Biomedical Engineering Reference
In-Depth Information
2.6
ΣΔ
LEARNING FRAMEWO
RK
the time delay
τ
(
p
j
) can be assumed to be a linear
projection between the vector
p
j
and the unit
vector u oriented toward the direction of arrival
of the acoustic wavefront (see
Figure 2.8
). Thus,
ΣΔ
learning is based on an optimization frame-
work that integrates
ΣΔ
modulation with sta-
tistical learning
[40]
. Given a random input
vector
X
∈ R
D
and a vector
V
∈ R
M
(where each
element of the vector
v
is the membrane poten-
tial of each neuron), a
ΣΔ
learner estimates
the parameters of the synaptic-weight matrix
W
∈
R
D
×
R
M
according to the following opti-
mization criterion:
τ(
P
j
) =
1
c
U
T
·
P
j
,
(2.10)
where
c
is the speed of sound waves in air.
Ignoring the higher-order terms in the series
expansion (under far-field assumptions), we
simplify Eq.
(2.9)
as
max
W
∈C
(
min
V
f
(
V
,
W
))
,
(2.13)
x
(
P
j
,
t
)
≈
s
(
t
)
−
τ(
P
j
)
s
(1)
(
t
),
(2.11)
thereby implying that under far-field conditions,
the signals recorded at the microphone array
are linear with respect to the bearing parameter
τ
(
p
j
). Thus, the differential signal is given by
where
E
denotes a constraint space of the trans-
formation matrix
W
and.
f
(
V
,
W
)
= ||
V
||
1
−
V
T
E
x
{
W
T
X
}.
(2.14)
Here,
C
x
{.} denotes an expectation operator with
respect to the random vector
x
(or rate encoding
of
x
).
The term
||
V
||
bears similarity to a regulariza-
tion operator that is extensively used in machine
learning algorithms to prevent over-fitting
[41,
42]
. However, the
L
1
norm in Eq.
(2.14)
forms an
important link in connecting the cost function in
Eq.
(2.13)
to spike generation. This is illustrated
in
Figure 2.9
a, which shows an example of a
one-dimensional regularization function
||
V
||
.
The piecewise behavior of
||
V
||
leads to discon-
tinuous gradient sgn(
v
), as shown in
Figure
2.9
b. The signum is a Boolean function that indi-
cates whether a spike is generated or not. The
minimization step in Eq.
(2.13)
ensures that the
membrane potential vector
v
is correlated with
the transformed input signal
Wx
(signal-tracking
step), and the maximization step in Eq.
(2.13)
adapts the parameters of
W
such that it mini-
mizes the correlation (decorrelation step), simi-
lar to the lateral inhibition observed in afferent
neurons. The uniqueness of the proposed
approach, compared to other optimization tech-
niques to solve Eq.
(2.13)
, is the use of bounded
gradients to generate SA limit cycles about a
x
(
t
)
=
x
(
P
2
,
t
)
−
x
(
P
1
,
t
)
=
2
s
(1)
(
t
)
d
c
c
os
θ ,
(2.12)
whose envelope is a function of the bearing
angle
θ
, and
s
(1)
(
t
) is the first-order temporal
derivative of
s
(
t
). The fly's neural circuitry uses
this differential signal to accurately extract the
amplitude and hence estimate of the bearing.
Adaptation and noise exploitation play a critical
role in this process and directly affect the pre-
cision of the estimated bearing angle. Thus, the
signal-measurement process is integrated with
the statistical learning and adaptation process
that alleviates the effects of sensor artifacts and
noise.
This integration served as an inspiration for a
novel online learning framework called
ΣΔ
learning
[39]
that integrates the noise-shaping
principles with the statistical learning and adap-
tation process. The framework exploits the struc-
tural similarities between an integrate-and-fire
neuron and a
ΣΔ
modulator for which stochas-
tic resonance and noise shaping have been dem-
onstrated in literature. Let us now show how
ΣΔ
learning can be used to resolve acute bearing
cues in subwavelength microphone arrays.
