Interval-valued Matrix Factorization with Applications
Zhiyong Shen
1,3
,LiangDu
2,1
, Xukun Shen
3
, Yidong Shen
2
1
Hewlett Packard Labs China, zhiyongs@hp.com
2
State Key Laboratory of Computer Science, China,{duliang,ydshen}@ios.ac.cn
3
State Key Laboratory of Virtual Reality Technology and system,China, xkshen@vrlab.buaa.edu.cn
Abstract—In this paper, we propose the Interval-valued
Matrix Factorization (IMF) framework. Matrix Factorization
(MF) is a fundamental building block of data mining. MF
techniques, such as Nonnegative Matrix Factorization (NMF)
and Probabilistic Matrix Factorization (PMF), are widely used
in applications of data mining. For example, NMF has shown
its advantage in Face Analysis (FA) while PMF has been
successfully applied to Collaborative Filtering (CF). In this
paper, we analyze the data approximation in FA as well
as CF applications and construct interval-valued matrices to
capture these approximation phenomenons. We adapt basic
NMF and PMF models to the interval-valued matrices and
propose Interval-valued NMF (I-NMF) as well as Interval-
valued PMF (I-PMF). We conduct extensive experiments to
show that proposed I-NMF and I-PMF significantly outperform
their single-valued counterparts in FA and CF applications.
Keywords -Matrix factorization, uncertainty
I. INTRODUCTION
Exploring data approximation has attracted much atten-
tion in uncertain data mining [1] and privacy preserving
data mining [2]. Data approximation might be caused by
limitations of measuring, delayed data update or intensional
data perturbation. When traditional data mining techniques
are employed, the consideration of data approximation may
improve the quality of results. Thus, various data mining
techniques, such as clustering, classification, association
mining have been adapted to handling data approximation. In
this paper, we devote to inject data approximation into Ma-
trix Factorization (MF) techniques. MF, also known as ma-
trix decomposition, underlies many data mining techniques
including clustering, dimensionality reduction and missing
data prediction etc.. It decomposes an input data matrix into
a number of low-rank factor matrices, which leads to a more
compact linear approximation for the original data matrix.
Variations MF have been extensively studied in literatures.
In this paper, we pay special attention to Nonnegative Ma-
trix Factorization (NMF) [3], [4] and Probabilistic Matrix
Factorization (PMF) [5]. Each of these MF techniques is
suited for a particular class of applications. For example,
NMF has shown its advantage in Face Analysis (FA) [4].
In FA applications, each face is represented by a feature
vector. NMF factorizes the matrix of multiple face feature
vectors into factor matrices and then achieve a more compact
representation of the original face data. On the other hand,
PMF has been successfully applied to Collaborative Filtering
(CF) [6]. CF is one of the most successful techniques
for automatic recommendation systems which need only
an observed r ating matrix as input. PMF decomposes the
sparse rating matrix into user profile matrix and item profile
matrix, and then makes predictions for the unknown entries.
However, traditional NMF and PMF ignore the following
data approximation phenomenons in FA and CF.
Alignment approximation in FA: The faces need to be
rotated and aligned to make sure that same columns in the
data matrix are corresponding to the same positions in faces.
Such alignment is hardly to be perfect in practice, i.e. there
is approximation with the alignment in FA applications (see
Section II-A for details).
Rating approximation in CF: When a user rates an
item in a real-life rating system she/he usually selects a
discretized rating value which is close to the ideal numerical
preference value (the exact preference degree). Thus, the
rating matrix does contain approximations to some degree
(see Section II-B for details).
Interval bounds are better than single-valued variables to
describe the above phenomenons of approximation. Many
application areas have taken advantage of interval-valued
data analysis (see for instance [7]), such as object tracking,
market analysis, quantitative economics and so on. In tra-
ditional MF techniques, input data matrices might be real
values, non-negative values or binary values etc., all of
which are single-valued. In this paper, we introduce a new
type of data matrix − interval-valued matrix to MF, which
captures approximation in the observed data matrix. Then,
we propose a novel MF framework − Interval-valued Matrix
Factorization (IMF) to decompose such matrices. Under the
IMF framework, we inject data approximation i nto NMF and
PMF and extend them to interval-valued NMF (I-NMF for
short) and interval-valued PMF (I-PMF for short). Therefore,
our work is a marriage between interval-valued data analysis
[7] and MF, and our contributions on both sides of research
area are summarized as follows
∙ We analyze the alignment approximation in FR as well
as the rating approximation in CF, and formalize them
with interval-valued matrices (Section II).
∙ We propose the IMF framework, under which we
extend two representative basic MF techniques NMF
2010 IEEE International Conference on Data Mining
1550-4786/10 $26.00 © 2010 IEEE
DOI 10.1109/ICDM.2010.115
1037
and PMF to I-NMF and I-PMF which are capable of
handling interval-valued matrices (Section IV).
∙
We conduct extensive experiments to show that the the
proposed I-NMF and I-PMF significantly outperform
their traditional single-valued counterparts in FA and
CF applications (Section V).
II. I
NTERVAL-VALUED MATRIX AND DATA
APPROXIMATION
In this section we formalize the approximation in CF and
FR problems with interval-valued matrices. First of all, we
give formal definitions of interval-valued matrix.
Let 𝑿 ∈ ℝ
𝑛×𝑑
denote the input data matrix, with
entries denoted as 𝑋
𝑖𝑗
.Let𝐼(𝑿) denote the interval-valued
matrix corresponding to 𝑿, and we have the following two
equivalent representations for 𝐼(𝑿).
Definition 1 (Center-radius representation). We denote the
interval with center 𝑋
𝑖𝑗
and radius 𝛿
𝑖𝑗
as
𝐼(𝑋
𝑖𝑗
)=⟨𝑋
𝑖𝑗
,𝛿
𝑖𝑗
⟩ (1)
For entire matrices, we have 𝐼(𝑿)=⟨𝑿, 𝜹⟩.
Definition 2 (Min-max representation). We denote the in-
terval bounds as 𝑋
low
𝑖𝑗
= 𝑋
𝑖𝑗
− 𝛿
𝑖𝑗
and 𝑋
up
𝑖𝑗
= 𝑋
𝑖𝑗
+ 𝛿
𝑖𝑗
.
𝐼(𝑋
𝑖𝑗
)=[𝑋
low
𝑖𝑗
,𝑋
up
𝑖𝑗
] (2)
For entire matrices, we have 𝐼(𝑿)=[𝑿
low
, 𝑿
up
].
In practice, we might only observe single-valued data
matrices rather than interval-valued ones. In the following
subsections we’ll give the empirical method to construct
𝑰(
𝑿) based on 𝑿. The above definitions have already been
adopted in interval-valued data analysis [8]. In our work,
we’ll use the center-radius representation (Definition 1) to
formalize the rating approximation in CF and alignment
approximation in FR and then construct interval-valued
matrices. The min-max representation (Definition 2) will be
used as input for the proposed IMF models introduced in
Section IV.
A. Alignment Approximation in FR
In many FA techniques, we need to align the faces image
such that, ideally, the pixels with the same coordinates
correspond to the identical positions of a face. In Figure
1, we take the position of the nose tip as an example to
show that the alignment is not perfect. Although the same
position of a face is not exactly aligned, they should be
near to each other. Take the first row as examples, the pixel
with coordinate (33,35) may corresponds to the face position
coordinated by (41,34) in the second image, or (33,40) in
the third and so on. Formally, the value of a pixel with
coordinates (𝑥, 𝑦),𝑥 ∈{1, ..., 𝑑
𝑥
},𝑦 ∈{1, ..., 𝑑
𝑦
}, might
correspond to a pixel with coordinates (𝑥 +Δ𝑥, 𝑦 +Δ𝑦),
0
≤ Δ𝑥, Δ𝑦 ≤ 𝑟.InMF,the𝑖’th face is represented by
(33,35)
20 40 60
20
40
60
(41,34)
20 40 60
20
40
60
(33,40)
20 40 60
20
40
60
(25,32)
20 40 60
20
40
60
(44,33)
20 40 60
20
40
60
(34,32)
20 40 60
20
40
60
(29,38)
20 40 60
20
40
60
(32,36)
20 40 60
20
40
60
(35,38)
20 40 60
20
40
60
(37,37)
20 40 60
20
40
60
Figure 1. Illustration o f alignment approximation.
20 40 60
20
40
60
20 40 60
20
40
60
20 40 60
20
40
60
20 40 60
20
40
60
20 40 60
20
40
60
20 40 60
20
40
60
20 40 60
20
40
60
20 40 60
20
40
60
20 40 60
20
40
60
20 40 60
20
40
60
Figure 2. An example of 𝜹 matrix corresponding to faces in Figure 1.
a vector 𝑿
𝑖⋅
with dimensionality 𝑑 = 𝑑
𝑥
× 𝑑
𝑦
.Weuse
(𝑥
(𝑖,𝑗)
,𝑦
(𝑖,𝑗)
) to denote the coordinates of pixels in the 𝑖’th
image which corresponds to the 𝑗’th element in vector 𝑿
𝑖⋅
,
namely 𝑋
𝑖𝑗
. Then, we define the following set of the entries
in 𝑿 for each 𝑋
𝑖𝑗
𝒮
FA( 𝑟)
𝑖𝑗
= {𝑋
𝑖𝑗
′
∣∣𝑥
(𝑖,𝑗
′
)
−𝑥
(𝑖,𝑗)
∣≤𝑟 ∧∣𝑦
(𝑖,𝑗
′
)
−𝑦
(𝑖,𝑗)
∣≤𝑟}
(3)
The elements in 𝒮
FA( 𝑟)
𝑖𝑗
correspond to pixels around
(𝑥
(𝑖,𝑗)
,𝑦
(𝑖,𝑗)
) in a range 𝑟. Intuitively, 𝑋
𝑖𝑗
may corresponds
to a value in the interval of [min(𝒮
FA ( 𝑟)
𝑖𝑗
),max(𝒮
FA ( 𝑟)
𝑖𝑗
)],
which coincides the min-max definition (Definition 2).
However, min-max statistics are not robust in practice and
alternatively, we construct 𝐼(𝑋
𝑖𝑗
) based on the standard
deviation to capture the variation in 𝒮
FA( 𝑟)
𝑖𝑗
. According to
Definition 1, we set 𝑋
𝑖𝑗
as the center of 𝐼(𝑋
𝑖𝑗
) and calculate
the radius via
𝛿
FA( 𝑟)
𝑖𝑗
:= 𝛼 ⋅ std(𝒮
FR(𝑟)
𝑖𝑗
) (4)
where 𝛼 ∈ ℝ
+
is a multiplicative scale coefficient. Based
on Defintion 2, it’s easy to calculate the bounds of interval-
valued input for I-NMF according to min-max representation
(Definition 2). Examples of the 𝜹
FR(r)
𝑖⋅
corresponding to
the faces in Figure 1 are shown in Figure 2, where lighter
gray level represents larger radius. In Figure 2, we can see
positions such as eyes or nose have larger radius. These
positions are more sensitive to alignment error that may
hurt the performance of single-valued techniques. With a
relatively large radius, the interval-valued techniques may
be more tolerant to such alignment errors.
1038
Table I
E
XAMPLES OF SINGLE-VALUED AND INTERVAL-VALUED RATING MATRICES FOR CF
(a) A single-valued rating matrix: 𝑿
𝑚
1
𝑚
2
𝑚
3
𝑚
4
𝑚
5
𝑢
1
14 5
𝑢
2
312
𝑢
3
14
𝑢
4
5
𝑢
5
142
𝑢
6
325
(b) A interval-valued rating matrix: 𝐼(𝑿)
𝑚
1
𝑚
2
𝑚
3
𝑚
4
𝑚
5
𝑢
1
[0.6,1.4] [3.5,4.5] [4.8,5.2]
𝑢
2
[2.8,3.2] [0.5,1.5] [1.5,2.5]
𝑢
3
[0.7,1.3] [3.5,4.5]
𝑢
4
[4.5,5.5]
𝑢
5
[0.4,1.6] [3.7,4.3] [1.8,2.2]
𝑢
6
[2.7, 3.3] [1.4, 2.6] [4.2, 5.8]
B. Rating Approximation in CF
In CF, the rating degree is actually an approximate to
its actual preference degree of a user 𝑢 over an item. For
example, a web site allows users to rate items from one star
to five stars. User 𝑢 may think the two items 𝑎 and 𝑏 are
beyond two stars while not worth four stars, and he may
prefer 𝑎 to 𝑏. Suppose the continuous preference degrees of
user 𝑢 on 𝑎 and 𝑏 are 3.4 and 2.8, respectively. However,
due to the constraint of the rating system, 𝑢 can only rate
both 𝑎 and 𝑏 as three stars, and the difference between 𝑎
and 𝑏 disappears. It also indicates that the rating degree
actually represents a continuous interval, which may include
the i deal preference degree. Intuitively, the rating degree 𝑋
𝑖𝑗
is affected by both the 𝑖’th user and 𝑗’th item. Therefore,
we define the observations relevant to 𝑋
𝑖𝑗
with the set as
follows:
𝒮
CF
𝑖𝑗
= {𝑋
𝑖
′
𝑗
′
∣(𝑖
′
= 𝑖 ∨ 𝑗
′
= 𝑗) ∧ (𝑖
′
,𝑗
′
) ∈ (i, j)} (5)
𝒮
CF
𝑖𝑗
is actually constructed by the observed rating degrees
in the 𝑖-th row and 𝑗-th column of the rating matrix in CF.
Again, we calculate the radius 𝛿
CF
𝑖𝑗
for each observed r ating
degree 𝑋
𝑖𝑗
according to Definition 1 based on the standard
deviation of the ratings in 𝒮
CF
𝑖𝑗
:
𝛿
CF
𝑖𝑗
:= 𝛼 ⋅ std(𝒮
CF
𝑖𝑗
) (6)
where 𝛼 ∈ ℝ
+
is again a multiplicative scale coefficient.
Intuitively, a user’s ratings on different items (or the ratings
of a item from different users) vary greatly, we should assign
a big value of interval radius to this entry. Then, it’s easy
to calculate the bounds of interval-valued input for I-PMF
according to min-max representation (Definition 2). A exam-
ple of interval-valued rating matrix with its corresponding
single-valued matrix in min-max representation are shown
in Table II-A
III. M
ATR I X FACTORIZATION WITH APPLICATIONS
In this section we briefly discuss the MF techniques with
their applications. We devote special attention the the NMF
and PMF techniques since they serve to be the single-valued
counterparts of the proposed IMF models.
MF is a linear approximation data representation for the
original data matrix 𝑿 ∈ ℝ
𝑛×𝑑
. Generally, we have
𝑿 → 𝑼𝑽 (7)
where 𝑼 ∈ ℝ
𝑛×𝑘
and 𝑽 ∈ ℝ
𝑘×𝑑
. Each data instance 𝑋
𝑖⋅
is approximated by a linear combination of the rows of 𝑽
with weight vector 𝑼
𝑖⋅
,the𝑖’th row of 𝑼 . Thus, we call
𝑼 as weight matrix and 𝑽 as basis matrix. The ranks of
𝑼 and 𝑽 are always much lower than the rank of 𝑿, i.e.
𝑘 ≪ 𝑚𝑖𝑛(𝑛, 𝑑). After learning 𝑼 and 𝑽 , we can reconstruct
𝑿 as follows
ˆ
𝑿 ← 𝑼𝑽 (8)
Various assumptions over 𝑼 and 𝑽 lead to different MF
models which have been widely used in data mining appli-
cations. The following two series of applications are relevant
to this paper:
Parts-based representation: MF naturally represent the
original data matrix 𝑿 by parts. The rows in 𝑽 , so-called
basis vectors, are optimized for the linear approximation
for 𝑿 , and 𝑼
𝑖⋅
could be regard as a representation for the
𝑋
𝑖⋅
with lower dimensionality. NMF has been successfully
applied to find addictive parts-based representations for face
images (see for detail in Section III-A).
Missing Data Prediction: The reconstructed matrix
ˆ
𝑿 is
a full matrix. Therefore, when 𝑿 is sparse, we can make
prediction for its missing entries based on
ˆ
𝑿. For example,
PMF is successfully applied to predict the missing entries
of the rating matrices in CF (see for detail in Section III-B).
A. Nonnegative Matrix Factorization
NMF aims to factorize a nonnegative matrix 𝑿 ∈ ℝ
𝑛×𝑑
+
with two nonnegative matrices 𝑼 ∈ ℝ
𝑛×𝑘
+
and 𝑽 ∈ ℝ
𝑘×𝑑
+
which minimize the following 𝐿
2
loss function
ℒ
NMF
= ∥𝑿 − 𝑼𝑽 ∥
2
F
s.t. 𝑼 ≥ 0, 𝑽 ≥ 0
(9)
where ∥⋅∥
F
denotes the Frobenius norm. The estimations
of 𝑼 and 𝑽 can be find via the multiplicative update rules
proposed in [3], which iteratively update 𝑼 and 𝑽 as follows
1039
𝑈
𝑖𝑗
← 𝑈
𝑖𝑗
(𝑿𝑽
𝑇
)
𝑖𝑗
(𝑼𝑽 𝑽
𝑇
)
𝑖𝑗
𝑉
𝑖𝑗
← 𝑉
𝑖𝑗
(𝑼
𝑇
𝑿)
𝑖𝑗
(𝑼
𝑇
𝑼𝑽 )
𝑖𝑗
(10)
The update rules in (10) can be deduced according to
Karush-Kunhn-Trucker optimal condition [9] of inequality
constraint (see for detail in [10]). In [3], it is proved that
the updates in (10) lead to a local minimum of (9). The
non-negative constraints on 𝑼 and 𝑽 only allow addictive
linear combination of basis vectors in 𝑽 , so-called parts-
based representation [4]. NMF is suited for many real world
applications such as human face analysis [4]. In human
face analysis, the resultant matrix 𝑼 constructs a optimized
representation for the original data instances. Many FA
algorithms, such as face recognition, face clustering, may
be directly applied on 𝑼 instead of the original data matrix
𝑿.
B. Probabilistic Matrix Factorization
In CF, the PMF model [5] assume that the ratings are
drawn from some Gaussian distribution.
𝑝(𝑋
𝑖𝑗
∣𝑖, 𝑗, 𝑼 , 𝑽 ,𝜎
2
)=G(𝑋
𝑖𝑗
∣𝑼
𝑖⋅
𝑽
⋅𝑗
,𝜎
2
) (11)
For 𝑼 and 𝑽 , they place zero-mean spherical Gaussian
priors
𝑝(𝑼 ∣𝜎
2
1
)=
∏
𝑖
G(𝑼
𝑖⋅
∣0,𝜎
2
1
𝑰),𝑝(𝑽 ∣𝜎
2
1
)=
∏
𝑗
G(𝑽
⋅𝑗
∣0,𝜎
2
1
𝑰)
(12)
The 𝑼 and 𝑽 are computed via over the observed ratings
ℒ
PMF
= ∥𝑿 − 𝑼𝑽 ∥
2
F
+ 𝜆
[
∥𝑼 ∥
2
F
+ ∥𝑽 ∥
2
F
]
(13)
where 𝜆 = 𝜎
2
/𝜎
2
1
. A local minimum of (13) can be found
via gradient decent in 𝑼
𝑖⋅
and 𝑽
⋅𝑗
∂ℒ
PMF
∂𝑼
𝑖⋅
=
∑
𝑗∈j
𝑖
(𝑼
𝑖⋅
𝑽
⋅𝑗
− 𝑋
𝑖𝑗
)𝑽
𝑇
⋅𝑗
+ 𝜆𝑼
𝑖⋅
∂ℒ
PMF
∂𝑽
⋅𝑗
=
∑
𝑖∈i
𝑗
(𝑼
𝑖⋅
𝑽
𝑇
⋅𝑗
− 𝑋
𝑖𝑗
)𝑼
𝑇
𝑖⋅
+ 𝜆𝑽
⋅𝑗
(14)
Based on the learning of 𝑼 and 𝑽 , we can estimate the
unknown ratings in 𝑿 via
ˆ
𝑋
𝑖𝑗
= 𝑼
𝑖⋅
𝑽
⋅𝑗
(15)
IV. I
NTERVAL-VALUED MATR I X FACTORIZATION
In this section, we introduce the IMF framework. The
proposed framework is based on the Min-Max representation
of the interval-valued matrix: 𝐼(𝑿 )=[𝑿
low
, 𝑿
up
]. We can
extend the original MF over 𝑋 to the joint MF over 𝑿
low
and 𝑿
up
. Firstly, we assume each 𝑋
𝑖𝑗
is drawn from a
uniform distribution with parameters 𝑋
low
𝑖𝑗
and 𝑋
up
𝑖𝑗
.
𝑋
𝑖𝑗
∼ uniform(𝑋
low
𝑖𝑗
,𝑋
up
𝑖𝑗
) (16)
Base on this assumption, we have
E(𝑋
𝑖𝑗
)=
1
2
(𝑋
low
𝑖𝑗
+ 𝑋
up
𝑖𝑗
) (17)
Therefore, we propose to estimate the bounds of 𝐼(𝑿) first
via the following joint MF
𝑿
low
→ 𝑼𝑽
low
, 𝑿
up
→ 𝑼𝑽
up
(18)
We fix the weight matrix 𝑼 to make a unique profile for
each data instance and use 𝑽
low
, 𝑽
up
to maintain the data
approximation. The reconstructions of 𝑿
low
and 𝑿
up
could
be calculated as follows
ˆ
𝑿
low
← 𝑼𝑽
low
,
ˆ
𝑿
up
← 𝑼𝑽
up
(19)
According to (17) and (19), we can reconstruct 𝑿 via
ˆ
𝑿 ←
1
2
(
ˆ
𝑿
low
+
ˆ
𝑿
up
) (20)
A. Interval-valued NMF
According to (9) and (18), the 𝐿
2
loss function of interval-
valued NMF (I-NMF f or short) is
ℒ
I−NMF
= ∥𝑿
low
− 𝑼𝑽
low
∥
2
F
+ ∥𝑿
up
− 𝑼𝑽
up
∥
2
F
s.t. 𝑼 ≥ 0, 𝑽
low
≥ 0, 𝑽
up
≥ 0
(21)
Similar to traditional NMF, we have the following multi-
plicative update rule for 𝑼 , 𝑽
low
and 𝑽
up
:
𝑈
𝑡+1
𝑖𝑗
← 𝑈
𝑡
𝑖𝑗
[𝑿
low
(𝑽
low
)
𝑇
+ 𝑿
up
(𝑽
up
)
𝑇
]
𝑖𝑗
[𝑼𝑽
low
(𝑽
low
)
𝑇
+ 𝑼𝑽
up
(𝑽
up
)
𝑇
]
𝑖𝑗
𝑉
low,𝑡+1
𝑖𝑗
← 𝑉
low,𝑡
𝑖𝑗
(𝑼
𝑇
𝑿
low
)
𝑖𝑗
(𝑼
𝑇
𝑼𝑽
low
)
𝑖𝑗
𝑉
up,𝑡+1
𝑖𝑗
← 𝑉
up,𝑡
𝑖𝑗
(𝑼
𝑇
𝑿
up
)
𝑖𝑗
(𝑼
𝑇
𝑼𝑽
up
)
𝑖𝑗
(22)
Similar to traditional NMF, we also have that the 𝐿
2
loss
function ℒ
I−NMF
as shown in (21) is nonincreasing under
the multiplicative update rules as shown in (22).
Traditional NMF decomposes the original data matrix
into two low-rank factor matrices: one profiles the data
instances while the other profiles the features. In I-NMF,
the proposed the joint matrix factorization framework makes
the feature profile factor matrices 𝑽
low
⋅𝑗
and 𝑽
up
⋅𝑗
contain the
data approximation while preserving a unique profile 𝑼
𝑖⋅
for
each data instance. We can directly apply the face analysis
techniques over 𝑼 .
B. Interval-valued PMF
In this section we introduce the interval-valued PMF (I-
PMF for short). Analogously to (13) and according to (18),
we have the following regularized 𝐿
2
loss
ℒ
I−PMF
= ∥𝑿
low
− 𝑼𝑽
low
∥
F
+ ∥𝑿
up
− 𝑼𝑽
up
∥
2
F
+𝜆
(
∥𝑼 ∥
2
F
+ ∥𝑽
low
∥
2
F
+ ∥𝑽
up
∥
2
F
)
(23)
1040
20 22 24 26 28 30 32 34 36 38 40
0.87
0.875
0.88
0.885
0.89
0.895
0.9
0.905
0.91
0.915
0.92
Face Recognition
F1 Measure
Number of Factors (k)
ORL32: Raw
ORL32: NMF
ORL32: I−NMF
20 22 24 26 28 30 32 34 36 38 40
13.5
14
14.5
15
15.5
16
16.5
17
Face Reconstruction
RE
Number of Factors (k)
20 22 24 26 28 30 32 34 36 38 40
0.62
0.63
0.64
0.65
0.66
0.67
0.68
0.69
0.7
0.71
Number of Factors (k)
ACC
Face Clustering
ORL64: Raw
ORL64: NMF
ORL64: I−NMF
20 22 24 26 28 30 32 34 36 38 40
0.78
0.785
0.79
0.795
0.8
0.805
0.81
0.815
0.82
0.825
0.83
Face Clustering
NMI
Number of Factors (k)
Figure 3. Performance comparison in face analysis.
It is easy to derive a gradient decent in 𝑼
𝑖⋅
, 𝑽
low
⋅𝑗
and
𝑽
up
⋅𝑗
to find a local minimum of (23).
∂ℒ
I−PMF
∂𝑼
𝑖⋅
=
∑
𝑗∈j
𝑖
[(𝑼
𝑖⋅
𝑽
low
⋅𝑗
− 𝑋
low
𝑖𝑗
)𝑽
low𝑇
⋅𝑗
+(𝑼
𝑖⋅
𝑽
up
⋅𝑗
− 𝑋
up
𝑖𝑗
)𝑽
up𝑇
⋅𝑗
]+𝜆𝑼
𝑖⋅
∂ℒ
I−PMF
∂𝑽
low
⋅𝑗
=
∑
𝑖∈i
𝑗
(𝑼
𝑖⋅
𝑽
low
⋅𝑗
− 𝑀
low
𝑢𝑚
)𝑼
𝑇
𝑖⋅
+ 𝜆𝑽
low
⋅𝑗
∂ℒ
I−PMF
∂𝑽
up
⋅𝑗
=
∑
𝑖∈i
𝑗
(𝑼
𝑖⋅
𝑽
up
⋅𝑗
− 𝑀
up
𝑢𝑚
)𝑼
𝑇
𝑖⋅
+ 𝜆𝑽
up
⋅𝑗
(24)
For CF application, we can used the learned 𝑼 , 𝑽
low
and
𝑽
up
to compute the unknown ratings via (19) and (20).
V. E
XPERIMENTAL RESULTS
We divide the experiments into two parts: In Section V-A
we conduct the comparison of I-NMF against the basic NMF
for FA applications, and in Section V-B we compare the
performance of I-PMF and PMF over CF applications.
A. Comparison of I-NMF against NMF
We compare the performance of NMF and I-NMF on
various FA applications including face recognition, face
reconstruction and face clustering.
1) Data Description and Evaluation Setting: We use the
Olivertti Research Laboratory (ORL) face data sets to evalu-
ate the NMF and I-NMF models, which contain ten different
images of each of 40 distinct persons, (𝑛 =10× 40 = 400
in total). Two versions of processed data sets
1
: one with res-
olution 32×32 (ORL32) and the other with 64×64 (ORL64),
are used for our experimental evaluation. In ORL32, each
face image is represented by a vector with dimensionality
𝑑 =32× 32 = 1024 while in ORL64, 𝑑 =64× 64 = 4096.
We implement I-NMF based on multiplicative update
rules introduced in Section IV-A. The experiments for NMF
1
http://www.cs.uiuc.edu/homes/dengcai2/Data/FaceData.html
and are based on the DTU NMF toolbox
2
. Various classifiers
has been adopted for face recognition and in this paper, we
apply the the nearest neighbor method for its simplicity. For
face clustering, we choose the popular K-means algorithm.
All the classification and clustering algorithms are applied
on the output weight matrices 𝑼 from NMF and I-NMF and
we also give the performance of these algorithms over the
raw data matrix 𝑿 as the baseline. In the construction of
interval-valued matrices (4), we set 𝑟 =5and 𝛼 =2.5.
We evaluate the proposed models in terms of the face
recognition and clustering effectiveness. Note that face
recognition is actually a classification problem. To evaluate
the effectiveness of classification (FR), we use the standard
𝐹 1 measure. We adopt two popular metrics Normalized
Mutual Information (NMI) [11] and Clustering Accuracy
(ACC) for cluster evaluation. Based on NMF, the faces
are reconstructed with the weighted summation of basis
vectors. We use the following Reconstruction Error (RE):
RE(
ˆ
𝑿, 𝑿 )=
√
∑
𝑛
𝑖=1
∑
𝑑
𝑗=1
(
ˆ
𝑋
𝑖𝑗
−𝑋
𝑖𝑗
)
2
𝑛×𝑑
to evaluate the good-
ness of reconstruction matrix
ˆ
𝑿 according to the original
data matrix 𝑿 .
Note that larger values of F1, NMI and ACC indicate bet-
ter face recognition or clustering results while small values
of RE indicate better performance of face reconstruction.
2) Evaluation Results: We compare the models with
varying rank of factor matrix 𝑘 and interval sizes.
Evaluation with varying 𝑘: The face clustering and face
reconstruction tasks are evaluated over entire data sets. For
the face recognition task, we make ten rounds of random
sampling of 50% data for training. In general, the perfor-
mance of NMF and I-NMF for all the face analysis tasks
varies with the number of latent factors (𝑘). For each value
of 𝑘, we run 100 rounds of NMF and I-NMF. The average
values of the performance metrics plotted for each model
as shown in Figure 3 where each sub-figure corresponds
to a face analysis task with the specific evaluation metric
and each line corresponds to a model on a specific data
set.From Figure 3, we see that I-NMF outperforms NMF
with statistical significance over all evaluation metrics on
both two data sets.
B. Comparison of I-PMF against PMF
1) Data Description and Evaluation Setting: In this part
of experiments, we also use two data sets for evaluation.
Movielens data set
3
is downloaded from the web-site of
GroupLens research group and we use the subset which
contains 100,000 ratings for 𝑑 = 1682 movies by 𝑛 = 943
users of the online movie recommender service. We name
this data set as Movielens-100K. Netflix data set
4
is the
official data set used in the Netflix Prize competition. Again,
2
http://isp.imm.dtu.dk/toolbox/nmf/nmf toolbox ver1.4.zip
3
http://www.grouplens.org/system/files/ml-data 0.zip
4
http://archive.ics.uci.edu/ml/datasets/Netflix+Prize
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