Biomedical Engineering Reference
In-Depth Information
Next we introduce the decomposition
d
ʳ
s
=
A
i
s
i
=
A
s
,
(6.15)
i
=
1
is selected such that
A
i
A
i
where each
A
i
=
C
i
and
A
[
A
1
...
A
d
ʳ
]
,
s
T
. Letting
L
[
L
1
,...,
L
d
ʳ
]=
s
1
...
s
d
ʳ
]
[
L
[
A
1
,...
A
d
ʳ
]
, this allows us to re-
express the original hierarchical Bayesian model as
1
2
||
−
L
2
Σ
−
1
p
(
y
|
s
)
∝
exp
(
−
y
s
||
)
(6.16)
1
2
2
p
(
s
i
|
ʳ
i
)
∝
exp
(
−
ʳ
i
||
s
i
||
F
),
∀
i
=
1
,...,
d
ʳ
,
(6.17)
is the standard Frobenius norm
trace
XX
T
where
. The hyperprior remains
unaltered. It is easily verified by the rules for transformation of random variables
that Eq. (
6.16
) and the original model are consistent. It also follows that
||
X
||
F
[
]
d
ʳ
d
ʳ
L
T
1
ʳ
i
L
i
Li
T
=
Σ
+
L
Σ
s
L
T
Σ
y
=
Σ
+
L
(
1
ʳ
i
C
i
)
=
Σ
+
(6.18)
i
=
i
=
where
-dependent prior covariance of the pseudo sources.
There are three ways to derive update rules for
Σ
s
is the diagonal,
ʳ
: EM iteration, Fixed-point
(MacKay) updates, and Convexity based updates. Although the EM iteration update
rules are the most straightforward to derive, they have empirically slow convergence
rates. In contrast, the fixed-point updates have fast convergence rates but no conver-
gence guarantees. In contrast, the convexity based updates have both fast convergence
properties as well as guaranteed convergence properties. Details of derivations and
properties of these update rules can be found elsewhere, but the three types of update
rules are listed here for completeness.
ʳ
1. EM-Updates
trace
1
1
r
i
L
T
L
T
ʳ
(
k
+
1
)
i
nr
i
||
ʳ
(
k
)
ʳ
(
k
)
i
−
ʳ
(
k
)
i
L
i
ʳ
(
k
)
i
(Σ
(
k
)
)
−
1
2
i
(Σ
(
k
)
)
−
1
ₒ
y
||
F
+
I
y
y
i
i
(6.19)
2. MacKay update:
1
L
T
L
T
ʳ
(
k
+
1
)
i
n
||
ʳ
(
k
)
[
ʳ
(
k
)
i
i
(Σ
(
k
)
)
−
1
2
i
(Σ
(
k
)
)
−
1
L
i
]
)
−
1
ₒ
y
||
F
(
trace
(6.20)
y
y
i
3. Convexity-based update
ₒ
ʳ
(
k
)
[
L
T
ʳ
(
k
+
1
)
i
i
n
||
L
i
(Σ
(
k
)
)
−
1
i
(Σ
(
k
)
)
−
1
L
i
]
)
−
1
/
2
y
||
F
(
trace
(6.21)
y
y
