning
mechanism to effectively and efficiently learn the mapping function as well as
recommender systems at the same time. In the final stage, we concatenate the
item embeddings with the user embeddings in the shared latent space for
subsequent rating estimation. We feed them into the Multi-Layer Perceptron
(MLP) network with dropout to provide final cross-domain recommendations due
to its superior representational capacity and ease of training [51].

Compared to the previous approaches, the proposed DML model has the following
advantages. It

  * •

utilizes dual learning mechanism to enable bidirectional transfer of user
preferences that improves recommendation performance simultaneously in both
domains over time;

  * •

combines dual learning with metric learning to effectively obtain latent
relations across different domains, thus significantly reducing the amount of
overlap users required to achieve strong performance;

  * •

transfers latent representations of features and user preferences, instead of
explicit information, between different domains to capture latent and complex
interactions between the users and items;

  * •

learns the latent orthogonal mapping function across the two domains, which is
capable of preserving similarities of user preferences and computing the
inverse mapping function efficiently.

We will show in the following section that the proposed model is capable of
providing state-of-the-art cross-domain recommendation performance and
significantly reducing the need for user overlap information.

##  4 Theoretical Analysis

In this section, we provide the theoretical foundations to justify the design
of our DML model. Specifically, we point out that DML reduces the need for
overlap information, guarantees to converge under certain conditions and could
be easily extended to multi-domain scenarios.

###  4.1 Reducing the need for overlap information

By framing the cross-domain recommendation problem as the metric learning
problem, we do not need to extract explicit user behaviors from both domains;
instead, we only need to learn the metric learning mapping that represents the
relations between the two domains. In addition, by mapping both user and item
features into the latent space, we overcome the problem of discretization in
the feature space and mathematically formulate the problem as follows: given a
set of $k$-dimensional feature vectors
$\\{W_{u_{A_{1}}},W_{u_{A_{2}}},\cdots,W_{u_{A_{n}}}\\}$ in domain A and
$\\{W_{u_{B_{1}}},W_{u_{B_{2}}},\cdots,W_{u_{B_{n}}\\}}$ in domain B, the goal
is to identify the matrix $X$ that $XW_{u_{A_{i}}}=W_{u_{B_{i}}}$ holds for
every $i$ from $1$ to $n$. In addition, we restricts this matrix $X$ to be
orthogonal, i.e., $XX^{T}=I$. From basic linear algebra [59], we know that the
unique solution exists when the number of restricted equations
$n=\frac{k(k-1)}{2}$. Therefore, when the dimension of feature embeddings is
$k$, we only need $\frac{k(k-1)}{2}$ number of overlap users to determine the
metric learning mapping. As shown in the results section, we conduct
sensitivity analysis to study the impact of different embedding dimensions
towards cross domain recommendation performance, which shows that we can
achieve reasonable and satisfying performance using 16 dimensions of latent
vectors. Therefore, our theoretical analysis shows that we could determine the
metric learning mapping using only 120 overlap users for our recommendation
tasks, and experiment results further confirm that we could provide satisfying
recommendation performance using as little as 8 overlap users.

###  4.2 Convergence Analysis

In this section, we present the convergence theorem of classical recommender
systems with dual learning mechanism. We denote the rating matrices as
$V_{A},V_{B}$ for the two domains A and B respectively. The goal is to find
the approximate non-negative factorization matrices that simultaneously
minimizes the reconstruction loss. We conduct multiplicative update algorithms
[60] for the two reconstruction loss iteratively given $\alpha$, $X$, $V_{A}$
and $V_{B}$.

$\begin{split}\displaystyle\min_{W_{A},W_{B},H_{A},H_{B}}&|V_{A}-(1-\alpha)W_{A}H_{A}-\alpha
XW_{B}H_{B}|\\\ +&|V_{B}-(1-\alpha)W_{B}H_{B}-\alpha
X^{T}W_{A}H_{A}|\end{split}$ (7)

######  Proposition 1.

Convergence of the iterative optimization of dual matrix factorization for
rating matrices $V_{A}$ and $V_{B}$ in (7) is guaranteed.

###### Proof.

Combining the two parts of objective functions and eliminating $W_{A}H_{A}$
term, we will get

$(1-\alpha)V_{B}-\alpha X^{T}V_{A}=(1-2\alpha)W_{B}H_{B}$ (8)

Similarly, when we eliminate $W_{B}H_{B}$ term, we will get

$(1-\alpha)V_{A}-\alpha XV_{B}=(1-2\alpha)W_{A}H_{A}$ (9)

Based on the analysis in [61] for the classical single-domain matrix
factorization, to show that the repeated iteration of update rules is
guaranteed to converge to a locally optimal solution, it is sufficient to show
that

$\left\\{\begin{array}[]{rcl}2\alpha-1<0&&(a)\\\ (1-\alpha)V_{B}-\alpha
X^{T}V_{A}\geq\textbf{0}&&(b)\\\ (1-\alpha)V_{A}-\alpha
XV_{B}\geq\textbf{0}&&(c)\\\ \end{array}\right.$

where 0 stands for the zero matrix. Condition (a) is an intuitive condition
which indicates that the information from the target domain dominantly
determines the user preference, while information transferred from the source
domain only serves as the regularizer in the learning period. To fulfill the
seemingly complicated condition (b) and (c), we recall that $V_{A}$ and
$V_{B}$ are ratings matrices, which are non-negative and bounded by the rating
scale $k$. We design two ”positive perturbation”
$V_{A}^{\prime}=V_{A}+mk\textbf{1}$ and $V_{B}^{\prime}=V_{B}+mk\textbf{1}$
where $m$ is the rank of the mapping matrix $X$. Condition (a) is independent
of the specific rating matrices, and we could check that $V_{A}^{\prime}$ and
$V_{B}^{\prime}$ satisfies condition (b) and (c). Thus, the matrix
factorization of $V_{A}^{\prime}$ and $V_{B}^{\prime}$ is guaranteed
convergence; to reconstruct the original matrix $V_{A}$ and $V_{B}$, we only
need to deduce the ”positive perturbation” term $m$, so the original problem
is also guaranteed convergence. ∎

In this paper, to capture latent interactions between users and items, we use
the neural network based recommendation approach, therefore it still remains
unclear if this convergence happens to the proposed model. However, our
hypothesis is that even in our case we will experience similar convergence
process guaranteed in the classical matrix factorization case, as stated in
the proposition. We test and validate this hypothesis in the result section.

###  4.3 Extension to Multiple Domains

In previous sections, we describe the proposed model with special focus on the
idea of combining dual metric learning mechanism with latent embedding
approaches. We point out that the proposed model not only works for cross
domain recommendations between two domains, but can easily extend to
recommendations between multiple domains as well. Consider $N$ different
domains $D_{1},D_{2},D_{3},\cdots,D_{N}$. To provide cross-domain
recommendations for these domains, we only need to obtain the $N-1$ orthogonal
mapping learning mapping $X_{12},X_{23},\cdots,X_{(N-1)N}$, and the latent
orthogonal transfer matrix $X_{jk}$ between domain $D_{j}$ and $D_{k}$ could
be obtained as $X_{jk}=X_{j(j+1)}*X_{(j+1)(j+2)}*\cdots*X_{(k-1)k}$.
Therefore, the proposed model is capable of providing recommendations for
multiple-domain application effectively and efficiently.

##  5 Experiment

To demonstrate the superiority of our proposed method, we conducted extensive
experiments on two real-world datasets in this paper to study the following
research questions: (1) the cross-domain recommendation performance of
different approaches; (2) the impact of different architectures of neural
networks; (3) the influence of including and excluding explicit feature
information during the representation learning stage; (4) the impact of
different numbers of overlap users included in the model; (5) the sensitivity
of hyperparameters, particularly the dimension of latent embeddings; (6) the
scalability of the proposed approach; and (7) the convergence behavior of the
proposed approach. Our models and codes have been made publicly available
111https://github.com/lpworld/DOML.

###  5.1 Dataset

We evaluate the cross-domain recommendation performance of the proposed on two
sets of large-scale industrial datasets: the Imhonet dataset [62], which is
obtained directly from the user logs of the European online recommendation
service Imhonet, containing three domains of user and item features as well as
rating information, namely Book, Movie and Music; and the Amazon dataset [63]
222https://nijianmo.github.io/amazon/index.html which consists of user
purchase actions and rating information collected from the Amazon platform,
and we select three domains with sufficient amount of overlap users for
conducting our experiments: Beauty, Fashion and Art Crafts. Overlap users
across different domains can be identified by the same user ID. We normalize
the scale of ratings to between 0 and 1 for both datasets. The basic
statistics about these three datasets are shown in Table I and Table II. There
are around 2,000 overlap users between each domain pair in the Imhonet
dataset, and around 30,000 overlap users between each domain pair in the
Amazon dataset.

Datasets | Book | Movie | Music
---|---|---|---
# Users | 804,285 | 959,502 | 45,962
# Items | 182,653 | 79,866 | 183,114
# Records | 223,007,805 | 51,269,130 | 2,536,273
Sparsity | 0.0157% | 0.0669% | 0.0301%
TABLE I: Descriptive Statistics for the Imhonet Dataset Domain | Beauty | Fashion | Art Crafts
---|---|---|---
#Users | 324,038 | 749,233 | 1,579,230
#Items | 32,586 | 186,189 | 302,809
#Records | 371,345 | 883,636 | 2,875,917
Sparsity | 0.0035% | 0.0006% | 0.0006%
TABLE II: Descriptive Statistics of the Amazon Dataset

###  5.2 User and Item Features

Besides the records of interactions between users and items in the Imhonet
dataset, the online platform also collects user information by asking them
online questions. Typical question examples include preferences between two
items, ideas about recent events, life styles, demographic information, etc..
From all these questions, we select eight most-popular questions and use the
corresponding answers from users as user features. Meanwhile, although the
online platform does not directly collect item features, we obtain these
information through Google Books API333https://developers.google.com/books/,
IMDB API444http://www.omdbapi.com/ and Spotify
API555https://developer.spotify.com/documentation/web-api/, following the
method in [14]. To ensure the correctness of the collected item features, we
validate the retrieved results with the timestamp included in the dataset. To
sum up, for the Imhonet dataset, we include the user features of Age, Movie
Taste, Residence, Preferred Category, Recommendation Usage, Marital Status and
Personality, the book features of Category, Title, Author, Publisher,
Language, Country, Price and Date, the movie features of Genre, Title,
Director, Writer, Runtime, Country, Rating and Votes, and the music features
of Listener, Playcount, Artist, Album, Tag, Release, Duration and Title in the
recommendation process to construct user and item embeddings correspondingly.

For the Amazon dataset used in our study, we also collect the meta-information
of users and items in each domain, as provided by the Amazon platform [63]. In
particular, we include the user features of rating, purchase history, review
and timestamp, and the item features of category, description, title and price
to generate user and item embedding respectively for the three domains.

###  5.3 Baseline Models

To conduct experiments and evaluate our model, we utilize the 5-fold cross
validation and evaluate the recommendation performance based on RMSE, MAE,
Precision and Recall metrics [64]. We compare the performance with a group of
state-of-the-art methods and report the results in the next section.

  * •

DDTCDR [14] Deep Dual Transfer Cross Domain Recommendation (DDTCDR)
efficiently transfers user preferences across domain pairs through dual
learning mechanism.

  * •

CML[23] Collaborative Metric Learning (CML) learns a joint metric space to
encode users’ preferences and similarity information.

  * •

CCFNet[13] Cross-domain Content-boosted Collaborative Filtering neural NETwork
(CCCFNet) utilizes factorization to tie CF and content-based filtering
together with a unified multi-view neural network.

  * •

CDFM[8] Cross Domain Factorization Machine (CDFM) proposes an extension of FMs
that incorporates domain information in interaction patterns.

  * •

CoNet[12] Collaborative Cross Networks (CoNet) enables knowledge transfer
across domains by cross connections between base networks.

  * •

CMF[5] Collective Matrix Factorization (CMF) simultaneously factor several
matrices, sharing parameters among factors when a user participates in
multiple domains.

  * •

NCF[51] Neural Collaborative Filtering (NCF) models latent features of users
and items using collaborative filtering method and obtains satisfying
recommendation performance. In this baseline, we train the NCF model
separately to provide recommendations for each domain.

  * •

NCF-joint[51] In this baseline, we merge the data records of both domains and
jointly train the NCF model to provide recommendations for both domains
simultaneously.

###  5.4 Parameter Settings

To obtain the superior recommendation performance, we conduct Bayesian
Optimization [65] to identify the best suitable hyperparameters for our
proposed DML model as well as baseline models. Specifically, we select the
dimension of feature embeddings as 16 and the batch size as 64. We construct
both the encoder and the decoder as the Multi-Layer Perceptron with two hidden
layers of 32 and 16 neurons respectively. The orthogonal constraint of the
metric learning mapping is achieved through the Gram-Schmidt orthogonalization
mechanism [66, 67].

##  6 Results

###  6.1 Cross-Domain Recommendation Performance

Since we have three different domains in each dataset, this results in three
domain pairs for evaluation in each recommendation task. We report the
experimental results of cross domain recommendation performance for DML and
other baselines for the Imhonet dataset in Tables V, V and V, and the results
for the Amazon dataset in Tables VIII, VIII and VIII respectively. As shown in
these tables, the proposed DML model significantly and consistently
outperforms all the baseline models across all six recommendation tasks and
all the evaluation metrics. In particular in the Book-Movie recommendation
scenario for the Imhonet dataset, we have observed increases of 3.01% in the
RMSE metric, 4.96% in MAE, 2.28% in Precision and 2.81% in Recall over the
second-best baseline approach. We have also observed similar significant
improvements in other experimental settings. Therefore, we have shown that the
proposed dual metric learning method works well in practice and achieves
significant performance improvements vis-a-vis other state-of-the-art cross
domain recommendation approaches.

Algorithm | Book Dataset | Movie Dataset
---|---|---
RMSE | MAE | Pre@5 | Rec@5 | RMSE | MAE | Pre@5 | Rec@5
DML | 0.2184* | 0.1646* | 0.8826* | 0.9850* | 0.2109* | 0.1606* | 0.9002* | 0.9962*
Improved % | (+1.31%) | (+3.63%) | (+2.69%) | (+3.71%) | (+4.70%) | (+6.30%) | (+0.86%) | (+0.92%)
DDTCDR | 0.2213 | 0.1708 | 0.8595 | 0.9594 | 0.2213 | 0.1714 | 0.8925 | 0.9871
CML | 0.2408 | 0.1927 | 0.8150 | 0.8864 | 0.2342 | 0.1924 | 0.8520 | 0.9380
CCFNet | 0.2639 | 0.1841 | 0.8102 | 0.8872 | 0.2476 | 0.1939 | 0.8545 | 0.9300
CDFM | 0.2494 | 0.2165 | 0.7978 | 0.8610 | 0.2289 | 0.1901 | 0.8498 | 0.9312
CMF | 0.2921 | 0.2478 | 0.7972 | 0.8523 | 0.2738 | 0.2293 | 0.8324 | 0.9012
CoNet | 0.2305 | 0.1892 | 0.8328 | 0.8990 | 0.2298 | 0.1903 | 0.8680 | 0.9601
NCF | 0.2315 | 0.1887 | 0.8357 | 0.8924 | 0.2276 | 0.1895 | 0.8428 | 0.9495
NCF-joint | 0.2378 | 0.1920 | 0.8275 | 0.8880 | 0.2254 | 0.1880 | 0.8410 | 0.9478

TABLE III: Comparison of Cross-Domain Recommendation Performance between Book
and Movie Datasets. * stands for 95% significance.

Algorithm | Book Dataset | Music Dataset
---|---|---
RMSE | MAE | Pre@5 | Rec@5 | RMSE | MAE | Pre@5 | Rec@5
DML | 0.2162* | 0.1615* | 0.8649* | 0.9645* | 0.2689* | 0.2255* | 0.8475* | 0.9070*
Improved % | (+2.13%) | (+5.22%) | (+0.92%) | (+0.44%) | (+4.94%) | (+2.04%) | (+0.99%) | (+1.59%)
DDTCDR | 0.2209 | 0.1704 | 0.8570 | 0.9602 | 0.2753 | 0.2302 | 0.8392 | 0.8928
CML | 0.2398 | 0.1902 | 0.8112 | 0.9120 | 0.3075 | 0.2478 | 0.7922 | 0.8402
CCFNet | 0.2630 | 0.1842 | 0.8150 | 0.9108 | 0.3090 | 0.2422 | 0.7902 | 0.8388
CDFM | 0.2489 | 0.2155 | 0.8104 | 0.9102 | 0.3252 | 0.2463 | 0.7895 | 0.8365
CMF | 0.2921 | 0.2478 | 0.8072 | 0.8978 | 0.3478 | 0.2698 | 0.7820 | 0.8324
CoNet | 0.2307 | 0.1897 | 0.8230 | 0.9300 | 0.2801 | 0.2410 | 0.7912 | 0.8428
NCF | 0.2315 | 0.1887 | 0.8357 | 0.8924 | 0.2828 | 0.2423 | 0.7930 | 0.8450
NCF-joint | 0.2377 | 0.1870 | 0.8388 | 0.8890 | 0.2844 | 0.2440 | 0.7912 | 0.8428

TABLE IV: Comparison of Cross-Domain Recommendation Performance between Book
and Music Datasets. * stands for 95% significance.

Algorithm | Movie Dataset | Music Dataset
---|---|---
RMSE | MAE | Pre@5 | Rec@5 | RMSE | MAE | Pre@5 | Rec@5
DML | 0.2111* | 0.1647* | 0.9013* | 0.9980* | 0.2718* | 0.2231* | 0.8487* | 0.9050*
Improved % | (+2.90%) | (+4.24%) | (+0.97%) | (+1.12%) | (+1.45%) | (+3.46%) | (+1.40%) | (+1.66%)
DDTCDR | 0.2174 | 0.1720 | 0.8926 | 0.9869 | 0.2758 | 0.2311 | 0.8370 | 0.8902
CML | 0.2259 | 0.1870 | 0.8578 | 0.9270 | 0.3018 | 0.2590 | 0.7875 | 0.8390
CCFNet | 0.2468 | 0.1932 | 0.8398 | 0.9310 | 0.3090 | 0.2433 | 0.7952 | 0.8498
CDFM | 0.2289 | 0.1895 | 0.8306 | 0.9382 | 0.3252 | 0.2467 | 0.7880 | 0.8460
CMF | 0.2738 | 0.2293 | 0.8278 | 0.9222 | 0.3478 | 0.2698 | 0.7796 | 0.8400
CoNet | 0.2302 | 0.1908 | 0.8450 | 0.9508 | 0.2811 | 0.2428 | 0.8010 | 0.8512
NCF | 0.2276 | 0.1895 | 0.8428 | 0.9495 | 0.2828 | 0.2423 | 0.7930 | 0.8450
NCF-joint | 0.2238 | 0.1834 | 0.8450 | 0.9508 | 0.2877 | 0.2445 | 0.7898 | 0.8438

TABLE V: Comparison of Cross-Domain Recommendation Performance between Movie
and Music Datasets. * stands for 95% significance.

Algorithm | Beauty | Fashion
---|---|---
RMSE | MAE | Pre@5 | Rec@5 | RMSE | MAE | Pre@5 | Rec@5
DML | 0.2930* | 0.1972* | 0.8354* | 0.8702* | 0.2997* | 0.2398* | 0.8001* | 0.8577*
Improved % | (+1.97%) | (+1.93%) | (+1.49%) | (+1.26%) | (+0.93%) | (+1.24%) | (+1.30%) | (+2.08%)
DDTCDR | 0.2989 | 0.2012 | 0.8231 | 0.8594 | 0.3025 | 0.2428 | 0.7898 | 0.8402
CML | 0.3033 | 0.2046 | 0.8125 | 0.8533 | 0.3077 | 0.2455 | 0.7860 | 0.8371
CCCFNet | 0.3045 | 0.2035 | 0.8134 | 0.8546 | 0.3089 | 0.2460 | 0.7852 | 0.8362
CDFM | 0.3022 | 0.2035 | 0.8130 | 0.8548 | 0.3102 | 0.2489 | 0.7834 | 0.8355
CMF | 0.3017 | 0.2028 | 0.8122 | 0.8533 | 0.3111 | 0.2501 | 0.7798 | 0.8328
CoNet | 0.2968 | 0.1982 | 0.8218 | 0.8590 | 0.3044 | 0.2433 | 0.7883 | 0.8402
NCF | 0.3010 | 0.2023 | 0.8191 | 0.8577 | 0.3077 | 0.2472 | 0.7860 | 0.8355
NCF-joint | 0.2988 | 0.2010 | 0.8205 | 0.8582 | 0.3065 | 0.2466 | 0.7877 | 0.8364

TABLE VI: Comparison of Cross-Domain Recommendation Performance between Beauty
and Fashion Datasets. ‘*’ stands for 95% significance.

Algorithm | Beauty | Art Crafts
---|---|---
RMSE | MAE | Pre@5 | Rec@5 | RMSE | MAE | Pre@5 | Rec@5
DML | 0.2941* | 0.1986* | 0.8341* | 0.8725* | 0.2633* | 0.2011* | 0.8905* | 0.9203*
Improved % | (+1.70%) | (+1.39%) | (+1.48%) | (+1.60%) | (+1.98%) | (+3.87%) | (+2.16%) | (+1.19%)
DDTCDR | 0.2992 | 0.2014 | 0.8219 | 0.8588 | 0.2680 | 0.2092 | 0.8717 | 0.9095
CML | 0.3065 | 0.2044 | 0.8122 | 0.8533 | 0.2729 | 0.2130 | 0.8644 | 0.9011
CCCFNet | 0.3044 | 0.2031 | 0.8137 | 0.8540 | 0.2717 | 0.2119 | 0.8655 | 0.9039
CDFM | 0.3021 | 0.2037 | 0.8130 | 0.8548 | 0.2708 | 0.2127 | 0.8652 | 0.9042
CMF | 0.3021 | 0.2029 | 0.8127 | 0.8532 | 0.2726 | 0.2133 | 0.8637 | 0.9013
CoNet | 0.2977 | 0.1982 | 0.8211 | 0.8592 | 0.2689 | 0.2080 | 0.8723 | 0.9049
NCF | 0.3017 | 0.2029 | 0.8197 | 0.8570 | 0.2707 | 0.2111 | 0.8699 | 0.9071
NCF-joint | 0.3001 | 0.2009 | 0.8203 | 0.8588 | 0.2695 | 0.2099 | 0.8711 | 0.9092

TABLE VII: Comparison of Cross-Domain Recommendation Performance between
Beauty and Art Crafts Datasets. ‘*’ stands for 95% significance.

Algorithm | Fashion | Art Crafts
---|---|---
RMSE | MAE | Pre@5 | Rec@5 | RMSE | MAE | Pre@5 | Rec@5
DML | 0.2984* | 0.2375* | 0.8023* | 0.8514* | 0.2622* | 0.2030* | 0.8895* | 0.9233*
Improved % | (+1.29%) | (+2.22%) | (+1.56%) | (+1.37%) | (+2.46%) | (+2.96%) | (+2.09%) | (+1.60%)
DDTCDR | 0.3023 | 0.2429 | 0.7900 | 0.8399 | 0.2688 | 0.2092 | 0.8713 | 0.9088
CML | 0.3092 | 0.2477 | 0.7839 | 0.8337 | 0.2725 | 0.2136 | 0.8637 | 0.9010
CCCFNet | 0.3087 | 0.2462 | 0.7849 | 0.8357 | 0.2711 | 0.2116 | 0.8664 | 0.9046
CDFM | 0.3100 | 0.2488 | 0.7831 | 0.8359 | 0.2709 | 0.2129 | 0.8651 | 0.9040
CMF | 0.3112 | 0.2506 | 0.7801 | 0.8333 | 0.2731 | 0.2137 | 0.8637 | 0.9015
CoNet | 0.3048 | 0.2439 | 0.7886 | 0.8394 | 0.2683 | 0.2071 | 0.8726 | 0.9048
NCF | 0.3079 | 0.2477 | 0.7859 | 0.8357 | 0.2707 | 0.2120 | 0.8693 | 0.9062
NCF-joint | 0.3061 | 0.2449 | 0.7870 | 0.8381 | 0.2691 | 0.2103 | 0.8705 | 0.9074

TABLE VIII: Comparison of Cross-Domain Recommendation Performance between
Fashion and Art Crafts Datasets. ‘*’ stands for 95% significance.

###  6.2 Impact of the Number of Overlap Users

As we have discussed in this paper, one important advantage of the proposed
model lies in its ability to significantly reduce the requirement of overlap
user information for the cross-domain recommendation tasks. To empirically
demonstrate this point, we conduct additional experiments where we include
different numbers of overlap users into the recommendation process, and
discard the records from other unselected overlap users from the dataset.
These included overlap users are randomly selected from the pool of overlap
users. Our results in Figure 3 show that the proposed DML model indeed works
well with very few overlap users. When we do not include any overlap users
into the learning process, the cross-domain recommendation performance is not
particularly satisfying. When we start to include a few overlap users into the
cross-domain recommendation model, we would observe rapid and significant
improvements of recommendation performance. After we add more overlap users
into the model, we would witness the phenomenon of ”elbow point”, where
additional overlap users only incrementally improve the recommendation
performance. In particular, when we only include 8 overlap users and use DML
model to provide cross-domain recommendations, we still achieve satisfying
recommendation performance and outperform all the baseline models that utilize
all the overlap users, though the improvements are not as much as those when
we also include all overlap information in the DML model. Eventually when we
utilize all the overlap users in the datasets, we achieve the state-of-the-art
recommendation performance and outperform all other baselines consistently and
significantly. This observation validates our claim and significantly enhances
the practicality of our proposed model.

(a) Book-Movie Domains

(b) Beauty-Fashion Domains

Figure 3: Cross-Domain Recommendation Performance versus Number of Overlap
Users Included

###  6.3 Impact of Including/Excluding Explicit Feature Information

In our experiments, we collect and include explicit feature information of
users and items into the recommendation process, and we manage to achieve
satisfying recommendation performance. However, in some recommendation
scenarios, explicit features might not be available, which could potentially
jeopardize the recommendation performance. To test for its impact, we conduct
an ablation study in this paper in which we exclude all the explicit features
in the learning process, and the latent embeddings are generated directly from
the user and item IDs correspondingly. We also exclude explicit features from
our selected baselines during their training process for the fair comparison.
As we show in Table IX, in the case when no feature information is used, our
proposed DML model still significantly and consistently outperforms the
baseline approaches in terms of all four evaluation metrics, illustrating the
superiority and the robustness of our proposed approach. In addition, we
observe that the recommendation performance is significantly worse than that
in Table V, demonstrating the importance of including explicit feature
information for solving cross domain recommendation tasks.

Algorithm | Book Dataset | Movie Dataset
---|---|---
RMSE | MAE | Pre@5 | Rec@5 | RMSE | MAE | Pre@5 | Rec@5
DML | 0.2395* | 0.1801* | 0.8644* | 0.9535* | 0.2382* | 0.1888* | 0.8714* | 0.9622*
Improved % | (+6.73%) | (+9.81%) | (+1.97%) | (+3.21%) | (+5.10%) | (+5.46%) | (+1.46%) | (+3.27%)
DDTCDR | 0.2568 | 0.1997 | 0.8477 | 0.9238 | 0.2510 | 0.1997 | 0.8589 | 0.9317
CML | 0.2712 | 0.2114 | 0.8011 | 0.8710 | 0.2689 | 0.2115 | 0.8322 | 0.9008
CDFM | 0.2791 | 0.2326 | 0.7739 | 0.8499 | 0.2571 | 0.2169 | 0.8180 | 0.8999
CMF | 0.3112 | 0.2701 | 0.7739 | 0.8345 | 0.2984 | 0.2500 | 0.8009 | 0.8733
CoNet | 0.2611 | 0.2099 | 0.8150 | 0.8866 | 0.2498 | 0.2124 | 0.8374 | 0.9298
NCF | 0.2635 | 0.2135 | 0.8293 | 0.8798 | 0.2588 | 0.2121 | 0.8222 | 0.9079
TABLE IX: Comparison of Cross-Domain Recommendation Performance Excluding
Feature Information. * stands for 95% significance.

###  6.4 Impact of Different Recommendation Networks for Recommendations

To validate that the improvement gained in our model is not sensitive to the
specific selection of neural recommendation models, we also conduct additional
experiments to examine the change of recommendation performance corresponding
to different choices of $RS_{A}$ and $RS_{B}$, including MLP that we use in
this paper, CNN [68], RNN [69], Neural Attention [70] and Adversarial Network
[71] to learn user preferences and produce final recommendations. As shown in
Table X, the cross-domain recommendation performance between book and movie
domains is generally consistent across all the different types of
autoencoders, albeit some small numerical differences in recommendation
performance take place where the neural attention model performs slightly
better than other models - due to its capability to capture dynamic user
preferences. Nevertheless, the proposed DML model works well in all these
experimental settings, and significantly outperforms all the considered cross-
domain recommendation baselines.

Recommendation Network | Book Dataset | Movie Dataset
---|---|---
RMSE | MAE | Pre@5 | Rec@5 | RMSE | MAE | Pre@5 | Rec@5
Multi-Layer Perceptron | 0.2184 | 0.1646 | 0.8826 | 0.9850 | 0.2109 | 0.1606 | 0.9002 | 0.9962
Convolutional Neural Network | 0.2186 | 0.1648 | 0.8823 | 0.9848 | 0.2107 | 0.1609 | 0.8998 | 0.9962
Recurrent Neural Network | 0.2184 | 0.1648 | 0.8830 | 0.9850 | 0.2111 | 0.1606 | 0.9004 | 0.9952
Neural Attention | 0.2165 | 0.1632 | 0.8847 | 0.9858 | 0.2097 | 0.1587 | 0.9020 | 0.9980
Adversarial Network | 0.2192 | 0.1653 | 0.8815 | 0.9848 | 0.2119 | 0.1609 | 0.8998 | 0.9952

TABLE X: Comparison of Different Recommendation Networks for Cross-Domain
Recommendations in the Imhonet dataset

###  6.5 Impact of Different Types of AutoEncoder for Feature Embeddings

To validate that the improvement gained in our model is not sensitive to the
specific selection of the autoencoder network, we also conduct additional
experiments to examine the change of recommendation performance corresponding
to different types of autoencoders, including AE [72], VAE [73], AAE [74], WAE
[75] and HVAE [76] to construct the feature embeddings for users and items. As
shown in Table XI, the cross-domain recommendation performance remains
consistent across all the different selection of autoencoder models, where we
do not witness any significant differences in any evaluation metrics. This
observation also holds true other recommendation tasks in our experiments.
Therefore, the choice of particular autoencoder method is not relevant to the
cross-domain recommendation performance in our proposed model.

Types of AutoEncoder | Book Dataset | Movie Dataset
---|---|---
RMSE | MAE | Pre@5 | Rec@5 | RMSE | MAE | Pre@5 | Rec@5
AutoEncoder | 0.2184 | 0.1646 | 0.8826 | 0.9850 | 0.2109 | 0.1606 | 0.9002 | 0.9962
Variational AutoEncoder | 0.2190 | 0.1654 | 0.8796 | 0.9848 | 0.2107 | 0.1609 | 0.8998 | 0.9948
Adversarial AutoEncoder | 0.2186 | 0.1641 | 0.8805 | 0.9836 | 0.2115 | 0.1599 | 0.9008 | 0.9952
Wasserstein AutoEncoder | 0.2186 | 0.1649 | 0.8822 | 0.9832 | 0.2119 | 0.1597 | 0.9002 | 0.9952
Hyperspherical Variational AutoEncoder | 0.2190 | 0.1641 | 0.8816 | 0.9848 | 0.2104 | 0.1606 | 0.8998 | 0.9960

TABLE XI: Comparison of Autoencoder Settings for Cross-Domain Recommendations
in the Imhonet dataset

###  6.6 Sensitivity Analysis

As we theoretically discussed in Section 4.1, the number of overlap users
required by the proposed model for producing cross-domain recommendations
would be determined by the dimension of latent embeddings. If the size of the
embedding is very large, we might still need to obtain a large number of
overlap users. Therefore, we conduct additional analysis in this section to
explore sensitivity of the dimension of latent embeddings towards cross domain
recommendation performance. In particular, we present the recommendation
performance for the Imhonet dataset in Figure 4 using the dimension sizes of
4, 8, 16, 32, 64, 128 and 256 respectively to learn the user and item
embeddings. We observe that increasing the size of the latent dimensions would
indeed improve the recommendation performance in general. However, these
improvements are only marginal in their absolute values, and we can achieve
reasonable and satisfying cross-domain recommendation using 16 or 32
dimensions of latent vectors. Therefore, our theoretical analysis could be
applied to our cross-domain recommendation tasks, and ablation studies in the
previous section further confirm that we could provide satisfying
recommendation performance using as little as 8 overlap users. To sum up, our
proposed model would not be sensitive to the specific selection of the
dimension size.

(a) Book-Movie Domains

(b) Beauty-Fashion Domains

Figure 4: Sensitivity Analysis of Different Sizes of Latent Embeddings of the
DML model.

###  6.7 Scalability Analysis

To test for scalability, we provide cross-domain recommendations using DML
with aforementioned hyperparameter values for the Imhonet dataset of Book,
Movie and Music domains with increasing data sizes from 100 to 1,000,000
records. As shown in Figure 5, we empirically observe that the proposed DML
model scales linearly with the increase in the number of data records for the
model training process. The training procedure comprises of generating feature
embeddings for users and items, as well as the dual metric learning stage
jointly optimized across different domains, as we describe in Section 3. The
optimization phase is made efficient using batch normalization [77] and
distributed training. As our experiments confirm, DML is capable of providing
cross-domain recommendations efficiently and indeed scales well.

(a) Book-Movie Domains

(b) Beauty-Fashion Domains

Figure 5: Scalability Analysis of the DML model with increasing data sizes
from 100 to 1,000,000 records.

###  6.8 Convergence Analysis

While we have provided the technical foundations of the convergence
proposition of the dual learning method under the classical setting in Section
4.2, this proposition might not be directly applicable to our proposed model,
as we utilize deep learning techniques for which the optimization process is
different. Therefore, we test the convergence empirically in our study, and
conjecture that it should still happen even though our method does not
explicitly satisfy the condition. In particular, we train the model
iteratively for 100 epochs until the change of loss function is less than
1e-5. We plot the training loss over time for our cross-domain recommendation
tasks in Figure 6. The key observation is that, DML starts with relatively
higher loss due to the noisy initialization. As times goes by, DML manages to
stabilize quickly and significantly achieves satisfying recommendation
performance only after 10 epochs. Therefore, we empirically validate that the
proposed DML indeed converges in our experimental settings.

(a) Book-Movie Domains

(b) Beauty-Fashion Domains

Figure 6: Convergence Analysis of the DML model with Different Training
Epochs.

##  7 Conclusion

In this paper, we propose a novel dual metric learning based model DML that
significantly improves recommendation performance across different domains and
reduces the requirement of overlapping user information. We accomplish this by
proposing a novel latent orthogonal metric mapping and iteratively going
through the learning loop until the models stabilize for both domains. We also
prove that this convergence is guaranteed under certain conditions and
empirically validate the hypothesis for our model across different
experimental settings. We show that the proposed model could be extended for
multi-domain tasks.

Note that the proposed approach provides several benefits, including that it
(a) transfers information about latent interactions instead of explicit
features from the source domain to the target domain; (b) utilizes the dual
learning mechanism to enable the bidirectional training that improves
performance measures for both domains simultaneously; (c) learns the latent
orthogonal mapping function across two domains, that (i) preserves similarity
of user preferences and thus enables proper transfer learning and (ii)
computes the inverse mapping function efficiently; (d) works well with very
few overlap users (as low as 8 in our experiments) and obtains the results
comparable to the state-of-the-art baselines. This makes our approach more
practical vis-a-vis the previously proposed cross-domain recommendation models
because many business applications have very few overlapping users.

As the future work, we plan to further study the application of metric
learning to provide even better cross-domain recommendations. In addition, we
plan to extend the convergence proposition to more general settings.

## References

  * [1] I. Cantador, I. Fernández-Tobías, S. Berkovsky, and P. Cremonesi, “Cross-domain recommender systems,” in _Recommender systems handbook_. Springer, 2015, pp. 919–959.
  * [2] I. Fernández-Tobías, I. Cantador, M. Kaminskas, and F. Ricci, “Cross-domain recommender systems: A survey of the state of the art,” 2012.
  * [3] A. I. Schein, A. Popescul, L. H. Ungar, and D. M. Pennock, “Methods and metrics for cold-start recommendations,” in _Proceedings of the 25th annual international ACM SIGIR conference on Research and development in information retrieval_. ACM, 2002, pp. 253–260.
  * [4] G. Adomavicius and A. Tuzhilin, “Toward the next generation of recommender systems: A survey of the state-of-the-art and possible extensions,” _IEEE Transactions on Knowledge & Data Engineering_, no. 6, pp. 734–749, 2005.
  * [5] A. P. Singh and G. J. Gordon, “Relational learning via collective matrix factorization,” in _Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining_. ACM, 2008, pp. 650–658.
  * [6] B. Li, Q. Yang, and X. Xue, “Can movies and books collaborate? cross-domain collaborative filtering for sparsity reduction,” in _Twenty-First International Joint Conference on Artificial Intelligence_ , 2009.
  * [7] L. Hu, J. Cao, G. Xu, L. Cao, Z. Gu, and C. Zhu, “Personalized recommendation via cross-domain triadic factorization,” in _Proceedings of the 22nd international conference on World Wide Web_ , 2013.
  * [8] B. Loni, Y. Shi, M. Larson, and A. Hanjalic, “Cross-domain collaborative filtering with factorization machines,” in _European conference on information retrieval_. Springer, 2014, pp. 656–661.
  * [9] I. Fernández-Tobías, I. Cantador, P. Tomeo, V. W. Anelli, and T. Di Noia, “Addressing the user cold start with cross-domain collaborative filtering: exploiting item metadata in matrix factorization,” _User Modeling and User-Adapted Interaction_ , vol. 29, no. 2, pp. 443–486, 2019.
  * [10] Y. Wang, C. Feng, C. Guo, Y. Chu, and J.-N. Hwang, “Solving the sparsity problem in recommendations via cross-domain item embedding based on co-clustering,” in _Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining_ , 2019, pp. 717–725.
  * [11] S. J. Pan and Q. Yang, “A survey on transfer learning,” _IEEE Transactions on knowledge and data engineering_ , 2010.
  * [12] G. Hu, Y. Zhang, and Q. Yang, “Conet: Collaborative cross networks for cross-domain recommendation,” in _Proceedings of the 27th ACM International Conference on Information and Knowledge Management_ , 2018.
  * [13] J. Lian, F. Zhang, X. Xie, and G. Sun, “Cccfnet: a content-boosted collaborative filtering neural network for cross domain recommender systems,” in _Proceedings of the 26th international conference on World Wide Web companion_ , 2017, pp. 817–818.
  * [14] P. Li and A. Tuzhilin, “Ddtcdr: Deep dual transfer cross domain recommendation,” in _Proceedings of the 13th International Conference on Web Search and Data Mining_ , 2020, pp. 331–339.
  * [15] C. Gao, X. Chen, F. Feng, K. Zhao, X. He, Y. Li, and D. Jin, “Cross-domain recommendation without sharing user-relevant data,” in _The World Wide Web Conference_ , 2019, pp. 491–502.
  * [16] T. Qin, _Dual Learning_. Springer Nature, 2020.
  * [17] E. Zhong, W. Fan, J. Peng, K. Zhang, J. Ren, D. Turaga, and O. Verscheure, “Cross domain distribution adaptation via kernel mapping,” in _Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining_. ACM, 2009.
  * [18] M. Long, J. Wang, G. Ding, W. Cheng, X. Zhang, and W. Wang, “Dual transfer learning,” in _Proceedings of the 2012 SIAM International Conference on Data Mining_. SIAM, 2012.
  * [19] H. Wang, H. Huang, F. Nie, and C. Ding, “Cross-language web page classification via dual knowledge transfer using nonnegative matrix tri-factorization,” in _Proceedings of the 34th international ACM SIGIR conference on Research and development in Information Retrieval_. ACM, 2011, pp. 933–942.
  * [20] D. He, Y. Xia, T. Qin, L. Wang, N. Yu, T.-Y. Liu, and W.-Y. Ma, “Dual learning for machine translation,” in _Advances in Neural Information Processing Systems_ , 2016, pp. 820–828.
  * [21] P. Cremonesi and M. Quadrana, “Cross-domain recommendations without overlapping data: myth or reality?” in _Proceedings of the 8th ACM Conference on Recommender systems_ , 2014, pp. 297–300.
  * [22] L. Yang and R. Jin, “Distance metric learning: A comprehensive survey,” _Michigan State Universiy_ , vol. 2, no. 2, p. 4, 2006.
  * [23] C.-K. Hsieh, L. Yang, Y. Cui, T.-Y. Lin, S. Belongie, and D. Estrin, “Collaborative metric learning,” in _Proceedings of the 26th international conference on world wide web_ , 2017, pp. 193–201.
  * [24] B. Kulis _et al._ , “Metric learning: A survey,” _Foundations and trends in machine learning_ , vol. 5, no. 4, pp. 287–364, 2012.
  * [25] Q. Do, W. Liu, J. Fan, and D. Tao, “Unveiling hidden implicit similarities for cross-domain recommendation,” _IEEE Transactions on Knowledge and Data Engineering_ , 2019.
  * [26] F. Zhu, C. Chen, Y. Wang, G. Liu, and X. Zheng, “Dtcdr: A framework for dual-target cross-domain recommendation,” in _Proceedings of the 28th ACM International Conference on Information and Knowledge Management_ , 2019, pp. 1533–1542.
  * [27] S. Sahebi, P. Brusilovsky, and V. Bobrokov, “Cross-domain recommendation for large-scale data,” in _CEUR Workshop Proceedings_ , vol. 1887, 2017, pp. 9–15.
  * [28] H. Wu, Z. Zhang, K. Yue, B. Zhang, J. He, and L. Sun, “Dual-regularized matrix factorization with deep neural networks for recommender systems,” _Knowledge-Based Systems_ , vol. 145, pp. 46–58, 2018.
  * [29] C. Zhao, C. Li, and C. Fu, “Cross-domain recommendation via preference propagation graphnet,” in _Proceedings of the 28th ACM International Conference on Information and Knowledge Management_ , 2019, pp. 2165–2168.
  * [30] H. Kanagawa, H. Kobayashi, N. Shimizu, Y. Tagami, and T. Suzuki, “Cross-domain recommendation via deep domain adaptation,” in _European Conference on Information Retrieval_. Springer, 2019, pp. 20–29.
  * [31] W. Fu, Z. Peng, S. Wang, Y. Xu, and J. Li, “Deeply fusing reviews and contents for cold start users in cross-domain recommendation systems,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 33, 2019, pp. 94–101.
  * [32] G. Hu, Y. Zhang, and Q. Yang, “Transfer meets hybrid: A synthetic approach for cross-domain collaborative filtering with text,” in _The World Wide Web Conference_ , 2019, pp. 2822–2829.
  * [33] F. Zhuang, P. Luo, Z. Shen, Q. He, Y. Xiong, Z. Shi, and H. Xiong, “Collaborative dual-plsa: mining distinction and commonality across multiple domains for text classification,” in _Proceedings of the 19th ACM international conference on Information and knowledge management_. ACM, 2010, pp. 359–368.
  * [34] M. Long, J. Wang, G. Ding, J. Sun, and P. S. Yu, “Transfer feature learning with joint distribution adaptation,” in _Proceedings of the IEEE ICCV_ , 2013\.
  * [35] T. Song, Z. Peng, S. Wang, W. Fu, X. Hong, and S. Y. Philip, “Based cross-domain recommendation through joint tensor factorization,” in _International Conference on Database Systems for Advanced Applications_. Springer, 2017, pp. 525–540.
  * [36] C. Zhao, C. Li, R. Xiao, H. Deng, and A. Sun, “Catn: Cross-domain recommendation for cold-start users via aspect transfer network,” in _Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval_ , 2020, pp. 229–238.
  * [37] A. Krishnan, M. Das, M. Bendre, H. Yang, and H. Sundaram, “Transfer learning via contextual invariants for one-to-many cross-domain recommendation,” in _Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval_ , 2020, pp. 1081–1090.
  * [38] Y. Xia, T. Qin, W. Chen, J. Bian, N. Yu, and T.-Y. Liu, “Dual supervised learning,” in _Proceedings of the 34th International Conference on Machine Learning-Volume 70_ , 2017.
  * [39] Y. Wang, Y. Xia, L. Zhao, J. Bian, T. Qin, G. Liu, and T.-Y. Liu, “Dual transfer learning for neural machine translation with marginal distribution regularization,” in _Thirty-Second AAAI Conference on Artificial Intelligence_ , 2018.
  * [40] E. P. Xing, M. I. Jordan, S. J. Russell, and A. Y. Ng, “Distance metric learning with application to clustering with side-information,” in _Advances in neural information processing systems_ , 2003, pp. 521–528.
  * [41] D. Kedem, S. Tyree, F. Sha, G. R. Lanckriet, and K. Q. Weinberger, “Non-linear metric learning,” in _Advances in neural information processing systems_ , 2012, pp. 2573–2581.
  * [42] J. Wang, H. T. Do, A. Woznica, and A. Kalousis, “Metric learning with multiple kernels,” in _Advances in neural information processing systems_ , 2011, pp. 1170–1178.
  * [43] M. Khoshneshin and W. N. Street, “Collaborative filtering via euclidean embedding,” in _Proceedings of the fourth ACM conference on Recommender systems_ , 2010, pp. 87–94.
  * [44] S. Chen, J. L. Moore, D. Turnbull, and T. Joachims, “Playlist prediction via metric embedding,” in _Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining_ , 2012, pp. 714–722.
  * [45] S. Feng, X. Li, Y. Zeng, G. Cong, Y. M. Chee, and Q. Yuan, “Personalized ranking metric embedding for next new poi recommendation,” in _Twenty-Fourth International Joint Conference on Artificial Intelligence_ , 2015.
  * [46] Y. Tay, L. Anh Tuan, and S. C. Hui, “Latent relational metric learning via memory-based attention for collaborative ranking,” in _Proceedings of the 2018 World Wide Web Conference_ , 2018, pp. 729–739.
  * [47] C. Park, D. Kim, X. Xie, and H. Yu, “Collaborative translational metric learning,” in _2018 IEEE International Conference on Data Mining (ICDM)_. IEEE, 2018, pp. 367–376.
  * [48] L. Vinh Tran, Y. Tay, S. Zhang, G. Cong, and X. Li, “Hyperml: A boosting metric learning approach in hyperbolic space for recommender systems,” in _Proceedings of the 13th International Conference on Web Search and Data Mining_ , 2020, pp. 609–617.
  * [49] S. Zhang, L. Yao, A. Sun, and Y. Tay, “Deep learning based recommender system: A survey and new perspectives,” _ACM Computing Surveys (CSUR)_ , vol. 52, no. 1, p. 5, 2019.
  * [50] H. Wang, N. Wang, and D.-Y. Yeung, “Collaborative deep learning for recommender systems,” in _Proceedings of the 21th ACM SIGKDD_. ACM, 2015, pp. 1235–1244.
  * [51] X. He, L. Liao, H. Zhang, L. Nie, X. Hu, and T.-S. Chua, “Neural collaborative filtering,” in _Proceedings of the 26th International Conference on World Wide Web_ , 2017, pp. 173–182.
  * [52] S. Sedhain, A. K. Menon, S. Sanner, and L. Xie, “Autorec: Autoencoders meet collaborative filtering,” in _Proceedings of the 24th International Conference on World Wide Web_. ACM, 2015\.
  * [53] Y. Wu, C. DuBois, A. X. Zheng, and M. Ester, “Collaborative denoising auto-encoders for top-n recommender systems,” in _Proceedings of the Ninth ACM WSDM_ , 2016.
  * [54] X. Li and J. She, “Collaborative variational autoencoder for recommender systems,” in _Proceedings of the 23rd ACM SIGKDD_. ACM, 2017, pp. 305–314.
  * [55] D. Liang, R. G. Krishnan, M. D. Hoffman, and T. Jebara, “Variational autoencoders for collaborative filtering,” in _Proceedings of the 2018 World Wide Web Conference on World Wide Web_. International World Wide Web Conferences Steering Committee, 2018\.
  * [56] P. Li and A. Tuzhilin, “Latent multi-criteria ratings for recommendations,” in _Proceedings of the 13th ACM Conference on Recommender Systems_ , 2019, pp. 428–431.
  * [57] P. Li and A. Tuzhilin, “Learning latent multi-criteria ratings from user reviews for recommendations,” _IEEE Transactions on Knowledge and Data Engineering_ , pp. 1–1, 2020.
  * [58] P. Li, M. Que, Z. Jiang, Y. Hu, and A. Tuzhilin, “Purs: Personalized unexpected recommender system for improving user satisfaction,” in _Fourteenth ACM Conference on Recommender Systems_ , 2020, pp. 279–288.
  * [59] C. D. Meyer, _Matrix analysis and applied linear algebra_. Siam, 2000, vol. 71.
  * [60] C.-J. Lin, “On the convergence of multiplicative update algorithms for nonnegative matrix factorization,” _IEEE Transactions on Neural Networks_ , vol. 18, no. 6, pp. 1589–1596, 2007.
  * [61] D. D. Lee and H. S. Seung, “Algorithms for non-negative matrix factorization,” in _Advances in neural information processing systems_ , 2001, pp. 556–562.
  * [62] V. Bobrikov, E. Nenova, and D. I. Ignatov, “What is a fair value of your recommendation list?” in _Proceedings of the Third Workshop on Experimental Economics and Machine Learning co-located with the 13th International Conference on Concept Lattices and Their Applications (CLA 2016), Moscow, Russia, July 18, 2016_ , 2016, pp. 1–12.
  * [63] J. Ni, J. Li, and J. McAuley, “Justifying recommendations using distantly-labeled reviews and fine-grained aspects,” in _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , 2019, pp. 188–197.
  * [64] F. Ricci, L. Rokach, and B. Shapira, “Introduction to recommender systems handbook,” in _Recommender systems handbook_. Springer, 2011, pp. 1–35.
  * [65] J. Snoek, H. Larochelle, and R. P. Adams, “Practical bayesian optimization of machine learning algorithms,” in _Advances in Neural Information Processing Systems_ , F. Pereira, C. J. C. Burges, L. Bottou, and K. Q. Weinberger, Eds. Curran Associates, Inc., 2012, pp. 2951–2959.
  * [66] Å. Björck, “Numerics of gram-schmidt orthogonalization,” _Linear Algebra and Its Applications_ , vol. 197, pp. 297–316, 1994.
  * [67] N. Bansal, X. Chen, and Z. Wang, “Can we gain more from orthogonality regularizations in training deep networks?” in _Advances in Neural Information Processing Systems_ , 2018, pp. 4261–4271.
  * [68] R. Ying, R. He, K. Chen, P. Eksombatchai, W. L. Hamilton, and J. Leskovec, “Graph convolutional neural networks for web-scale recommender systems,” in _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, 2018, pp. 974–983.
  * [69] B. Hidasi, A. Karatzoglou, L. Baltrunas, and D. Tikk, “Session-based recommendations with recurrent neural networks,” _arXiv preprint arXiv:1511.06939_ , 2015.
  * [70] Q. Zhu, X. Zhou, Z. Song, J. Tan, and L. Guo, “Dan: Deep attention neural network for news recommendation,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 33, no. 01, 2019, pp. 5973–5980.
  * [71] Q. Wang, H. Yin, Z. Hu, D. Lian, H. Wang, and Z. Huang, “Neural memory streaming recommender networks with adversarial training,” in _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, 2018, pp. 2467–2475.
  * [72] I. Sutskever, O. Vinyals, and Q. V. Le, “Sequence to sequence learning with neural networks,” in _Advances in neural information processing systems_ , 2014, pp. 3104–3112.
  * [73] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” _arXiv preprint arXiv:1312.6114_ , 2013.
  * [74] A. Makhzani, J. Shlens, N. Jaitly, I. Goodfellow, and B. Frey, “Adversarial autoencoders,” _arXiv preprint arXiv:1511.05644_ , 2015.
  * [75] I. Tolstikhin, O. Bousquet, S. Gelly, and B. Schölkopf, “Wasserstein auto-encoders,” in _6th International Conference on Learning Representations (ICLR 2018)_. OpenReview. net, 2018.
  * [76] T. R. Davidson, L. Falorsi, N. De Cao, T. Kipf, and J. M. Tomczak, “Hyperspherical variational auto-encoders,” _arXiv preprint arXiv:1804.00891_ , 2018.
  * [77] S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in _International conference on machine learning_ , 2015, pp. 448–456.

|  Pan Li Pan Li is a PhD student in the department of Technology, Operation
and Statistics, Stern School of Business, New York University. His research
interests include recommender systems and business analytics. He has published
papers in international conferences and journals including WSDM, EMNLP,
RecSys, AAAI, IEEE TKDE, ACM TIST and ACM TMIS.
---|---
|  Alexander Tuzhilin Alexander Tuzhilin is the Leonard N. Stern Professor of
Business in the Department of Technology, Operations and Statistics at the
Stern School of Business, NYU. His research interests include personalization,
recommender systems, machine learning and AI, where he has produced over 150
research publications. He has served on the organizing committees of numerous
conferences, including as the Program and the General Chair of ICDM, and as
the Program and the Conference Chair of RecSys. He has also served on the
editorial boards of several journals, including as the Editor-in-Chief of the
ACM TMIS.
---|---