Digital Signal Processing Reference
In-Depth Information
where
κ
i
represents the kurtosis of the
i
th source. Inserting this expression into
Eq. (
9.27
), and after some algebraic manipulations keeping in mind that
g
=
1,
we easily arrive at
νκ
i
g
i
.
(9.29)
Hence, a relationship of the form (
9.26
), and thus FastICA's cubic convergence,
can be achieved only if
ν
g
i
=
(
1
+
3
ν)g
i
+
1
/
3, which corresponds precisely to the coefficient
used in update rule (
9.23
) obtained by the simplified Newton iteration with iden-
tity Hessian matrix. Besides the specific step size value used in its update equa-
tion, another key ingredient of FastICA's excellent convergence properties is the
apparently trivial normalization step (
9.24
). It is indeed this step that allows the
global filter to fulfill
=−
=
1 after each iteration, thus allowing the simplification of
Eq. (
9.27
) through Eq. (
9.28
) into Eq. (
9.29
). Remark, however, that the objective
function (
9.19
) implicitly optimized by FastICA is a valid contrast for real-valued
sources and mixtures under prewhitening. The use of prewhitening imposes some
performance bounds on further higher-order processing, as analyzed in [
8
].
The desirable convergence properties of FastICA only hold asymptotically, i.e.,
under infinite sample size conditions, when the noiseless observation model is per-
fectly satisfied [
29
,
36
]. Nevertheless, when processing short observation windows
the fourth-order contrast presents higher sample variance than the kurtosis for a
range of source distributions (including sub- and super-Gaussian), leading to sep-
aration estimates farther from the optimal solution [
5
,
6
]. A simple numerical ex-
ample helps to illustrate this limitation. Figure
9.3
plots the contrast function val-
ues against angle
Δθ
parametrizing the global filter
g
g
T
for
an instantaneous orthogonal mixture of two zero-mean unit-variance uniformly dis-
tributed sources, with an observation window of just
T
=
=[
cos
(Δθ),
sin
(Δθ)
]
50 samples. Separation so-
lutions are defined by integer multiples of
π/
2 rad, recovering the sources up to the
sign and permutation ambiguities inherent to blind processing (Sect.
9.2
). Clearly,
the local minima of the sample fourth-order moment (dashed line) lie farther away
from the separation solutions than the maxima of the sample absolute kurtosis (solid
line); in addition, the minima near 0 and
π
rad become saddle points, which tend
to slow down the algorithm's convergence. Comparing panels (a) and (b) shows
that FastICA converges to different solutions depending on the initial value of the
extracting vector, requiring in each case nearly 30 iterations [
74
]. The theoretical
large-sample performance of FastICA has been analyzed in [
62
], including a solu-
tion to prevent the detrimental effects of saddle points. This solution, however, is
only valid in the version of the algorithm designed for joint or simultaneous source
separation rather than single source extraction.
The version of the FastICA algorithm reviewed above is designed for real-valued
sources and mixtures only. An extension to complex signals was carried out in [
7
],
and was later shown to inherit the cubic global convergence property of its real-
valued counterpart [
56
]. Such an extension, however, is valid only for sources sat-
isfying the second-order circularity condition E
±
s
i
}=
0. The non-circular source
scenario is specifically addressed in [
30
,
38
,
48
,
49
], all under the prewhitening
assumption. Interestingly, the alternative version of the algorithm independently de-
veloped a decade earlier in [
40
] comprised the complex non-circular case, too.
{
Search WWH ::
Custom Search