Multi-view Anomaly Detection via Robust
Probabilistic Latent Variable Models
Tomoharu Iwata
NTT Communication Science Laboratories
iwata.tomoharu@lab.ntt.co.jp

Makoto Yamada
Kyoto University
makoto.m.yamada@ieee.org

Abstract
We propose probabilistic latent variable models for multi-view anomaly detection, which is the task of finding instances that have inconsistent views given
multi-view data. With the proposed model, all views of a non-anomalous instance
are assumed to be generated from a single latent vector. On the other hand, ananomalous instance is assumed to have multiple latent vectors, and its different
views are generated from different latent vectors. By inferring the number of latent vectors used for each instance with Dirichlet process priors, we obtain multiview anomaly scores. The proposed model can be seen as a robust extension ofprobabilistic canonical correlation analysis for noisy multi-view data. We present
Bayesian inference procedures for the proposed model based on a stochastic EM
algorithm. The effectiveness of the proposed model is demonstrated in terms of
performance when detecting multi-view anomalies.

1 Introduction
There has been great interest in multi-view learning, in which data are obtained from various information sources. In a wide variety of applications, data are naturally comprised of multiple views.
For example, an image can be represented by color, texture and shape information; a web page canbe represented by words, images and URLs occurring on in the page; and a video can be represented
by audio and visual features. In this paper, we consider the task of finding anomalies in multi-view
data. The task is called horizontal anomaly detection [13], or multi-view anomaly detection [16].
Anomalies in multi-view data are instances that have inconsistent features across multiple views.
Multi-view anomaly detection can be used for many applications, such as information disparity management [9], purchase behavior analysis [13], malicious insider detection [16], and user aggregationfrom multiple databases. In information disparity management, multiple views can be obtained fromdocuments written in different languages such as Wikipedia. Multi-view anomaly detection tries tofind documents that contain different information across different languages, which would be helpful for editors to select documents to be updated, or beneficial for cultural anthropologists to analyzesocial difference across different languages. In purchase behavior analysis, multiple views for each
item can be defined as its genre and its purchase history, i.e. a set of users who purchased the item.
Multi-view anomaly detection can find movies inconsistently purchased by users based on the movie
genre, which would assist creating marketing strategies.
Multi-view anomaly detection is different from standard (single-view) anomaly detection. Singleview anomaly detection finds instances that do not conform to expected behavior [6]. Figure 1 (a)shows the difference between a multi-view anomaly and a single-view anomaly in a two-view data
set. ‘M’ is a multi-view anomaly since ‘M’ belongs to different clusters in different views (‘A–D’cluster in View 1 and ‘E–J’ cluster in View 2) and views of ‘M’ are not consistent. ‘S’ is a singleview anomaly since ‘S’ is located far from other instances in each view. However, both views of‘S’ have the same relationship with the others (they are far from the other instances), and then ‘S’
29th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.

Latent spaceBA
MCD

F GI M
J
HE
S

W1 ւ
BD
MC
A

ց W2

a
b

α

w

S

F GI
J
H E

γ
M

S

Observed view 1

I EH
J G

F

θ

s

x

D

z

∞N

r

AD
C
B

Observed view 2
(a)

(b)

Figure 1: (a) A multi-view anomaly ‘M’ and a single-view anomaly ‘S’ in a two-view data set. Eachletter represents an instance, and the same letter indicates the same instance. Wd is a projection
matrix for view d. (b) Graphical model representation of the proposed model.
is not a multi-view anomaly. Single-view anomaly detection methods, such as one-class support
vector machines [18] or tensor-based anomaly detection [11], consider that ‘S’ is anomalous. On
the other hand, we would like to develop a multi-view anomaly detection method that detects ‘M’ asanomaly, but not ‘S’. Note that although single-view anomalies are uncommon instances, multi-view
anomalies can be majority if they are inconsistent across multiple views.
We propose a probabilistic latent variable model for multi-view anomaly detection. With the proposed model, there is a latent space that is shared across all views. We assume that all views of a
non-anomalous (normal) instance are generated using a single latent vector. On the other hand, ananomalous instance is assumed to have multiple latent vectors, and its different views are generatedusing different latent vectors, which indicates inconsistency across different views of the instance.
Figure 1 (a) shows an example of a latent space shared by the two-view data. Two views of every
non multi-view anomaly can be generated from a latent vector using view-dependent projection matrices. On the other hand, since two views of multi-view anomaly ‘M’ are not consistent, two latentvectors are required to generate the two views using the projection matrices.
Since the number of latent vectors for each instance is unknown, we automatically infer it from thegiven data by using Dirichlet process priors. The inference of the proposed model is based on astochastic EM algorithm. In the E-step, a latent vector is assigned for each view of each instanceusing collapsed Gibbs sampling while analytically integrating out latent vectors. In the M-step,
projection matrices for mapping latent vectors into observations are estimated by maximizing thejoint likelihood. By alternately iterating E- and M-steps, we infer the number of latent vectors usedin each instance and calculate its anomaly score from the probability of using more than one latentvector.

2 Proposed ModelD
Suppose that we are given N instances with D views X = {Xn }N
n=1 , where Xn = {xnd }d=1
Md
is a set of multi-view observation vectors for the nth instance, and xnd ∈ R
is the observation
vector of the dth view. The task is to find anomalous instances that have inconsistent observation
features across multiple views. We propose a probabilistic latent variable model for this task. The
proposed model assumes that each instance has potentially a countably infinite number of latentK
vectors Zn = {znj }∞j=1 , where znj ∈ R . Each view of an instance xnd is generated depending
on a view-specific projection matrix Wd ∈ RMd ×K and a latent vector znsnd that is selected froma set of latent vectors Zn . Here, snd ∈ {1, · · · , ∞} is the latent vector assignment of xnd . Whenthe instance is non-anomalous and all its views are consistent, all of the views are generated froma single latent vector. In other words, the latent vector assignments for all views are the same,
sn1 = sn2 = · · · = snD . When it is an anomaly and some views are inconsistent, different views2are generated from different latent vectors, and some latent vector assignments are different, i.e.
snd 6= snd′ for some d 6= d′ .
Specifically, the proposed model is an infinite mixture model, where the probability for the dth viewof the nth instance is given byp(xnd |Zn , Wd , θn , α) =

∞
X
j=1

θnj N (xnd |Wd znj , α−1 I),(1)where θn = {θnj }∞j=1 are the mixture weights, θnj represents the probability of choosing the jthlatent vector, α is a precision parameter, N (µ, Σ) denotes the Gaussian distribution with mean µand covariance matrix Σ, and I is the identity matrix. Information of non-anomalous instances thatcannot be handled by a single latent vector is modeled in Gaussian noise which is controlled by α.
Since we assume the same observation noise α across different views, the observations need to benormalized. We use a Dirichlet process for the prior of mixture weight θn . Its use enables us toautomatically infer the number of latent vectors for each instance from the given data.
The complete generative process of the proposed model for multi-view instances X is as follows,
1. Draw a precision parameter α ∼ Gamma(a, b)
2. For each instance: n = 1, . . . , N(a) Draw mixture weights θn ∼ Stick(γ)(b) For each latent vector: j = 1, . . . , ∞i. Draw a latent vector znj ∼ N (0, (αr)−1 I)(c) For each view: d = 1, . . . , D
i. Draw a latent vector assignment snd ∼ Discrete(θn )ii. Draw an observation vector xnd ∼ N (Wd znsnd , α−1 I)
Here, Stick(γ) is the stick-breaking process [19] that generates mixture weights for a Dirichlet
process with concentration parameter γ, and r is the relative precision for latent vectors. α is sharedfor observation and latent vector precision because it makes it possible to analytically integrate out α
as shown in (4). Figure 1 (b) shows a graphical model representation of the proposed model, where
the shaded and unshaded nodes indicate observed and latent variables, respectively.
N
The joint probability of the data X and the latent vector assignments S = {{snd }D
d=1 }n=1 is givenbyp(X, S|W , a, b, r, γ) = p(S|γ)p(X|S, W , a, b, r),(2)where W = {Wd }D
d=1 . Because we use conjugate priors, we can analytically integrate out mixture
weights Θ = {θn }Nn=1 , latent vectors Z, and precision parameter α. Here, we use a Dirichlet
process prior for multinomial parameter θn , and a Gaussian-Gamma prior for latent vector znj . By
integrating out mixture weights Θ, the first factor is calculated byQ nNYγ Jn Jj=1(Nnj − 1)!
p(S|γ) =,(3)γ(γ + 1) · · · (γ + D − 1)n=1where Nnj represents the number of views assigned to the jth latent vector in the nth instance, andJn is the number of latent vectors of the nth instance for which Nnj > 0. By integrating out latentvectors Z and precision parameter α, the second factor of (2) is calculated byp(X|S, W , a, b, r) = (2π)−

N

P
d Md
2

r

K

P

n Jn
2

N Jn
1
ba Γ(a′ ) Y Y|Cnj | 2 ,′′ab Γ(a) n=1 j=1

(4)

where
′

a =a+

N

PD

d=1 Md2,b′ = b +N D
N Jn
1 XX ⊤1 XXµ⊤ C −1 µnj ,xnd xnd −2 n=1
2 n=1 j=1 nj nj
d=1

3

(5)

µnj = Cnj

X

−1
Cnj
=

Wd⊤ xnd ,d:snd =jX

Wd⊤ Wd + rI.

(6)

d:snd =j

The posterior for the precision parameter α and that for the latent vector znj are given byp(α|X, S, W , a, b) = Gamma(a′ , b′ ),p(znj |X, S, W , r) = N (µnj , α−1 Cnj ),(7)respectively.

3 Inference
We describe inference procedures for the proposed model based on a stochastic EM algorithm, inwhich collapsed Gibbs sampling of latent vector assignments S and the maximum joint likelihood
estimation of projection matrices W are alternately iterated while analytically integrating out thelatent vectors Z, mixture weights Θ and precision parameter α. By integrating out latent vectors,
we do not need to explicitly infer the latent vectors, leading to a robust and fast-mixing inference.
Let ℓ = (n, d) be the index of the dth view of the nth instance for notational convenience. In the
E-step, given the current state of all but one latent assignment sℓ , a new value for sℓ is sampled from
{1, · · · , Jn\ℓ + 1} according to the following probability,p(sℓ = j|X, S\ℓ , W , a, b, r, γ) ∝p(sℓ = j, S\ℓ |γ) p(X|sℓ = j, S\ℓ , W , a, b, r)·,p(S\ℓ |γ)p(X\ℓ |S\ℓ , W , a, b, r)(8)where \ℓ represents a value or set excluding the dth view of the nth instance. The first factor is given
by
(
Nnj\ℓ
p(sℓ = j, S\ℓ |γ)
if j ≤ Jn\ℓ
D−1+γ
=
(9)
γ
if j = Jn\ℓ + 1,p(S\ℓ |γ)D−1+γusing (3), where j ≤ Jn\ℓ is for existing latent vectors, and j = Jn\ℓ + 1 is for a new latent vector.
By using (4), the second factor is given by′a′\ℓb\ℓ Γ(a′sℓ =j ) |Cj,sℓ =j | 2
Mdp(X|sℓ = j, S\ℓ , W , a, b, r)
K= (2π)− 2 rI(j=Jn\ℓ +1) 2 ′a′,s =jp(X\ℓ |S\ℓ , W , a, b, r)
Γ(a′\ℓ ) |Cj\ℓ | 12
b ℓ
1

(10)

sℓ =j

where I(·) represents the indicator function, i.e. I(A) = 1 if A is true and 0 otherwise, and subscriptsℓ = j indicates the value when xℓ is assigned to the jth latent vector as follows,1
1
1b′sℓ =j = b′\ℓ + x⊤
xℓ + µ⊤C −1 µnj\ℓ − µ⊤C −1
µnj,sℓ =j ,2 ℓ
2 nj\ℓ nj\ℓ2 nj,sℓ =j nj,sℓ =ja′sℓ =j = a′ ,

(11)

−1
µnj,sℓ =j = Cnj,sℓ =j (Wd⊤ xℓ + Cnj\ℓµnj\ℓ ),

(12)

−1
−1
Cnj,s
= Wd⊤ Wd + Cnj\ℓ
.
ℓ =j

(13)

Intuitively, if the current view cannot be modeled well by existing latent vectors, a new latent vector
is used, which indicates that the view is inconsistent with the other views.
In the M-step, the projection matrices W are estimated by maximizing the logarithm of the joint
likelihood (2) while fixing cluster assignment variables S. By setting the gradient of the joint log
likelihood with respect to W equal to zero, an estimate of W is obtained as follows,
Wd =N
 a′ Xb′ n=1xnd µ⊤
nsnd

N X
Jn
X
n=1 j=1

N

Cnj +

−1
a′ X
⊤
µ
µ
.
nsnd nsnd
b′ n=1(14)

When we iterate the E-step that samples the latent vector assignment snd by employing (8) for
each view d = 1, . . . , D in each instance n = 1, . . . , N , and the M-step that maximizes the joint
likelihood using (14) with respect to the projection matrix Wd for each view d = 1, . . . , D, we
obtain an estimate of the latent vector assignments and projection matrices.
4

In Section 2, we defined that an instance is an anomaly when its different views are generated fromdifferent latent vectors. Therefore, for an anomaly score, we use the probability that the instanceuses more than one latent vector. It is estimated by using the samples obtained in the inference as
PH
(h)
(h)
follows, vn = H1 h=1 I(Jn > 1), where Jn is the number of latent vectors used by the nth
instance in the hth iteration of the Gibbs sampling after the burn-in period, and H is the numberof the iterations. The output of the proposed method is a ranked list of anomalies based on theiranomaly scores. An analyst would investigate top few anomalies, or use a threshold to select theanomalies [6]. The threshold can be determined based on a targeted false alarm and detection rate.
We can use cross-validation to select an appropriate dimensionality for the latent space K. Withcross-validation, we assume that some features are missing from the given data, and infer the modelwith a different K. Then, we select the smallest K value that has performed the best at predicting
missing values.

4 Related Work
Anomaly detection has had a wide variety of applications, such as credit card fraud detection [1],intrusion detection for network security [17], and analysis for healthcare data [3]. However, mostexisting anomaly detection techniques assume data with a single view, i.e. a single observation
feature set.
A number of anomaly detection methods for two-view data have been proposed [12, 20–22, 24].
However, they cannot be used for data with more than two views. Gao et al. [13] proposed a
HOrizontal Anomaly Detection algorithm (HOAD) for finding anomalies from multi-view data. In
HOAD, there are hyperparameters including a weight for the constraint that require the data to belabeled as anomalous or not for tuning, and the performance is sensitive to the hyperparameters. On
the other hand, the parameters with the proposed model can be estimated from the given multi-viewdata without label information by maximizing the likelihood. In addition, because the proposedmodel is a probabilistic generative model, we can extend it in a probabilistically principled manner,for example, for handling missing data and combining with other probabilistic models.
Liu and Lam [16] proposed multi-view anomaly detection methods using consensus clustering. They
found anomalies based on the inconsistency of clustering results across multiple views. Therefore,
they cannot find inconsistency within a cluster. Christoudias et al. [8] proposed a method for filtering
instances that are corrupted by background noise from multi-view data. The multi-view anomalies
considered in this paper include not only instances corrupted by background noise but also instances
categorized into different foreground classes across views, and instances with inconsistent viewseven if they belong to the same cluster. Recently, Alvarez et al. [2] proposed a multi-view anomaly
detection method. However, since the method is based on clustering, it cannot find anomalies when
there are no clusters in the given data.
The proposed model is a generalization of either probabilistic principal component analysis(PPCA) [23] or probabilistic canonical correlation analysis (PCCA) [5]. When all views are generated from different latent vectors for every instance, the proposed model corresponds to PPCA
that is performed independently for each view. When all views are generated from a single latentvector for every instance, the proposed model corresponds to PCCA with spherical noise.
PCCA, or canonical correlation analysis (CCA), can be used for multi-view anomaly detection. With
PCCA, a latent vector that is shared by all views for each instance and a linear projection matrix foreach view are estimated by maximizing the likelihood, or minimizing the reconstruction error of thegiven data. The reconstruction error for each instance can be used as an anomaly score. However, thereconstruction errors are not reliable because they are calculated from parameters that are estimatedusing data with anomalies by assuming that all of the instances are non-anomalous. On the other
hand, because the proposed model simultaneously estimates the parameters and infers anomalies,
the estimated parameters are not contaminated by the anomalies. With PPCA and PCCA, Gaussian
distributions are used for observation noise, which are sensitive to atypical observations. Robust
PPCA and PCCA [4] use Student-t distributions instead of Gaussian distributions, which are stable
to data containing single-view anomalies. The proposed model assumes Gaussian observation noise,
and its precision is parameterized by a Gamma distributed variable α. Since we marginalize out α
in the inference as written in (4), the observation noise becomes a Student-t distribution. Therefore,
the proposed model is robust to single-view anomalies.
5With some CCA-related methods, each latent vector is factorized into shared and private components
across different views [10]. They assume that every instance has shared and private parts that are thesame dimensionality for all instances. In contrast, the proposed model assumes that non-anomalous
instances have only shared latent vectors, and anomalies have private latent vectors. The proposedmodel can be seen as CCA with private latent vectors, where latent vectors across views are clustered for each instance. When CCA with private latent vectors are inferred without clustering, theinferred private latent vectors do not become the same even if it is generated from a single latent vector, because switching latent dimension or rotating the latent space does not change the likelihood.
Therefore, difference of the latent vectors cannot be used for multi-view anomaly detection.

5 Experiments
Data We evaluated the proposed model quantitatively by using 11 data sets, which we obtained
from the LIBSVM data sets [7]. We generated two views by randomly splitting the features, where
each feature can belong to only a single view, and anomalies were added by swapping views of
two randomly selected instances regardless of their class labels for each view. Splitting data does
not generate anomalies. Therefore, we can evaluate methods while controlling the anomaly rate
properly. By swapping, although single-view anomalies cannot be created since the distribution foreach view does not change, multi-view anomalies are created.
Comparing methods We compared the proposed model with probabilistic canonical correlation
analysis (PCCA), horizontal anomaly detection (HOAD) [13], consensus clustering based anomaly
detection (CC) [16], and one-class support vector machine (OCSVM) [18]. For PCCA, we used
the proposed model in which the number of latent vectors was fixed at one for every instance. The
anomaly scores obtained with PCCA were calculated based on the reconstruction errors. HOAD requires to select an appropriate hyperparameter value for controlling the constraints whereby different
views of the same instance are embedded close together. We ran HOAD with different hyperparameter settings {0.1, 1, 10, 100}, and show the results that achieved the highest performance for eachdata set. For CC, first we clustered instances for each view using spectral clustering. We set the
number of clusters at 20, which achieved a good performance in preliminary experiments. Then, we
calculated anomaly scores by the likelihood of consensus clustering when an instance was removed
since it indicates inconsistency of the instance across different views. OCSVM is a representative
method for single-view anomaly detection. To investigate the performance of a single-view method
for multi-view anomaly detection, we included OCSVM as a comparison method. For OCSVM,
multiple views are concatenated in a single vector, then use it for the input. We used Gaussian kernel. In the proposed model, we used γ = 1, a = 1, and b = 1 for all experiments. The number of
iterations for the Gibbs sampling was 500, and the anomaly score was calculated by averaging overthe multiple samples.
Multi-view anomaly detection For the evaluation measurement, we used the area under the ROC
curve (AUC). A higher AUC indicates a higher anomaly detection performance. Figure 2 shows
AUCs with different rates of anomalies using 11 two-view data sets, which are averaged over 50
experiments. For the dimensionality of the latent space, we used K = 5 for the proposed model,
PCCA, and HOAD. In general, as the anomaly rate increases, the performance decreases. The
proposed model achieved the best performance with eight of the 11 data sets. This result indicatesthat the proposed model can find anomalies effectively by inferring a number of latent vectors foreach instance. The performance of CC was low because it assumes that there are clusters for eachview, and it cannot find anomalies within clusters. The AUC of OCSVM was low, because it is asingle-view anomaly detection method, which considers instances anomalous that are different fromothers within a single view. Multi-view anomaly detection is the task to find instances that haveinconsistent features across views, but not inconsistent features within a view. The computationaltime needed for PCCA was 2 sec, and that needed for the proposed model was 35 sec with wine
data.
Figure 3 shows AUCs with different dimensionalities of latent vectors using data sets whose anomaly
rate is 0.4. When the dimensionality was very low (K = 1 or 2), the AUC was low in most of the data
sets, because low-dimensional latent vectors cannot represent the observation vectors well. With all
the methods, the AUCs were relatively stable when the latent dimensionality was higher than four.
6

(a) breast-cancer(b) diabetes

0.7(c) glass

0.6

Proposed
PCCA
HOAD
CC
OCSVM

0.65

0.58
0.65

AUC

AUC

AUC

0.6

0.56
0.54

0.6

0.55

0.52
0.55
0.4
0.6
anomaly rate

0.2

0.8

(d) heart

0.4
0.6
anomaly rate

0.8

0.2

(e) ionosphere

0.4
0.6
anomaly rate

0.8(f) sonar(g) svmguide2

0.85
0.8

0.6
AUC

0.56

AUC

0.75

0.58

0.7
0.65

0.54

0.6

0.52

0.55

0.9

0.56

0.8

0.54
AUC

0.62

AUC

0.5

0.5
0.2

0.7

0.52
0.5

0.6

0.2

0.4
0.6
anomaly rate

0.48

0.8

0.2

(h) svmguide4

0.4
0.6
anomaly rate

0.8

0.2

0.4
0.6
anomaly rate(i) vehicle

0.2

0.8

(j) vowel

0.4
0.6
anomaly rate

0.8(k) wine
0.8

0.85

0.75

0.8

0.7

0.7

0.7

0.75

AUC

AUC

AUC

0.8

0.7

0.55
0.4
0.6
anomaly rate

0.65
0.6

0.55

0.6
0.2

0.65
0.6

0.65
0.6

AUC

0.9

0.75

0.55

0.5

0.8

0.2

0.4
0.6
anomaly rate

0.8

0.2

0.4
0.6
anomaly rate

0.2

0.8

0.4
0.6
anomaly rate

0.8

Figure 2: Average AUCs with different anomaly rates, and their standard errors. A higher AUC isbetter.
(a) breast-cancer(b) diabetes(c) glass

0.58

0.6

0.65

0.6

AUC

AUC

0.56
AUC

Proposed
PCCA
HOAD

0.62

0.54

0.56
0.54

0.52

0.55

0.58

0.52
0.5
2

4
6
8
latent dimensionality

210(d) heart468
latent dimensionality

0.5

10

(e) ionosphere

0.6

4
6
8
latent dimensionality

10(f) sonar(g) svmguide2
0.55

0.9

0.8

0.58
0.7

AUC

0.56

AUC

0.8
AUC

AUC

2

0.7

0.5

0.54
0.6

0.6

0.52
0.5
4
6
8
latent dimensionality

102(h) svmguide4

4
6
8
latent dimensionality

0.5

10

(i) vehicle

0.9

0.85

0.8

0.75

0.45

10

2

0.65

10

0.75
0.7

0.7

0.7

4
6
8
latent dimensionality(k) wine

0.75

AUC

AUC

AUC

0.7

4
6
8
latent dimensionality(j) vowel

0.8

0.65

0.65

0.6

0.6

0.55

0.55

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0.55
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2

AUC

2

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2

4
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latent dimensionality

10

2

4
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latent dimensionality

10

2

4
6
8
latent dimensionality

10

2

4
6
8
latent dimensionality

10

Figure 3: Average AUCs with different dimensionalities of latent vectors, and their standard errors.

Single-view anomaly detection We would like to find multi-view anomalies, but woul not like todetect single-view anomalies. We illustrated that the proposed model does not detect single-view
anomalies using synthetic single-view anomaly data. With the synthetic data, latent vectors for7

Table 1: Average AUCs for single-view anomaly detection.
Proposed
0.117 ± 0.098

PCCA
0.174 ± 0.095

OCSVM
0.860 ± 0.232

Table 2: High and low anomaly score movies calculated by the proposed model.
Title
Score Title
Score
The Full Monty
0.98 Star Trek VI
0.04
Liar Liar
0.93 Star Trek III
0.04
The Professional
0.91 The Saint
0.04
Mr. Holland’s Opus 0.88 Heat
0.03
Contact
0.87 Conspiracy Theory 0.03√
single-view anomalies were generated from N (0, 10I), and those for non-anomalous instanceswere generated from N (0, I). Since each of the anomalies has only one single latent vector, it is
not a multi-view anomaly. The numbers of anomalous and non-anomalous instances were 5 and 95,
respectively. The dimensionalities of the observed and latent spaces were five and three, respectively.
Table 1 shows the average AUCs with the single-view anomaly data, which are averaged over 50
different data sets. The low AUC of the proposed model indicates that it does not consider singleview anomalies as anomalies. On the other hand, the AUC of the one-class SVM (OCSVM) was
high because OCSVM is a single-view anomaly detection method, and it leads to low multi-view
anomaly detection performance.
Application to movie data For an application of multi-view anomaly detection, we analyzed inconsistency between movie rating behavior and genre in MovieLens data [14]. An instance corresponds to a movie, where the first view represents whether the movie is rated or not by users, and thesecond view represents the movie genre. Both views consist of binary features, where some movies
are categorized in multiple genres. We used 338 movies, 943 users and 19 genres. Table 2 showshigh and low anomaly score movies when we analyzed the movie data by the proposed method with
K = 5. ‘The Full Monty’ and ‘Liar Liar’ were categorized in ‘Comedy’ genre. They are rated
by not only users who likes ‘Comedy’, but also who likes ‘Romance’ and ‘Action-Thriller’. ‘The
Professional’ was anomaly because it was rated by two different user groups, where a group prefers‘Romance’ and the other prefers ‘Action’. Since ‘Star Trek’ series are typical Sci-Fi and liked by
specific users, its anomaly score was low.

6 Conclusion
We proposed a generative model approach for multi-view anomaly detection, which finds instancesthat have inconsistent views. In the experiments, we confirmed that the proposed model could
perform much better than existing methods for detecting multi-view anomalies. There are several
avenues that can be pursued for future work. Since the proposed model assumes the linearity ofobservations with respect to their latent vectors, it cannot find anomalies when different views arein a nonlinear relationship. We can relax this assumption by using Gaussian processes [15]. We canalso relax the assumption that non-anomalous instances have the same latent vector across all viewsby introducing private latent vectors [10]. The proposed model assumes Gaussian observation noise.
Our framework can be extended for binary or count data by using Bernoulli or Poisson distributions
instead of Gaussian.
Acknowledgments
MY was supported by KAKENHI 16K16114.

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