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By: Douglas L. Jones
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By: Douglas L. Jones
Online:
CONNEXIONS Rice University, Houston, Texas
L2
L2 xk
dk
⇔ E [yk ] = E [yk+d ] ryz (l) = E [yk zk+l ] ∀k, l : ( ryy (0) < ∞)
E 2 dk
xk
M −1
yk =
(wl xk l ) −
l=0
2
argminE [2 ] = E (dk − yk ) wl
2
M −1 l=0
M −1 l=0
(wl E [dk xk l ]) + −
= E M −1 m=0
E 2 = rdd (0) − 2
2
(wl xk l ) −
((wl wmE [xk l xk
M −1
M −1 l=0
dk −
(wl rdx (l)) +
l=0
−
m
−
M −1
M −1
l=0
m=0
= E dk 2 −
]))
(wl wm rxx (l − m))
rdd (0) = E dk 2
rdx (l) = E [dk X k l ] −
rxx (l − m) = E [xk xk+l
m]
−
E 2 = rdd (0) − 2PWT + WT RW
P
R=
=
rdx (0) rdx (1)
rdx (M − 1)
rxx (0)
rxx (1)
rxx (1)
rxx (0)
rxx (M − 1)
...
.
∇=
...
rxx (M − 1)
rxx (0)
rxx (1)
rxx (1)
rxx (0)
...
...
W
∂ ∂w 0 ∂ ∂w 1
∂ ∂w M −1
2 2
2
∇ = − (2P) + 2 RW d dW
R
T
W = AT
A
d dW
(WM W) = 2M W
W opt R = P ⇒ W opt = R
⇒
M 1
−
P
R
W opt = R
1
−
P
xk
dk
R
P
xk dk
k ˆ(l ) rxx
N →
→
ˆ opt W
x d N
N > M
O M 2
R M O (M ) Rk+1
Rk
O (M )
O (M ) O M 2
N
E 2 = rdd (0) − 2PT W + WT RW R
E 2 (w0 , w1 , . . . , wM
1)
−
N
R
R P
L
1
−
P
ε
E ∇ = ∂ ∂w i
2
E
M
R
W
i
∇ W
µ
i+1
= Wi − µ∇i
µ
µ µ W 0
i = 1, ∞
∇i = − (2P ) + 2 RW i W i+1 = W i − µ∇i
W opt = W
∞
O M 2
O (M )
N −1
1 rxxˆ(l) ≈
N
rxdˆ(l) ≈
(xk xk+l )
k=0 N −1
1 N
ˆ opt = Rˆ 1 = P ˆ W
(dk xk l ) −
k=0
−
k
rxxˆ(l)
rdxˆ(l)
ˆk W opt k ≈ R
1
−
k
=
=
1 N
1 N
N −1
(xk
m xk −m−l )
−
m=0 N −1
(xk
m−l dk−m )
−
m=0
ˆk P
k k 1 rxx (l) = rxx (l) + xk xk −
l
−
− xk
N xk−N −l
−
k 1 (1 − α) rxx k (l) = αrxx (l) + xk xk −
l
−
{xk } {dk } W
E k 2
M −1
k = dk − yk = dk −
T
(wi xk i ) = dk − X k W k −
i=0
X k =
xk xk
xk
1
−
M +1
−
k
W =
w0k w1k
k wM
1
−
=0
∇k
∂ ∂W
E k 2
=
= E 2k −X k
T
= E −2 dk − X k W k X k = − 2E dk X k
+ E X k
T
W
= −2P + 2RW
⇒ W opt = R W opt
1
−
P
W k+1 = W k − µ∇k R
P
P ⇒
R
∇k = k 2 k ∂ ∇ˆk =
∂W
T
E k 2
2
k 2 = 2 k
∂ E k 2 ∂W
∂ T dk − W k X k ∂W
M −1
= 2 k −X k = − 2k X k
yk = W k X k = i=0 wik xk i k = dk − yk W k+1 = W k − µ∇ˆk = W k − µ −2k X k = W k + 2µk X k wik+1 = wik + 2µk xk
V k+1 = V k − 2µRV k + (− (2µRW opt )) + 2 µP W opt = R
1
−
V k+1 = V k − 2µRV k + − 2µRR V k
R
R Q
1
−
ΛQ
→∞
1
−
P
→0
+ 2µP = ( I − 2µR) V k
Λ
λi
R
Q
R V k+1 = I − 2µ Q
1
−
Q
ΛQ
V k
QV k+1 = ( Q − 2µΛQ) V k = (1 − 2µΛ) QV k V = QV V
V k+1 = (1 − 2µΛ) V k m
V
V
R
1 − 2µΛ
V
→0
M
∀i, i = [1 , 2, . . . , M ] : V i k+1 = (1 − 2µλi ) V i k |1 − 2µλi | < 1
∀i : (|µλi | < 1) µ
∀i : µ <
V
1
λi
µ<
1 λmax λmax
M
tr (R) =
M
rii =
i=1
i=1
λi ≥ λmax
λi
∀i, i ∈ {1, M } : (rii = r (0)) r (0)
tr (R) = M r (0) = M E [xk xk ]
O (1)
1
µ<
ˆ M r (0)
µ
(1 − 2µλi )k 1 − 2µλmax 1 − 2µλmin λmin R
2
E k
= E
k T
dk − W
2
k
X
T
T
T
= E dk 2 − 2dk X k W k − W k X k X k W k T
T
= rdd (0) − 2W k P + W k RW k
W opt = R min 2
1
−
P
= E 2 =
rdd (0) − 2P T R
= rdd (0) − P T R
1 P
−
+ P T R
1 RR−1 P
−
1 P
−
V = Q [W − W opt ] k T
2
k T
E [k ] = rdd (0) − 2W P + W
k
k T
RW + − W
T
RW opt
Q
1
−
ΛQ = R
− W opt T RW k + T
W opt T RW opt + W k RW opt + W opt T RW k − W opt T RW opt = rdd (0) + V k RV k − T T T P T R 1 P = min 2 + V k RV k = min 2 + V k Q 1 QRQ 1 QV k = min 2 + V k ΛV k −
−
−
N −1
E k
E vjk
2
2
2
= min +
2
λj E vjk
j =0
V k
E V k V k
W k+1 = W k + 2µk X k V k+1 = W k + 2µk QX k
k V +1
= E V k+1 V k+1
T
T
= E 4µ2 k 2 QX k X k QT =
k
V
k
+ 2µ k QX V
k T
k T
k
+ 2µ k V X
T
Q
2
2
k
k T
+ 4µ E k QX X
T
T
Q
T
k = dk − W k X k = dk − W opt T − V k QX k
k
E k QX V
k T
k
= E dk QX V
k T
T
= 0 + 0 − QX k X k T
= − QE X X = − ΛV k
T
E k 2 QX k X k QT
k
− W opt X QX V k T
k
k
QT V k V
T
k T
k
Q E V V
k T
k T
− V
k T
k
QX V
k T
k 2 T
k = dk − W k X k
X W k ≈ W opt W k
2
k
k T
E k QX X
T
Q
= E k
2
X k X k k T
k
E QX X
Q
T
T
= E k 2 Λ
k 2 = min 2 + V k ΛV k 2
E k
k+1
V
2
= min + E = min 2 +
= ( I − 4µΛ) V k + 4µ2
λj V j k
λj V jjk
λj V jjk Λ + 4µ2 min 2 Λ V ∞
⇒ 4µΛV ⇒
∞
∞
V
= 4 µ2
=µ
2
= V
+1
∞
(λj V jj ) + min 2 Λ
(λj V jj ) + min 2 I
X k
T
∞
V ii
=µ
∞
V ii
V ii
1−µ ∞
V ii
2
∞
=
µmin 2 1 − µ λj
= min
2
=
2
2
∞
∞
∞
µmin 2P (λj ) λj 1−µ
= min 2 +
1 − µN σx 2
= µmin 2
λj
= min 2 1
E
2
= min 2 + E V λV
E
λj + min
∞
µ
−
1P
P
λj
1
1−µtr(R) 1 min 1−µrxx (0)N
= min 2
1 1 − µN σx 2 µ
µ
µ
k+1
V
= ( I − 4µΛ) V k + 4µ2
∀i : (4 µλi < 2) k+1 V ii
µ<
= (1 −
k+1
V ii
λj V jjk Λ + 4µ2 min 2 Λ
k+1
k +1
V ii
4µλi ) V iik
2
+ 4µ λi
λj
= (1 − 4µλi ) V iik + 4µ2 λi
k ≤ (1 − 4µλi ) V iimax + 4µ2 λi k
V jjmax
= 1 − 4µλi V ijk
1 2λmax
1 − 4µλi V ii
k+1
V ij
λj
k
V jjmax
1 − 4µλi + 4µ2 λi 4µ2 λi
k
V jj
+ 4µ2 min 2 λi
λj
k
V jj
= 1 − 4µλi + 4µ2 λi
λj < 1
λj < 4µλi
λj
k
V jjmax
µ<
P1
λj
1
=
tr(R)
=
Nr xx (0)
=
Nσ x 2
1 1
µ µ= xk dk
µ 3Nσ x 2
R
1 P
−
= W ∞
dk = xk ∗ hk =
(xk i hi ) −
i=0
=
P
∞
i=0
E [dk xk
(hi E [xk i xk j ]) −
−
j
−
]
=
E [(
=
∞
i=0
∞
i=0
(xk i hi )) xk j ] −
−
(rxx ( j − i))
r (M − 1) | r (M ) r (M + 1) . . .
rxx (0)
r (1)
r (1)
r (0)
|
...
r (2)
r (1)
|
...
...
...
. . . r (0)
r (1)
|
r (2)
r (3)
...
r (M − 1) r (M − 2) . . . r (1)
r (0)
|
r (1)
r (2)
...
H
h (m) = h (m + 1) = · · · = 0
M
P = Rh
W opt = R
1
−
1
−
P = R
1
−
(Rh) =
h
M
= =
h (0) h (1) h (2)
W H ≈ z δ (n − ∆)
∆
−
W ≈
z −∆ H
W
sk
sk sˆk sk
∆0
−
∆0
−
sˆk
< 75%
dk
nk nk
nk
E k 2 = E (sk + nk − yk ) sk nk
nk
2
2 = E sk 2 + 2E [sk (nk − yk )] + E (nk − yk )
sk
E [sk (nk − yk )] = E [sk ] E [nk − yk ] = 0
nk
nk
E k 2 = E sk 2 + E (nk − yk ) sk
W = Rn n
1
−
P nn
2
E k 2
∆
sk
∆ sk
sk
∆
−
• • •
W opt = R
1
−
P
R
1
P
−
O N 2
O (N )
R R
QRQ
1
−
W opt
Q
R R R O (NlogN )
Q N O (logN )
N
L
yk =
L
vnk yk−n
n=1
W k =
U k =
+
wnk xk
n=0
v1k v2k
k vL
w0k w1k
k wL
yk
−
yk
−
yk
1 2
L
−
xk
xk
xk
n
−
1
−
L
−
k = dk − yk = dk − W k T U k
E k 2 = k 2
ˆk = ∇
∂ ∂ k 2 = 2 k ( k ) = 2 k ∂W k ∂W k
∂ ∂ (yk ) = k ∂v i ∂v ik
L
ˆk = ∇
β 1k
β 2k
k . . . βL
∂ ∂v 1k
∂ ∂ k
= −2k
w1k
L
vnk yk−n
L
wnk xk−n
+
n=1
L
n
−
+
...
L
wnk xk−n
+
k αL
n=1
∂ ∂w ik
( yk )
= xk
−i
+
k−j
vjk αi
L
β ik = yk
i +
−
k−j
vjk β i
j =1
( yk )
∂ k ∂v L
(yk )
∂ k ∂w 0
(yk )
∂ k ∂w L
( yk )
∂ (yk ∂v ik
∂ ∂w ik
n)
−
n)
−
+ xk
+0
n
−
(yk ) β ik =
∂ ∂v ik
yk = W k T U k
j =1
ˆ k = −2k ∇
∂ ∂v 1k
∂ ( yk ∂w ik
αik =
( yk )
L
αki
vnk
=
n=0
∂ ∂v ik T
vnk
n=1
L
n=1
αk0
= yk
n=0
vnk yk−n
(k )
∂ ∂ (yk ) = k ∂w i ∂w ik
k
β 1
k
β 2
...
k
α0
k
α1
...
αkL
T
ˆk W k+1 = W k − U ∇ µ
dk
(yk )
2
k = ( |yk |) − A2
yk = W kH X k ek = dk − yk W k+1 = W k + 2µek X k ∗
