x=y+W
x
y
W
y
^=Gx
y
G
x
H=argminGE(y−Gx)T(y−Gx)
H
G
y
G
x
y
G
x
where
E(y−Gx)T(y−Gx)
y
G
x
y
G
x
is the MSE.

MS=E(y−Gx)T(y−Gx)=Etr(y−Gx)(y−Gx)T=trE(y−Gx)(y−Gx)T
MS
y
G
x
y
G
x
tr
y
G
x
y
G
x
tr
y
G
x
y
G
x

(1)
Minimizing MSE is equivalent to minimizing

ε2=E(y−Gx)(y−Gx)T=
R
yy
−G
R
xy
−
R
yx
GT+G
R
xx
GT
ε
2
y
G
x
y
G
x
R
yy
G
R
xy
R
yx
G
G
R
xx
G

(2)
Taking the derivative with repsect to

GG
∂ε2∂
G
=2
R
yx
+2G
R
xx
=0
G
ε
2
-2
R
yx
2
G
R
xx
0
This implies

H=
R
yx
R
xx
-1=
R
yy
R
yy
+
R
WW
-1
H
R
yx
R
xx
R
yy
R
yy
R
WW

(3)