LOW-RANK DATA MATRIX RECOVERY WITH MISSING VALUES AND FAULTY SENSORS
Roberto L
´
opez-Valcarce
∗
Universidade de Vigo, Spain
Josep Sala-Alvarez
†
Universitat Polit
`
ecnica de Catalunya, Spain
ABSTRACT
In practice, data gathered by wireless sensor networks often
belongs in a low-dimensional subspace, but it can present
missing as well as corrupted values due to sensor malfunc-
tioning and/or malicious attacks. We study the problem
of Maximum Likelihood estimation of the low-rank factors
of the underlying structure in such situation, and develop
an Expectation-Maximization algorithm to this purpose, to-
gether with an effective initialization scheme. The proposed
method outperforms previous schemes based on an initial
faulty sensor identification stage, and is competitive in terms
of complexity and performance with convex optimization-
based matrix completion approaches.
Index Terms— Low-rank approximation, matrix comple-
tion, outliers, faulty sensors, wireless sensor networks.
1. INTRODUCTION
The fact that many high-dimensional data structures can be
accurately represented as lying in a low-dimensional sub-
space has spurred a number of approaches to exploit such
low-rank structure [1]. In particular, in wireless sensor net-
works (WSNs), the measurements taken by K sensors over
N sensing slots can be organized into a K × N data matrix,
which in many practical applications will be close to having
rank r min{K, N}. This is because environmental data
usually exhibits strong correlation across space and time [2].
When applied to WSNs, matrix completion techniques [1, 3],
which exploit this property in order to recover the whole data
matrix from a subset of its samples, may result in significant
savings in sensing and communication [4, 5]. Missing data
may also occur if a transmission from a sensor to the Fusion
Center (FC) is affected by a deep channel fade.
WSNs consist of low-cost, battery-powered devices often
operating in harsh environments, and therefore prone to sen-
sor malfunction [6, 7], making data generated by faulty sen-
sors unreliable. Additionally, sensors are usually unattended,
∗
Supported by Agencia Estatal de Investigaci
´
on (Spain) and the European
Regional Development Fund (ERDF) under project WINTER (TEC2016-
76409-C2-2-R), and by Xunta de Galicia (Agrupaci
´
on Estrat
´
exica Conso-
lidada de Galicia accreditation 2016-2019).
†
Supported by Agencia Estatal de Investigaci
´
on (Spain) and ERDF under
project WINTER (TEC2016-76409-C2-1-R), and the Catalan administration
(AGAUR) under 2017 SGR 578.
which makes them vulnerable to data manipulation by mali-
cious external agents [8,9]. Regardless of its origin, corrupted
samples may significantly degrade data analysis at the FC.
In this paper we investigate algorithms that estimate the
low-rank factors of the underlying data matrix in the presence
of both missing data and unreliable sensors. In particular, we
adopt a non-informative model under which faulty sensors,
whose number and identities are not known, produce i.i.d.
data samples uniformly distributed in some unknown interval.
In this setting, Maximum Likelihood (ML) estimators are at-
tractive because of their good asymptotic performance prop-
erties [15]. The presence of hidden variables (i.e., missing
data, identities of faulty sensors) precludes a closed-form so-
lution for the ML estimators and motivates the application of
the Expectation-Maximization (E-M) algorithm [16], which
iteratively refines the estimates of the low-rank factors as well
as a posteriori probabilities of individual sensor faults.
Matrix completion methods have become increasingly
popular in applications in which a low-rank matrix is to be
recovered from a limited number of observations. Typically,
they seek to minimize the nuclear norm (which is a convex
surrogate of the rank) under some constraint on the fidelity
of the reconstruction to account for missing data and/or mea-
surement noise [17,18]. When the underlying rank is assumed
known, non-convex approaches based on the so-called Burer-
Monteiro factorization [19] are also available [20, 21]. In the
presence of corrupted data, convex minimization techniques
have also been proposed by exploiting the fact that such
anomalous data can be assumed to be sparse [22–24]. Many
of these methods enjoy theoretical performance guarantees
under suitable conditions [1, 3]; nevertheless, they invariably
include regularization terms whose parameters need appro-
priate tuning. This is not the case with the proposed E-M
approach, which iteratively estimates all unknown param-
eters; in addition, simulation results show that E-M may
outperform sparsity-based methods even with optimal tuning.
2. PROBLEM STATEMENT
Consider a network of K devices collecting data from their
environment. Each device, say sensor i, gathers N data points
{y
ij
, j = 1, . . . , N}, to be collected at the FC in the K × N
data matrix Y . It is assumed that the underlying physical
phenomenon gives rise to a matrix LR having low rank r
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min{K, N}, with L ∈ R
K×r
and R ∈ R
r×N
. Measure-
ments are noisy, so that under ideal conditions the data matrix
would be given by LR + N , where N is a noise matrix with
i.i.d. zero-mean Gaussian entries with variance σ
2
. However,
a number of sensors may be malfunctioning, in which case
their collected data do not adhere to the above model. Thus,
let a
i
= 0 if sensor i is faulty, and a
i
= 1 otherwise, and
collect these in a =
a
1
· · · a
K
T
. Whether a given
device is faulty or not is unknown at the FC. We assume that
sensor faults take place randomly and independently, with the
same a priori probability q for all devices, i.e., the a
i
’s are
i.i.d. Bernoulli random variables with Pr{a
i
= 0} = q. We
adopt a non-informative model under which the readings from
a defective device, say sensor i, are i.i.d. and uniformly dis-
tributed in some unknown interval [A
i
, B
i
].
Due to channel fading, device battery outages, etc., a sub-
set of the observations is missing at the FC. We denote by
Ω
i
the set of pairs (i, j) for which y
ij
is available, with car-
dinality |Ω
i
| = N
i
, and define Ω = Ω
1
∪ · · · ∪ Ω
K
, with
|Ω| =
P
K
i=1
N
i
, M. Then, arranging the available data in
the vector y ∈ R
M
, the data model can be expressed as
y = P
Ω
(A(LR + N ) + (I − A)W ) , (1)
where P
Ω
(X) extracts the entries of X with indices in Ω,
A , diag{a}, and W is a random matrix whose i-th row
entries are i.i.d., uniformly distributed in [A
i
, B
i
].
Our goal is to estimate the low-rank factors L, R given
y and Ω. If all sensors were working properly, and in the
absence of missing data, the ML estimates should minimize
kY − LRk
2
F
, where Y = unvec(y). By the Eckart-Young
theorem [10], these can be readily obtained from the SVD of
Y as
ˆ
L = U B
−1
and
ˆ
R = BSV
T
, where S is diagonal
with the r largest singular values, the columns of U, V com-
prise the corresponding left and right singular vectors, and
B is r × r invertible but otherwise arbitrary. However, with
(unknown) faulty sensors and missing data, the ML estimates
cannot be obtained in closed form.
3. E-M ALGORITHM DERIVATION
The E-M algorithm provides a computationally efficient
means to iteratively seek a maximum of the likelihood func-
tion in the presence of incomplete observations [16]. In
this context, we regard y as the incomplete dataset, and
z = {T , a} as the complete dataset, with T , A(LR +
N) + (I − A)W . The unknown parameters are collected in
θ = {L, R, σ
2
, q, A
1
, . . . , A
K
, B
1
, . . . , B
K
}.
The general form of the E-M algorithm is as follows.
Given the incomplete dataset y and an initial guess
ˆ
θ
0
, at
iteration k one performs the following:
• E-step: Compute the conditional expectation Q(θ;
ˆ
θ
k
) ,
E
z|y
{ln p(z; θ) | y ;
ˆ
θ
k
}, where p(z; θ) is the pdf of z pa-
rameterized by θ.
• M-step: the estimate is updated by maximizing this con-
ditional expectation, i.e.,
ˆ
θ
k+1
= arg max
θ
Q(θ;
ˆ
θ
k
).
3.1. Expectation step
Let t
i
be the i-th column of T . Due to independence, and
using the Bernoulli pdf p(a
i
; θ) = q
1−a
i
(1 − q)
a
i
, one has
p(z; θ) =
K
Y
i=1
p(t
i
, a
i
; θ) =
K
Y
i=1
p(t
i
| a
i
; θ)p(a
i
; θ)
=
K
Y
i=1
[qp
0
(t
i
; θ)]
1−a
i
[(1 − q)p
1
(t
i
; θ)]
a
i
, (2)
where for convenience we denote p
b
(t
i
; θ) , p(t
i
|a
i
= b; θ),
b ∈ {0, 1}. Now let us rewrite Q(θ;
ˆ
θ
k
) as
Q(θ;
ˆ
θ
k
) = E
T ,a|y
{ln p(z; θ) | y;
ˆ
θ
k
}
= E
a|y
n
E
T |a,y
{ln p(z; θ) | a, y;
ˆ
θ} | y;
ˆ
θ
k
o
. (3)
Using (2), the inner conditional expectation in (3) becomes
E
T |a,y
{ln p(z; θ) | a, y;
ˆ
θ
k
} = K ln q +
K
X
i=1
a
i
!
ln
1 − q
q
+
K
X
i=1
(1 − a
i
)E
t
i
|a
i
=0,y
{ln p
0
(t
i
; θ) | a
i
= 0, y ;
ˆ
θ
k
}
+
K
X
i=1
a
i
E
t
i
|a
i
=1,y
{ln p
1
(t
i
; θ) | a
i
= 1, y ;
ˆ
θ
k
}, (4)
since for any binary r.v. a ∈ {0, 1}, one has aE
t|a
{f(t)} =
a[(1 − a)E
t|a=0
{f(t)} + aE
t|a=1
{f(t)}] = aE
t|a=1
{f(t)};
analogously, (1−a)E
t|a
{f(t)} = (1−a)E
t|a=0
{f(t)}. Now,
the entries t
ij
of t
i
are independent under both a
i
= 0 and
a
i
= 1, so that p
b
(t
i
; θ) =
Q
N
j=1
p
b
(t
ij
; θ), b ∈ {0, 1}. Then,
for b ∈ {0, 1},
g
b
i
(y; θ;
ˆ
θ
k
) , E
t
i
|a
i
=b,y
{ln p
b
(t
i
; θ) | a
i
= b, y ;
ˆ
θ
k
}
=
N
X
j=1
E
t
ij
|a
i
=b,y
{ln p
b
(t
ij
; θ) | a
i
= b, y ;
ˆ
θ
k
} (5)
=
X
j∈Ω
i
ln p
b
(y
ij
; θ) +
X
j /∈Ω
i
f
b
ij
(θ;
ˆ
θ
k
), (6)
where f
b
ij
(θ;
ˆ
θ
k
) , E
t
ij
|a
i
=b
{ln p
b
(t
ij
; θ) | a
i
= b;
ˆ
θ
k
}.
Under our model, p
0
(t
ij
; θ) = U(t
ij
; A
i
, B
i
) is a uniform
pdf over [A
i
, B
i
], whereas p
1
(t
ij
; θ) = N (t
ij
; m
ij
, σ
2
) is
a Gaussian pdf with mean m
ij
, (LR)
ij
and variance σ
2
.
Hence, upon defining ˆm
ij,k
, (
ˆ
L
k
ˆ
R
k
)
ij
, it follows that
f
0
ij
(θ;
ˆ
θ
k
) =
− ln(B
i
− A
i
), A
i
≤
ˆ
A
i,k
, B
i
≥
ˆ
B
i,k
,
−∞, else,
(7)
f
1
ij
(θ;
ˆ
θ
k
) = −
1
2
ln 2πσ
2
−
1
2σ
2
[ˆσ
2
k
+ ( ˆm
ij,k
− m
ij
)
2
]. (8)
Thus, letting ˆa
i,k
, E
a
i
|y
{a
i
|y;
ˆ
θ
k
} = Pr{a
i
= 1|y;
ˆ
θ
k
},
Q(θ;
ˆ
θ
k
) = K ln q +
K
X
i=1
ˆa
i,k
!
ln
1 − q
q
+
K
X
i=1
h
(1 − ˆa
i,k
)g
0
i
(y; θ;
ˆ
θ
k
) + ˆa
i,k
g
1
i
(y; θ;
ˆ
θ
k
)
i
. (9)
Note that ˆa
i,k
can be interpreted as the a posteriori probabil-
ity of sesor i being non-defective, given the data y and the
estimate
ˆ
θ
k
. Using Bayes’ theorem, it is obtained as
ˆa
i,k
=
(1 − ˆq
k
)
Q
j∈Ω
i
p
1
(y
ij
;
ˆ
θ
k
)
ˆq
k
Q
j∈Ω
i
p
0
(y
ij
;
ˆ
θ
k
) + (1 − ˆq
k
)
Q
j∈Ω
i
p
1
(y
ij
;
ˆ
θ
k
)
.
(10)
3.2. Maximization step
In order to maximize Q(θ;
ˆ
θ
k
) in (9) w.r.t. θ, we proceed step
by step. First, the optimum value of q is readily found:
ˆq
k+1
= 1 −
1
K
K
X
i=1
ˆa
i,k
. (11)
Second, note that the values of A
i
, B
i
maximizing Q(θ;
ˆ
θ
k
)
are those maximizing g
0
i
(y; θ;
ˆ
θ
k
). This function evaluates to
−∞ unless A
i
≤
ˆ
A
i,k
, B
i
≥
ˆ
B
i,k
, and A
i
≤ y
ij
, B
i
≥ y
ij
∀j ∈ Ω
i
, in which case it evaluates to −N ln(B
i
− A
i
). Sub-
ject to the above constraints, this is maximized at
ˆ
A
i,k+1
=
min{
ˆ
A
i,k
} ∪{y
ij
, j ∈ Ω
i
},
ˆ
B
i,k+1
= max{
ˆ
B
i,k
} ∪{y
ij
, j ∈
Ω
i
}. Therefore, it is clear that we can simply take, for all k,
ˆ
A
i,k
= min
j∈Ω
i
{y
ij
},
ˆ
B
i,k
= max
j∈Ω
i
{y
ij
}. (12)
Third, letting c
i
, 1 −
N
i
N
, the values of L, R maximizing
Q(θ;
ˆ
θ
k
) in (9) are seen to be those minimizing the term
−2
K
X
i=1
ˆa
i,k
g
1
i
(y; θ;
ˆ
θ
k
) = N
K
X
i=1
ˆa
i,k
ln 2πσ
2
+ c
i
ˆσ
2
k
σ
2
+
1
σ
2
K
X
i=1
ˆa
i,k
X
j∈Ω
i
(y
ij
− m
ij
)
2
+
X
j /∈Ω
i
( ˆm
ij,k
− m
ij
)
2
(13)
Only the last term in the right-hand side of (13) depends on
L, R via {m
ij
}. If we define
ˆ
Y
k
∈ R
K×N
with entries
ˆy
ij,k
= y
ij
, j ∈ Ω
i
, ˆy
ij,k
= ˆm
ij,k
, j /∈ Ω
i
, (14)
and the diagonal matrix
ˆ
A
k
= diag{
ˆa
1,k
· · · ˆa
K,k
},
then such term can be rewritten so as to yield
min
L,R
1
σ
2
ˆ
A
1/2
k
(
ˆ
Y
k
− LR)
2
F
, (15)
which is a particular case of the weighted low-rank approx-
imation (WLRA) problem [12]. Although general WLRA
problems do not generally admit closed-form solutions, the
special structure of (15), in which the weighting is on a row-
by-row basis, constitutes an exception, as shown in [13]. This
can be seen by noticing in (15) that
ˆ
A
1/2
k
L and R are the
factors in the (unweighted) rank-r approximation of
ˆ
A
1/2
k
ˆ
Y
k
.
Therefore, we first compute the SVD of
ˆ
A
1/2
k
ˆ
Y
k
and trun-
cate it to its r principal components, say U
k
S
k
V
T
k
, where
U
k
∈ R
K×r
, V
k
∈ R
N×r
, and S
k
is r × r diagonal with the
largest singular values. Then we set
ˆ
L
k+1
=
ˆ
A
−1/2
k
U
k
,
ˆ
R
k+1
= S
k
V
T
k
. (16)
Finally, the value of σ
2
maximizing Q(θ;
ˆ
θ
k
) is the one min-
imizing (13), and is given by
ˆσ
2
k+1
=
1
N
ˆ
A
1/2
k
(
ˆ
Y
k
−
ˆ
L
k+1
ˆ
R
k+1
)
2
F
+ ˆσ
2
k
P
K
i=1
ˆa
i,k
c
i
P
K
i=1
ˆa
i,k
.
(17)
4. INITIALIZATION
Since the E-M iteration may in general converge to a local
maximum of the likelihood function, its initialization is a crit-
ical issue. Next we describe an initialization scheme which
appears to be quite effective. It is based on the PCA-based
“row-structure fault detection algorithm” from [14], a statis-
tical test to check if a given sensor is faulty. Specifically, let
the zero-padded data matrix
e
Y ∈ R
K×N
be given by
e
Y
ij
= y
ij
, (i, j) ∈ Ω,
e
Y
ij
= 0 (i, j) /∈ Ω, (18)
and let
e
µ
T
=
1
K
1
T
K
e
Y be the average of its rows. The
N × N sample covariance matrix is given by C =
1
K
(
e
Y −
1
K
e
µ
T
)
T
(
e
Y − 1
K
e
µ
T
). Let now Λ
r
∈ R
r×r
be diagonal
with the r largest eigenvalues of C, and let the columns of
U
r
∈ R
N×r
comprise the corresponding r eigenvectors.
According to [14], the test statistic
d
2
i
= kΛ
−1/2
r
U
T
r
(
e
y
i
−
e
µ)k
2
, (19)
where
e
y
T
i
is the i-th row of
e
Y , is approximately χ
2
r
-distributed
if the data from sensor i is not abnormal. Using this, the
authors of [14] declare sensor i as faulty if d
2
i
exceeds a
threshold, which is set to achieve a given probability of false
alarm P
FA
. Since d
2
i
will likely take large values when sensor
i is defective yielding corrupted data with high energy, we
propose to initialize the probabilities ˆa
i,0
proportionally to
these values after normalization. Specifically,
ˆa
i,0
= β ·
d
2
i
max
1≤j≤K
{d
2
j
}
, i = 1, . . . , K, (20)
where 0 < β ≤ 1 is a suitable constant. In practice, β = 0.95
has been found to give good results. Finally, the initial value
ˆ
L
0
ˆ
R
0
is taken as the best rank-r approximation of
e
Y .
0 5 10 15 20 25 30
iterations
0
0.1
0.2
0.3
0.4
0.5
0.6
NRMSE over healthy nodes
Naive
Hard fault detection
Clairvoyant
EM
Outlier Pursuit
Fig. 1. NRMSE vs. iterations (r = 10).
5. NUMERICAL RESULTS
Consider a setting with K = 100, N = 70. Entries of ground
truth matrices L ∈ R
K×r
, R ∈ R
r×N
are independently
drawn from a standard Gaussian distribution. The probabil-
ities of sensor malfunction and observing a given data point
are 0.1 and 0.65, respectively. The signal-to-noise ratio, de-
fined as kLRk
2
F
/(KNσ
2
), is set to 20 dB. The limits of the
uniform pdf for faulty sensor data are generated as A
i
= −B
i
with each B
i
independently drawn from a uniform distribu-
tion over [0, 65σ] to capture a wide range of sensor fault types.
In addition to the proposed E-M estimator, the following
schemes were also implemented for comparison:
• A “naive” scheme oblivious to sensor faults, which seeks a
minimum of ky−P
Ω
(LR)k
2
F
by alternatingly optimizing
over one of the factors while fixing the other [20], initial-
ized at the best rank-r approximation of the zero-padded
matrix
e
Y .
• A “clairvoyant” scheme with knowledge about which sen-
sors are faulty, which applies a similar alternating mini-
mization approach after discarding the corrupted data.
• A scheme based on a preliminary hard fault detection
stage declaring sensor i faulty if d
2
i
in (19) exceeds a
threshold. Data estimated as corrupted are then discarded
before applying the alternating minimization scheme
above. As in [14], the threshold is set for P
FA
= 0.025.
• A scheme based on the “noisy outlier pursuit” method
from [23], solving the following convex problem:
min
M,S
kMk
∗
+ λkS
T
k
1,2
s.to ky − P
Ω
(M + S)k ≤ ,
(21)
where k · k
∗
is the nuclear norm, and kBk
1,2
is the sum
of Euclidean norms of the columns of B. In (21), M is
0 5 10 15 20 25
rank r
0
0.1
0.2
0.3
0.4
0.5
NRMSE over healthy nodes
Naive
Hard fault detection
Clairvoyant
EM
Outlier Pursuit
Fig. 2. NRMSE for the different schemes vs. rank r.
an estimate of the low-rank term LR, whereas S is an es-
timate of the corrupted data, with the term kS
T
k
1,2
pro-
moting row-sparsity. The tuning parameters were set to
λ = 0.9 and = 0.01 by trial and error.
For r = 10, Fig. 1 shows the evolution of the normalized
root mean squared error (NRMSE) over clean data
1
, defined
as
NRMSE =
kA(LR −
ˆ
L
ˆ
R)k
F
kALRk
F
(22)
and averaged over 100 Monte Carlo runs (for the outlier pur-
suit scheme,
ˆ
L
ˆ
R in (22) is replaced by M ). Fig. 2 shows the
NRMSE as a function of the ground-truth rank r.
The performance of the clairvoyant estimator constitutes
a benchmark. The naive estimator behaves poorly, as could
be expected: the effect of corrupted data from faulty sensors
cannot be lightly dismissed. In this scenario, the improvement
by applying a data cleansing stage as suggested in [14] (hard
fault detection) is limited. It was observed that whenever the
corrupted data from a faulty sensor has small energy, the fault
detection mechanism based on (19) is not effective, but nev-
ertheless the impact on matrix recovery of such low-energy
abnormal data remains significant, which explains the poor
performance of this scheme. The outlier-pursuit method is
seen to provide reasonable performance, although it requires
fine tuning of the user-selectable parameters; in addition, its
computational load may be a bottleneck as matrix sizes grow.
In this sense, we note that the complexity of the proposed E-M
scheme is dominated by step (15), which requires the compu-
tation of the SVD of a K × N matrix at each iteration. The
E-M estimator gets close to the benchmark, showing the efec-
tiveness of the proposed initialization, with a graceful degra-
dation in performance as the rank r increases.
1
Corrupted data from faulty sensors is excluded from (22) since it is not
possible to recover a whole row of LR from the low-rank property alone.
We repeated the simulations under the same setting, but
generating the corrupted data for faulty sensor i as i.i.d. Gaus-
sian with mean (A
i
+B
i
)/2 and variance
1
12
(B
i
−A
i
)
2
, where
A
i
, B
i
were generated in the same way as above. Results
were roughly similar to those with uniformly distributed out-
liers, indicating that E-M is robust to deviations from the pro-
posed non-informative model for corrupted data.
6. CONCLUSIONS
We have presented an ML approach to the problem of matrix
completion under noisy data and unreliable sensors. Moti-
vated by the presence of unobserved variables, we developed
the E-M iterative algorithm for this problem, together with
an effective initialization strategy. E-M outperforms schemes
based on hard decisions about the sensor labels, at a fraction
of the computational cost of convex optimization approaches,
and without requiring parameter tuning.
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