A Compressive Sensing Based Compressed Neural
Network for Sound Source Localization
Mehdi Banitalebi Dehkordi
Speech Processing Research Lab
Elec. and Comp. Eng. Dept.,
Yazd University
Yazd, Iran
mahdi_Banitalebi@stu.yazduni.ac.ir
Hamid Reza Abutalebi
Elec. and Comp. Eng. Dept.,
Yazd University,
Yazd, Iran &
Idiap Research Institute,
Martigny, Switzerland
habutalebi@yazduni.ac.ir
Hossein Ghanei
Elec. and Comp. Eng. Dept.,
Yazd University,
Yazd, Iran
hghaneiy@yazduni.ac.ir
Abstract —Microphone arrays are today employed to specify the
sound source locations in numerous real time applications such
as speech processing in large rooms or acoustic echo cancellation.
Signal sources may exist in the near field or far field with respect
to the microphones. Current Neural Networks (NNs) based
source localization approaches assume far field narrowband
sources. One of the important limitations of these NN-based
approaches is making balance between computational complexity
and the development of NNs; an architecture that is too large or
too small will affect the performance in terms of generalization
and computational cost. In the previous analysis, saliency subject
has been employed to determine the most suitable structure,
however, it is time-consuming and the performance is not robust.
In this paper, a family of new algorithms for compression of NNs
is presented based on Compressive Sampling (CS) theory. The
proposed framework makes it possible to find a sparse structure
for NNs, and then the designed neural network is compressed by
using CS. The key difference between our algorithm and the
state-of-the-art techniques is that the mapping is continuously
done using the most effective features; therefore, the proposed
method has a fast convergence. The empirical work demonstrates
that the proposed algorithm is an effective alternative to
traditional methods in terms of accuracy and computational
complexity.
Keywords- compressive sampling; sound source; neural
network; pruning; multilayer Perceptron; greedy algorithms.
I. INTRODUCTION
Location of a sound source is an important piece of
information in speech signal processing applications. In the
sound source localization techniques, location of the source has
to be estimated automatically by calculating the direction of the
received signal [1]. Most algorithms for these calculations are
computationally intensive and difficult for real time
implementation [2]. Neural network based techniques have
been proposed to overcome the computational complexity
problem by exploiting their massive parallelism [3,4]. These
techniques usually assume narrowband far field source signal,
which is not always applicable [2].
In this paper, we design a system that estimates the
direction-of-arrival (DOA) (direction of received signal) for far
field and near field wide band sources. The proposed system
uses feature extraction followed by a neural network. Feature
extraction is the process of selection of the useful data for
estimation of DOA. The estimation is performed by the use CS.
The neural network, which performs the pattern recognition
step, computes the DOA to locate the sound source. The
important key insight is the use of the instantaneous cross-
power spectrum at each pair of sensors. Instantaneous cross-
power spectrum means the cross-power spectrum calculated
without any averaging over realizations. This step calculates
the discrete Fourier transform (DFT) of the signals at all
sensors. In the compressive sampling step K coefficients of this
DFT transforms are selected, and then multiplies the DFT
coefficients at these selected frequencies using the complex
conjugate of the coefficients in the neighboring sensors. In
comparison to the other cross-power spectrum estimation
techniques (which multiply each pair of DFT coefficients and
average the results), we have reduced the computational
complexity. After this step we have compressed the neural
network that is designed with these feature vectors. We
propose a family of new algorithms based on CS to achieve
this. The main advantage of this framework is that these
algorithms are capable of iteratively building up the sparse
topology, while maintaining the training accuracy of the
original larger architecture. Experimental and simulation
results showed that by use of NNs and CS we can design a
compressed neural network for locating the sound source with
acceptable accuracy.
The remainder of the paper is organized as follows. The
next section presents a review of techniques for sound source
localization. Section III explains feature selection and discusses
the training and testing procedures of our sound source
localization technique. Section IV describes traditional pruning
algorithms and compressive sampling theory and section V
contains the details of the new network pruning approach by
describing the link between pruning NNs and CS and the
introduction two definitions for different sparse matrices.
Experimental results are illustrated in Section VI while VII
concludes the paper.
II. S
OUND SOURCE LOCALIZATION
Sound source localization is performed by the use of DOA.
The assumption of far field sources remains true while the
distance between source and reference microphone is larger
than
[2] fig. 1. In this equation
is the minimum
wavelength of the source signal, and D is the microphone array
length. With this condition, incoming waves are approximately
planar. So, the time delay of the received signal between the
reference microphone and the −ℎ microphone would be
[15]:
=
(
−1
)
=(−1)
(1)
In (1) is the distance between two microphones, is the
DOA, and is the velocity of sound in air. Therefore,
is the
amount of time that the signal traverses the distance between
any two neighboring microphones, Fig. 1 and 2 illustrates this
fact.
ϕ
)(ts
)(
0
tts +
)2(
0
tts +
)3(
0
tts +
ϕ
ϕ
ϕ
Figure 1. Estimation of far-field source location
ϕ
)(ts
)(
0
tts +
)2(
0
tts +
)3(
0
tts +
Figure 2. Estimation of near-field source location
If the distances between source and microphones are not far
enough, then time delay of the received signal between the
reference microphone and the −ℎ microphone would be
[15] fig. 2:
=
(
)
(
(
)
)
(2)
where, r is the distance between source and the first (reference)
microphone [15].
III. F
EATURE SELECTION
The aim of this section is to compute the feature vectors
from the array data and use the MLP (Multi Layer Perceptron)
approximation property to map the feature vectors to the
corresponding DOA, as shown in Fig. 3 [6].
Figure 3. Multilayer Perceptron neural network for sound source
localization.
Feature vector must:
1. be able to be mapped to the desired output (DOA).
2. be independent in phase, frequency, bandwidth, and
amplitude of the source.
3. be able to be calculated computationally efficient.
Assume that
() is the signal received at the −ℎ
microphone and =1 is the reference microphone (
=0).
We can write the signal at the −ℎ microphone in terms of
the signal at the first microphone signal as follow:
(
)
=
(
+
)
⟹
(
)
=
()
(3)
Then the cross-power spectrum between sensor and
sensor +1 like below:
,
(
)
=
(
)
∗
(
)
=
|
()
|
(
)
(4)
The normalized version is:
,
(
)
=
(
)
(5)
This equation suggests that there exists a projection from
and
,
and
to
(for =1,2,…,N) and thus to the
DOA. Therefore our aim is to use an MLP neural network to
approximate this mapping.
We summarized our algorithm for computing a real-valued
feature vector of length (2(−1)+1), for dominant
frequencies and M sensors below:
Preprocessing algorithm for computing a real-valued feature
vector:
1. Calculate the -point FFT of the signal at each
sensor.
2. For =1,2,…,−1
2.1. Find the FFT coefficients in absolute value
for sensor with compressive sampling.
W
trai
n
[7].
F
sign
a
ran
d
sign
a
whe
r
the
f
the
mic
r
whi
t
[20
0
[0,
2
vect
o
inp
u
In t
r
feat
u
dec
r
rece
i
esti
m
erro
r
rela
t
rela
t
IV
G
thre
e
eac
h
2.2.
M
w
t
t
s
2.3.
N
a
3. Cons
and
coef
f
W
e utilized
n
ed it accordi
n
F
or training
n
a
ls. We mod
e
d
om frequenc
i
a
l at sensor
(
)
=
∑
r
e is the n
u
f
requency of
−ℎ cosin
e
r
ophone (
=
t
e Gaussian
n
0
,2000
2
]. We gen
e
or
, and then
ut
-output
p
airs
r
aining step,
a
u
re vectors,
w
r
ease feature
v
i
ved sample
d
m
ate DOA o
f
r
s in classi
fi
t
ion with nu
m
t
ions for far fi
e
Figure 4. Rel
a
. T
RADITIO
N
G
enerally sp
e
e
sub-
p
roced
u
h
element in t
h
M
ultiply the
w
ith the conj
u
t
he same indi
c
t
he instantane
o
s
pectrum.
N
ormalize all
a
bsolute valu
e
truct a featu
r
imaginary
p
f
icient and the
i
two-layer P
e
n
g to fast bac
k
n
etwork we us
e
e
led received
i
es and phas
e
as below:
cos(2
u
mber of cosi
n
the −ℎ c
o
e
, t
is the ti
m
=
1) and th
e
n
oise, and
] and
e
rate 100 in
d
calculate fe
a
are used to tr
a
a
fter making
l
w
e use com
p
v
ectors dime
n
d
signal, we
f
sound sour
c
fi
cation and
m
ber of hidd
e
e
ld and near
fi
a
tion
b
etween nu
m
N
AL
P
RUNIN
G
e
aking, netw
o
u
res: (i) defin
h
e network; (
i
FFT coeffi
c
u
gate of the
F
c
es for sensor
o
us estimate
o
the estimate
s
e
s.
r
e vector tha
t
p
art of cros
s
i
r correspond
i
e
rceptron ne
u
k
propagation
e
a simulated
signal as a s
u
e
s. We write
−
−2
n
es (we assu
m
o
sine,
is t
h
m
e dela
y
b
et
w
e
−ℎ mi
is uniforml
y
is uniforml
y
d
ependent set
s
a
ture vectors.
a
in the MLP.
l
earning data
s
p
ressive sam
p
n
sion. In testi
n
calculate fe
a
c
e. Our expe
r
in approxim
a
e
n neurons.
F
fi
eld sources.
m
ber of hidden n
G
A
LGORITHM
S
o
rk pruning
i
e and quanti
f
i
i) eliminate t
h
c
ient for sens
o
F
FT coeffici
e
+1 to cal
c
o
f the cross-
p
s
by dividing
t
contains th
e
s
-
p
ower spe
c
i
ng FFT indic
e
u
ral network
training algo
r
dataset of rec
e
u
m of cosines
received sa
m
)+[]
m
ed =10),
h
e initial pha
w
een the refe
r
crophone,
[
y
distribute
d
y
distributed
s
of 128 sa
m
A total of
s
et and calcu
l
p
ling algorith
m
n
g step for a
a
ture vecto
r
s
r
iments sho
w
a
tion have
d
F
ig.4 shows
t
eurons and erro
r
S
AND
CS
T
H
E
i
s often cas
t
e
f
y the salienc
y
h
e least signi
f
o
r
e
nt at
c
ulate
ower
there
e
real
c
trum
e
s.
and
r
ithm
e
ive
d
with
m
pled
(6)
is
se of
r
ence
]
is
over
over
m
pled
3600
l
ating
m
to
new
and
w
that
d
irect
t
hese
E
ORY
ed
as
y
for
f
icant
ele
m
kno
w
1
sig
n
2
min
i
3
M
cate
(O
B
Ma
g
p
ru
n
unit
Tes
t
A
rec
e
dim
e
stro
n
the
o
gua
r
so
m
A
p
ro
b
or
M
is e
x
and
the
a
I
of n
S
this
Ort
h
and
[16]
B
co
m
[11,
1
)
tha
n
2
)
a
m
−
W
mat
r
,
t
net
w
exp
a
m
ents; (iii)
r
w
ledge, the f
o
1
) What is t
h
n
ificance of el
e
2
) How to
e
i
mal increase
3
) How to ma
k
M
ethods for
c
gories: 1) w
e
B
D) [10],
O
g
nitude-
b
ased
n
ing, e.g.: S
k
s (NC) [10]
a
t
(EFAST) [2]
A
new theor
y
e
ntly emerge
d
e
nsionality
r
n
gly model-
b
o
ry states tha
t
r
antees the s
u
m
e conditions
f
A
ccording to
b
lem can be s
o
M
ultiple-Mea
s
x
pressed as f
o
a dictionary
a
toms), we se
e
I
n above equ
a
on-zero coef
fi
S
everal iterat
i
minimizati
o
h
ogonal Matc
h
Non-convex
.
V. P
ROB
B
efore we fo
r
m
pressive sam
p
10]:
)
If for all
c
n
S, then this
m
)
If the nu
m
m
atrix was s
m
−
ma
t
W
e assume t
h
r
ix , and the
t
hen the mat
h
w
ork can be
a
nsion:
min
‖
−
r
e-adjust the
o
llowing ques
t
h
e best criter
i
e
ments?
e
liminate th
o
in error?
k
e the metho
d
c
ompressing
N
e
ight prunin
g
O
ptimal Brai
n
pruning (M
A
k
eletonization
a
nd Extende
d
.
y
known as
d
that can al
s
r
eduction. Li
k
ased (relying
t
for a given
u
ccess of
r
ec
o
f
rom a small
n
the number
o
o
r
t
ed into Sin
g
s
urement Vec
t
o
llows. Given
∈
∗
(th
e
e
k a vector so
min
‖
‖
a
tion
‖
‖
(k
n
fi
cient of .
i
ve algorith
m
o
n problem
h
ing Pursuit (
O
local optimi
z
LEM
F
ORMUL
A
r
mulate the p
r
p
ling proble
m
c
olumns of a
m
atrix called
a
m
ber of rows t
h
m
aller than
t
rix.
h
at the traini
n
desired outp
u
h
ematical m
o
extrac
t
ed i
n
‖
..
remaining
t
t
ions may app
i
on to descri
b
o
se unimport
a
d
converge as
N
Ns can be
c
g
, e.g.: Opti
m
n Surgeon
A
G) [12]. An
d
(SKEL) [8]
d
Fourier A
m
Compressed
s
o be catego
r
k
e manifold
on sparsity
i
degree of re
o
vering the
g
n
umber of sa
m
o
f measurem
e
g
le-Measure
m
t
or (MMV).
T
a measurem
e
e
columns of
D
lution satis
f
..=
n
own as
n
o
m
s have been
(Greedy Al
g
O
MP) or Mat
c
z
ation like F
A
TION AND
M
r
oblem of ne
t
m
we introdu
c
matrix,
−
a
−
h
at contain n
o
then this
m
n
g input patte
r
u
t patterns ar
e
o
del for trai
n
n
the form
=
(
=
(
t
opology. By
ear in min
d
:
b
e the salien
c
a
nt ele
m
ents
fast as possib
l
c
lassified int
o
m
al Brain Da
m
(OBS) [4],
d
2) hidden n
e
, non-contri
b
m
plitude Sens
i
Sensing (CS
)
r
ized as a ty
p
learning,
C
i
n particular).
sidual erro
r
g
iven signal
u
m
ples [14].
e
nt vectors, t
h
m
ent Vector (
S
T
he SMV pr
o
e
nt sample
∈
D
are referre
d
f
ying:
o
r
m
), is the n
u
proposed to
g
orithms suc
h
c
hing Pursuit
O
CUSS algo
r
M
ETHODOLOG
Y
t
work prunin
g
e some defin
i
was s
m
matrix.
o
nzero eleme
n
m
atrix is cal
l
r
ns are sto
r
e
d
e
stored in a
m
n
ing of the
n
of the foll
o
+
)
+
)
this
c
y, or
with
l
e?
o
two
m
age
and
e
uron
b
uting
i
tivity
)
has
p
e of
C
S is
This
, CS
u
nder
h
e CS
S
MV)
o
blem
∈
d
to as
(7)
u
mber
solve
h
as
(MP)
r
ithm
Y
g
as a
i
tions
m
aller
n
ts in
l
ed a
d
in a
m
atrix
n
eural
o
wing
(8)
where
is the output matrix of neural network,
is the
output matrix of hidden layer,
,
are weight matrix of two
layers and
,
are bias terms.
In conclusion, our purpose is to design a neural network
with least number of hidden neurons (or weights) that has the
minimum increase in error given by
‖
−
‖
. When we
minimize a weight matrix (
or
), the behavior acts like
setting, in mathematical viewpoint, the relating elements in
or
to zero. Deduction from above shows that the goal of
finding the smallest number of weights in NNs within a range
of accuracy can consider to be equal to finding an S
−sparse
Matrix
or
. So we can write problem as below:
[
(
)
]
..
=
(
+
)
[
(
)
]
..
=
(
(
(
+
)
)+
)
(9)
This problem is equivalent to finding
which most of its
rows are zeros. So with definition of S
−sparse matrix we
can rewrite the problem as below:
[
(
)
]
..
=
(
(
(
+
)
)+
)(10)
In matrix form equation (9) and (10) can be written as:
[
(
)]
..[
(
)]
=(
)
(
)
[
(
)]
..[
(
)]
=(
∗
)
(
)
(11)
[
(
)]
..[
(
)]
=(
)
(
)
(12)
In which
∗
is input matrix of the hidden layer for the
compressed neural network. Comparing these equations with
(7) we can conclude that these minimization problems can be
written as CS problems. In these CS equations (
∗
)
, (
)
and (
)
was used as the dictionary matrixes and
(
)
and
(
)
are playing the role of the signal matrix. The process of
compressing NNs can be regarded as finding different sparse
solutions for weight matrix
(
)
or
(
)
.
VI. R
ESULTS AND DISCUSSION
As mentioned before, assuming that the received speech
signals are modeled with 10 dominant frequencies, we have
trained a two layer Perceptron neural network with 128
neurons in hidden layer and trained it with feature vectors that
are obtained with CS from the cross-power spectrum of the
received microphone signals. After computing network weights
we tried to compress network with our algorithms.
In order to compare our results with the previous
algorithms we have use SNNS (SNNS is a simulator for NNs
which is available at [19]). All of the traditional algorithms,
such as Optimal Brain Damage (OBD) [16], Optimal Brain
Surgeon (OBS) [17], and Magnitude-based pruning (MAG)
[18], Skeletonization (SKEL) [6], non-contributing units (NC)
[7] and Extended Fourier Amplitude Sensitivity Test
(EFAST)
[13], are available in SNNS (CSS1 is name of
algorithm that uses SMV for sparse representation and CSS2 is
another technique that uses MMV for sparse representation).
Table I and II demonstrate the results of the simulations.
Observing these results, in table I we compare algorithms on
classification problem and in table II we compare algorithms
on approximation problem. For classification problem we
compare sum of hidden neurons weights in different algorithms
with similar stopping rule in training neural networks. Another
thing that we compared in this table was classification error
and time of training epochs. In table II we compare number of
hidden neurons and error in approximation and time of training
epochs, where we have stopping rule in training neural
networks. With these outputs we can infer that CS algorithms
are faster than other algorithms and have smaller error in
compare with other algorithms. In comparison to other
algorithms CSS1 is faster than CSS2 and would achieve
smaller computational complexity. This means that, According
to the number of Measurement vectors, the algorithm that uses
single-measurement vector (SMV) is faster than another
algorithm that uses multiple-measurement vector (MMV) but
its achieve error is not smaller.
TABLE I. COMPARISON FOR DIFFERENT ALGORITHMS
(CLASSIFICATION)
Training epochs=50 MAG OBS OBD CSS1
Sum of neurons
weights
3261 3109 2401 780
classification erro
r
(s) 0.0537 0.0591 0.046 0.0043
Training epochs
time(s)
0.62 25.64 23.09 0.41
TABLE II. COMPARISON FOR DIFFERENT ALGORITHMS
(APPROXIMATION)
Training epochs=50 NC SKETL EFAST CSS2
Hidden neurons 127 128 7 6
Approximation erro
r
(s) 0.094 0.081 0.016 0.0023
Training epochs Time(s) 27.87 7.86 9.97 14.87
VII. C
ONCLUSION
In this paper, compressive sampling is utilized to designing
NNs. Particularly, using the pursuit and greedy methods in CS,
a compressing methods for NNs has been presented. The key
difference between our algorithm and previous techniques is
that we focus on the remaining elements of neural networks;
our method has a quick convergence. The simulation results,
demonstrates that our algorithm is an effective alternative to
traditional methods in terms of accuracy and computational
complexity. Results revealed this fact that the proposed
algorithm could decrease the computational complexity while
the performance is increased.
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