Mining User Interests for Personalized Tweet Recommendation on
Map-Reduce Framework
Guanyao Du
1
, Jianying Sun
1
, Xinglong Huang
2
and Jianjun Yu
1
1
Computer Network Information Center, Chinese Academy of Sciences, 4 Zhongguancun Nansijie,
Haidian District, Beijing 100190, China
2
MailBox No.28, 87th Xiangshannanlu, Beijing 100093, China
Keywords: Recommender System, User Interests, Topic-STG, SVD, Big Data, Map-Reduce.
Abstract: The tremendous growth of micro-blogging systems in recent years poses some key challenges for
recommender systems, such as how to process tweet big data under distributed environment, how to striking
a balance between high accurate recommendations and efficiency, and how to produce diverse
recommendations for millions of users. In our opinion, accurately, instantly, and completely capturing user
preferences over time is the key point for personalized tweet recommendation. Therefore, we introduce
three features to model personal user interests and its evolution for tweet recommendation, including textual
information, user behaviors, and time. We then offer two enhanced recommendation models: Topic-STG
(Session-based Temporal Graph) model and SVD (Singular Value Decomposition) model, combining these
features to learn user preference and recommend personalized tweet. To further improve the algorithm
efficiency for micro-blogging big data, we provide the parallel algorithm implementation for Topic-STG
and SVD models based on Hadoop Map-Reduce framework. Experiments on a large scale of micro-
blogging dataset illustrate the effectiveness of the proposed models and algorithms.
1 INTRODUCTION
As a convenient communication means, especially
with smart phones, the micro-blogging systems not
only act as the role of social relation between
people, but also as important sources for people to
obtain useful information. Currently, there are more
than 400 million messages generated on Twitter
from 500 million users, and 100 million Chinese
messages on Sina Weibo (a Chinese Twitter) each
day from 249 million users. Such enormous users
ceaselessly chase and produce a large amount of
information. It benefits the users but also can flood
users and hence puts them at the risk of information
overload.
Recommender system is a powerful tool to
address the information overload problem (Xavier
and Justin, 2016; Roberto et al., 2016). Much
previous work has been proposed to recommend
different objects on micro-blogging systems in
recent years. Some of them investigated the content-
based recommendation approaches, which were the
fundamental mechanisms on micro-blogging
systems, and were easy to be applied for hashtag
recommendation (Hannon et al., 2010). Others
considered the recommendation in social networks,
in which the social structure model was very useful
and helpful (Yigita et al., 2015). Currently,
combining several features and correspondingly
providing linear recommendation models is also a
prevailing way on tweet recommendation (Yin et al.,
2015). To easily incorporate information such as
temporal dynamics, neighborhood relationship, and
hierarchical information for recommendation, SVD
series (Koren, 2010), SVDFeature (Chen et al.,
2012), and other approaches (Jiang et al., 2014)
provided scalable framework to efficiently solve
large-scale collaborative filtering problems with
auxiliary information using matrix factorization
techniques for recommender systems.
Based on the probabilistic matrix factorization
technique (Salakhutdinov and Mnih, 2007) which
can offer a uniform and scalable framework to
model explicit user interests, some significant work
were presented to combine social factors (such as
personal interest, interpersonal interest similarity,
and interpersonal influence) together, and fused
them into a unified personalized recommendation
model. However, sometimes the user interests may
Du, G., Sun, J., Huang, X. and Yu, J.
Mining User Interests for Personalized Tweet Recommendation on Map-Reduce Framework.
DOI: 10.5220/0006274102010208
In Proceedings of the 19th International Conference on Enterprise Information Systems (ICEIS 2017) - Volume 1, pages 201-208
ISBN: 978-989-758-247-9
Copyright © 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
201

be implicit (like our circumstance), and should be
inferred from user behaviors (such as @, hashtags,
or retweeting). Also sometimes it's difficult to model
the short-term and long-term user interests. For
example, in (Xiang et al., 2010), Xiang et al.
proposed a novel recommendation approach named
STG to model users' long-term and short-term
preferences over time, whereas, they neglected the
textual information of tweets, which presented rich
sentiment information to predict user preference. To
solve this problem, (Yu and Shen, 2014; Yu et al.,
2014; Yu and Zhu, 2015) involving tweet
recommendation summarized three features to
model user interests, and constructed the hybrid
model or Topic-STG model to learn user preference
for tweet recommendation consequently. In this
paper, we would enhance the SVD model mainly to
model explicit user interests, and provide the Topic-
STG model for implicit and temporal tweet
recommendation.
At the same time, recommending tweet based on
massive micro-blogging datasets is a typical “Big
Data” application since the recommender models are
computation-intensive and time-consuming.
Furthermore, the capability of micro-blogging
datasets is ever-increasing that the traditional
approaches would be difficult to process such large
datasets. A parallel version for recommender models
is expected since most of big data applications are
developed with cloud computing technique which
enables convenient, on-demand network access to a
shared pool of configurable computing resources. As
a result, a new platform of “Big Data” tools has
arisen to handle sense making over large quantities
of data, as in the Apache Hadoop. With Hadoop
parallel Map-Reduce framework, recommender
algorithms can be distributed in different computers
to accelerate computation.
“Big Data” is a new term used to identify
datasets that cannot be managed with current
methodologies or data mining software tools due to
their large size and complexity (Fan and Bifet, 2012;
Kumar et al., 2013; Moens et al., 2014). Specifically,
Kumar et al. gave a Hazy system to build and
maintain big-data analytics with the latest statistical
and machine-learning techniques (Kumar et al.,
2013). Moens et al. introduced Frequent Itemset
Mining (FIM) approaches on the Map-Reduce
platform balancing data distribution and inter-
communication costs (Moens et al., 2014).
As for the “Big Data” in domain applications,
Roy et al. developed a system for end-to-end
processing of genomic data, including alignment of
short read sequences, variation discovery, and deep
analysis (Roy et al., 2012). Chawla et al. provided a
“Big Data” driven approach towards personalized
healthcare, and demonstrated its applicability to
patient-centered outcomes (Chawla and Davis,
2013). Zaiane built an agent that recommended on-
line learning activities or shortcuts in a course web
site based on learners' access history to improve
course material navigation as well as assist the
online learning process (Zaiane, 2012). However,
most of the previous work involving cloud
computing architectures for “Big Data” solutions
only focused on common data mining platform or
common services for data processing, and
considered less about data mining based on
recommender system, especially for micro-blogging
data. This motivated us to do this work to provide
special mining processes for recommender
techniques.
In this paper, we consider the tweet
recommendation with three features: the tweet
textual information, the user's behavior, and the time
factor, and focus the “Big Data” problem with
massive micro-blogging dataset under parallel
computation model. The contributions can be
summarized as follows:
(1) We offer two models for tweet
recommendation by considering the long-term and
the short-term aspects to extract the top N tweets,
one is the extended Topic-STG (Session-based
Temporal Graph) model (Yu et al., 2014), and the
other is the SVD (Singular Value Decomposition)
model.
(2) To further improve the algorithm efficiency,
we develop the parallel versions of Topic-STG and
SVD models under Hadoop Map-Reduce framework,
and conduct comprehensive experiments to evaluate
the two techniques on a real large dataset, i.e., Sina
Weibo.
(3) The experimental results illustrate that the
introduced strategies outperform the state-of-the-art
approach by a wide margin. It also shows that the
Topic-STG model is more suitable for short-term
user interests mining when users' behavior is not
easy to capture, whereas the SVD model has more
advantages for long-term user interests mining with
explicit ratings.
ICEIS 2017 - 19th International Conference on Enterprise Information Systems
202

1
U
2
U
2
Tw
5
E
1
Tw
3
Tw
4
Tw
111
(,)SU t
212
(,)SU t
321
(,)SU t
4
E
3
E
2
E
1
E
6
E
7
E
8
E
9
E
10
E
2
T
n
T
1
T

1/ ( , )
Tw
U
WNUTw 

(,)
''
/( )
UTw Tw Tw
Tw U U S U t
US
WW W W

(,) (,) (,)
''
/( )
Tw Tw Tw Tw
SUt SUt U SUt
US
WW W W

(,)
1/ ( ( , ), )
SUt
Tw
WNSUtTw 

(|)
T
U
WPTU 
(,)
''
/( )
UT T T
TU U SUt
US
WW W W



(| )
T
Tw
WPTTw
'
(| )/ (| )'
Tw
T
tw
W PTTw PTTw


(,)
(|(,))
T
SUt
WPTSUt

(,)
(,) (,)
''
/( )
SUt T T T
TSUt U SUt
US
WW W W

Figure 1: An example of Topic-STG and the weights of edges.
2 TWO ENHANCED
RECOMMENDATION MODELS
2.1 Recommendation on Topic-STG
Model
Topic-STG is an extended approach which bridges
STG (Xiang et al., 2010) and textual information by
adding a “topic-node”' into the existing bipartite
graph.
Once a user visits or operates (e.g. “retweet”,
“comment”, and “favorite”) a tweet, he/she would
show implicit interest to some extent. This implicit
interest can be summarized as a “topic preference”.
Similar to STG approach, if a user
1
U
operates on a
tweet
1
Tw
, two pairs of edges (
1
E
-
4
E
) which
represent the connection between
1
U
and
1
Tw
with
specific weights will be created as shown in Fig. 1.
In Topic-STG, the topic-node is generated by LDA
(Latent Dirichlet Allocation). We train the latent
topics to infer tweets' topic distribution, long-term
and short-term topic distribution of users,
consequently six new correspondingly topic-related
edges (
5
E
-
10
E
) will be created which are the links
between the topic-node and the corresponding user-
node, tweet-node, and session-node. We should
mention that a tweet may present several topics, here
we just select the topic with the highest probability
to present the implicit meaning of the tweet for
simplicity. Given the weight of each edge in Fig. 1,
we could recommend candidate tweets to a user U at
a timestamp t. Due to the space limitations, the
specific steps of the recommendation approach are
omitted here, please refer to (Yu et al., 2014) and
(Yu and Zhu, 2015) to get the details.
2.2 Recommendation on SVD Model
SVD is a matrix factorization technique commonly
used for producing low-rank approximations, which
produces results that are better than a traditional
collaborative filtering algorithm most of the time
when applied to the dataset with explicit ratings,
such as movie, music ratings datasets. SVD also
solves a general form of collaborative problems, and
thus allows develop new models just by defining
new features, which can easily incorporate
information such as temporal dynamics,
neighbourhood relationship, and hierarchical
information into the SVD model.
SVD maps both users and items (tweets) to a
joint latent factor space of dimensionality
f
, such
that user-item interactions are modeled as inner
products in that space, which approximates user
u 's
rating of item
i , and can be denoted by
ˆ
ui
r
.
ˆ
T
ui i u i u
rbbqp  
(1)
Here, the observed rating is broken down into its
four components: global average

, item bias
i
b
,
Mining User Interests for Personalized Tweet Recommendation on Map-Reduce Framework
203

user bias
u
b
, and user-item interaction
T
iu
qp
. SVD
gives a basic model for producing product
recommendations. We extend this model by
considering three features of tweets.
As for textual information, there are two
important sources to model user preference: 1)
Hashtag, which covers almost the whole user
interests; 2) the tweets' keywords, which can present
the user interest on a special topic. Hence we get:

||




1
1
2
(, ) ( ) () ()
ukw tg
kw K u n T u
pWukw ykwTu yu
(2)





1
1
2
(, ) ( ) | ()| ()
ikw tg
kw K i n T i
qWikw ykwTi yi
(3)
where
(*, )Wkw
gives the weights of keywords kw ,
(*)K
and
(*)T
present the keywords and hashtags
sets respectively.
As we observed, there are two types of
operations on the micro-blogging systems: the
attribute one and the non-attribute one. The attribute
one is the operation that may affect user interests via
the operation frequency, such as retweet, comment.

2
,
2
1/
uuujj
jAu
py 



(4)
where
,uj

is the normalized operation count.
The non-attribute operation is the one that the
operation frequency would not affect the user
interests, such as favorite, add friends.




3
11
rd
uj j
jNu jNu
py y
Nu Nu



 
(5)
Considering that a user's friends would also
affect his/her interest, we define a decay factor

to
model this relationship, and set
2rd 
.
According to our previous work, we found that
the short-term user interests may always change in
one week, and the minimum unit of this change is
one day. Thus the time factor can be modeled as
follows:
{}, [0,7)
day ti
bbi
(6)
Finally, we get an enhanced SVD model for
recommendation:
1123
ˆ
()( )
T
ui ui day i i u u u u
rbb qqpppp  
(7)
3 PARALLEL COMPUTING
IMPLEMENTATION
The two recommender parallel computing models
are developed with Map-Reduce component, which
is a programming model for processing large data
sets and used to do distributed computing on clusters
of computers. Since Map-Reduce is a common
parallel computing model, we need to encapsulate
data access interfaces and mining models to process
key/value pairs. We acquire micro-blogging data
stored in HDFS, and implement Map-Reduce
functions separately. It needs to be emphasized that
the mining algorithms are data-insensitive, which
means that the map function can read each record
randomly and do not affect the final result.
3.1 Topic-STG Map-Reduce
Implementation
The Topic-STG parallel implementation based on
Map-Reduce framework mainly develops map and
reduce functions. The map function scans each
records of micro-blogging datasets, calculates the
score of each path of Topic-STG, and finally
outcomes the preference of user on tweet. The
reduce function summarizes all preferences, and
ranks top N recommendation results. The details are
described in Algorithm 1.
The sentence must end with a period. As we
mentioned, Topic-STG would adopts LDA to
generate topic node, so we need to realize LDA
parallel version on Map-Reduce framework. The
LDA parallel version would use Mahout toolkit
since it builds a scalable machine learning library
based on Hadoop.
Algorithm 1. Map-Reduce implementation for Topic-
STG model.
1. set input file in HDFS and read into values
2. map(key, values, OuptputCollector output)
3. {
4. for score of each path:
()P 
do
5. computing preference of u on tweet tw:
Tw
U
p
selecting
shortest paths between
U and Tw
6. output.collect(key,{values.user.ID,
Tw
U
p
})
7. }
8.
9. reduce(key, values, OuptputCollector output)
10. {
11. for all users values.user.ID in {values} do
12. select Top N results from
Tw
U
p
13. output.collect(key,{values.user.ID,
Tw
U
p
})
14. }
15. set reduce result to output file
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204

3.2 SVD Map-Reduce Implementation
Mahout provides a standard SVD implementation,
we need to calculate a new
ˆ
ui
r
with modified
parameters:
ui
b
,
day
b
,
i
q
and
u
p
. The details are in
Algorithm 2.
Algorithm 2. Map-Reduce implementation for
enhanced SVD model.
1. set input file in HDFS and read into values
2. map(key, values, OuptputCollector output)
3. {
4. for each rating score do
5. calculating the difference between rating score and
predict score:
ui
e score preScore 
6. using
ui
e
to update the increments of
[],[],
ui
b uid b iid [][],[][]
iu
q uid k p iid k
7. output.collect({uid,iid}, flag(
ui
b
,
day
b
,
i
q
,
u
p
))
8. }
9.
10. reduce(key, values, OuptputCollector output)
11. {
12. for each key in{uid,iid} do
13.
ui
b
ui
b 

,
day day
bb

,
ii
qq

,
uu
pp

14. output.collect(uid,iid,
ui
b
,
day
b
, (
i
q
,
u
p
))
15. }
16. set reduce result to output file
4 EXPERIMENTAL
EVALUATION
We crawled 811,586 original tweets with 100 initial
users on Sina Weibo micro-blogging system after
filtering out those inactive accounts and spammers
to get a dense dataset from 2015/02/01 to
2015/11/01. There are 30,641 new-created tweets,
780,945 retweets, 1,852,562 comments, 68,327
favorites, 99,762 friendships, 41,127 hashtags,
19,023 2-degree users, 82,435 keywords in the
collected dataset. We chose our training dataset from
2015/02/01 to 2015/07/31, and the remains as the
test dataset. Considering the particularity of the
Chinese micro-blogging system, we generate these
Chinese terms from several basic corpuses,
including Sogou Pinyin input dict, NLPIR micro-
blogging corpus. Moreover, to avoid the possible
bias of training user preference, we chose 30 users
from our dataset, and ''@'' them with those
personalized recommendations. These volunteers are
active users in Weibo from different majors, jobs,
and ages with different interest. The volunteers show
their best effort helping to feedback whether they
were really interested in the recommendations.
Figure 2: Tweet recommendation precision with different
approaches.
When training the LDA parameters, we figure out
40-200 topics manually from the training tweets, and
set the parameters
0.5


and
0.1


.
The Map-Reduce environment was constructed
under a infrastructure with 128 virtual machines
(each with 1 core of multi-core CPU, 16GB memory,
300G disk), 1PB distributed storage. We constructed
two HDFS clusters for file-based micro-blogging
datasets with 64 virtual machines respectively, one
for Topic-STG, and the other for SVD model. We
then built two parallel recommender models on
virtual machines, and repeated each experiment with
100 times to calculate the average results.
4.1 The Precision of Recommendation
MAP@N (Mean Average Precision) evaluates the
prediction accuracy of the top N recommendations
for users, which is a popular rank evaluation method
to evaluate the recommendation accuracy. We gave
the MAP@N evaluation with the Topic-STG model,
the SVD model, and a baseline--content similarity
based approach. As shown in Fig.2, both the Topic-
STG model and the SVD model outperform the
baseline since we consider more features.
As we observed, MAP@N is positively
correlated with the length N of the recommendations,
and the MAP@N values of three approaches are very
close when N is set to 50. We guess that though
users have a wide range of interests, top 50
recommendations would cover almost all of the user
interests. Also it means the time factor and the users'
behaviors would reduce their effect when the
recommendation list is enough long. Moreover, we
found that the Topic-STG outperforms the SVD
model when
[3, 30]N

. Maybe it’s due to the fact
that the Topic-STG captures the long-term and
short-term user interests better than the SVD's since
users are always interested in those hot topics.
Mining User Interests for Personalized Tweet Recommendation on Map-Reduce Framework
205

Figure 3: Diversity of top 20 recommendations.
Figure 5: The time consumption comparison of different
periods of Map-Reduce.
Besides, Topic-STG considers the implicit tweet
content to make the scores of the hot topics higher
when the list is short. Whereas when the list
becomes longer, i.e.
30N  , the effect of the
hashtag textual feature is highlighted.
4.2 Diversity of Top N
Recommendation
It often happens that the products on the
recommendation list are highly similar to each other
and lack of diversity. For example, when Tianjin
Port explosion happened on Aug. 12, 2015, most of
the recommended tweets were associated with the
“Tianjin Port explosion” event. Consequently, those
tweets contributed for long-term user interests would
not be ranked for recommendation. Diversity would
increase the probability of retrieving unusual or
novel items which are relevant to the user. We use
the metric introduced in (Hurley and Zhang, 2011)
to evaluate the diversity.
We present the diversity results with the
proposed approaches comparing with the baseline
approach as shown in Fig. 3, and evaluate the
Figure 4: Efficiency comparison for the two Map-Reduce
based models.
Figure 6: MAP@N comparisons with topic classification
in SVD model.
diversity metric on Top 20 recommendation with 30
random users. Totally, the diversities of our
proposed models are better than the baseline since
both of them take into account the time factor. When
it refers to 7 days' diversity, our models perform
significantly better than the baseline. This is because
that, for each day, our models give different
recommendation results on different topics, which
makes more diversity for 7 days totally, whereas the
baseline approach uses the content similarity which
would recommend relatively the same topics when
time changes, and thus leads to fewer diversity for 7
days'. Also we find that Topic-STG's gets bigger
variance than the SVD's, which means that Topic-
STG shows more data fluctuation, and reflects that
it's time-sensitive and better to learn short-term user
interests.
4.3 Efficiency of Map-Reduce
We borrowed the concept of “speedup” to evaluate
the efficiency of Map-Reduce based recommender
models. In parallel computing, speedup refers to
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206

how much a parallel algorithm is faster than a
corresponding sequential algorithm.
1
/
pp
STT
(9)
where
p
is the number of map-reduce functions,
1
T
is the execution time of the sequential mining
algorithm, and
p
T
is the execution time of Map-
Reduce based recommender models.
We present
p
S
efficiency comparison for Map-
Reduce based Topic-STG and SVD models as
shown in Fig.4. We set the experiment with data
servers ranging from 10 to 60 with increment of 5. It
can be shown that
p
S
increases almost linearly with
the increase of data nodes, and the more data nodes
are online, the higher is
p
S
.
Map-Reduce based Topic-STG has better
p
S
than Map-Reduce based SVD model, this is because
that Map-Reduce based Topic-STG needs more
execution time since it includes LDA map-reduce
function, which would make the ratio of time for
resource preparation smaller. On the contrary, Map-
Reduce based SVD model needs less execution time
than Topic-STG, whereas the time for resource
preparation is the same.
We then present the experiment on time-
consumption of different steps in Map-Reduce based
Topic-STG process with map nodes ranging from 30
to 85 with increment of 5. As shown in Fig.5, the
total execution time fist reduces when map nodes
increase, and achieve the least when the number of
map nodes is equal to 55. And then the total
execution time would increase when map nodes
increase. This phenomenon reveals that the
efficiency of Map-Reduce is affected with three
steps and two factors. It is also shown in Fig.5 that,
the execution time of reduce function does not
change significantly, which means that the reduce
function is not the primary step affecting efficiency.
Map and shuffle functions would change
significantly when map nodes change. When more
map nodes are added, the execution time of map
function would decrease, and the execution time of
shuffle function would increase, this is because
when more map nodes are added, the task would be
distributed to more map nodes, and reduce the
execution time. Whereas this step needs more
network transmission, and increases communication
cost. We need to balance the parallel and the
communication cost, which inspires us to set the
number of map node as 55.
4.4 Discussion
4.4.1 The Effect of Topic Sensitiveness
General speaking, the evolution of tweet's topic may
affect the user interests. As we observed, a topic can
also be categorized into two classifications: time-
sensitive (such as “Tianjin Port Explosion”) and
time-insensitive (such as “Data Mining”). The time-
sensitive topic would help to model short-term user
interests, whereas the time-insensitive topic captures
the long-term user interests. The classifications
would reduce the computational cost of SVD model
since those time-insensitive hashtags and tweets are
no longer calculated with
day
b
in Equation (6).
We addressed the concept of “topic velocity”
which is measured by the increasing count of tweets
or hashtags during a period of time to refine
temporary time-sensitive tweets and hashtags from
the time-insensitive ones. We checked the topic
velocity for each hour, and set two thresholds
1000
ispeed
 
and
1000
dspeed
 
to find those
temporary time-sensitive tweets and hashtag
respectively. It can be observed from Fig.6 that
higher precision for the final recommendation
results will achieve when refine the temporary time-
sensitive tweets and hashtag.
4.4.2 The Effect of Topic Number
As we observed, the number of topics would also
influence MAP@N and diversity metrics of Topic-
STG model. As shown in Table 1, if we use the
Topic-STG model, the performance of MAP@N
tends to be stable when
( ) [150,200]Num topic 
,
nodes play an important role to optimize the
recommendation results.
As shown in Table 1, the more topics are added,
the better performance of the 7 days' diversity would
achieve. This is because that more topics would
Table 1: MAP@N and 7 days’ diversity influenced by latent topics in Topic-STG model.
Metrics The number of latent topics
40 50 80 100 120 150 200
MAP@N 0.258 0.319 0.333 0.367 0.385 0.412 0.425
Diversity 0.413 0.436 0.467 0.490 0.517 0.532 0.549
Mining User Interests for Personalized Tweet Recommendation on Map-Reduce Framework
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extend user interests, and bring more choices.
Another reason is that those new generated topics
may be sub topics of the original one. For example,
topics “football”, “basketball”, and “NBA” can all
be included in the topic “sports”, whereas the new
topics would bring more details on user interests.
Therefore, it's important to select suitable scope and
theme of topics for personalized recommendation.
Indeed, different background, culture, and
mutual influence among users, as potential and
implicit features, may all affect the user interests,
since different approaches capture user interests
from different profiles and granularity. The results
also reveal that the micro-blogging systems should
select suitable length of N for personalized
recommendation.
5 CONCLUSIONS
In this paper, we comprehensively considered three
aspects of the information: the textual information,
the users' behavior, and the time factor to model the
user interests, and constructed Topic-STG model
and SVD model for tweet recommendation. Also the
parallel versions of Topic-STG and SVD models
based on Map-Reduce framework were provided to
achieve better performance. Experiments on massive
Sina Weibo dataset show the effectiveness of the
proposed models and algorithms. Still there are
several issues should be solved. The first one is that
the Topic-STG model brings more computational
cost comparing with the original STG model. We
should utilize some pruning strategies to improve the
performance. The second problem is that retweets
and comments present clear attitude to represent
user's strong interest or hate. We need to adopt
opinion mining approach to identify the subjective
information for the SVD model.
ACKNOWLEDGEMENTS
This work is supported by the National Key
Research Program of China ( No.2016YFB0501900).
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