The document discusses various papers on conditional density estimation and reproducing kernel Hilbert spaces, including papers by Song, Gretton, Fukumizu, and others. It presents methods for learning conditional distributions based on maximum mean discrepancy and evaluating predictive performance through conditional density estimation. The papers covered use kernels and regularization to nonparametrically estimate conditional distributions from data.

1. 1 1
B
. 6 1 1
I + L
1 1
1 12
0 1 1
6 1 +

2. •
-
u
w x
•
- p
- u p p
•
- w x
- u
p
ʼt

3. u
u
p u
( O XO 3KbO d C O
u
( w Y 9 x
•
- ö p ~
p
– |p2ZZOXNSa
– W J J!W
(

4. u
C 9D!
u w x
k(x, y) = exp( ||x y||2
/ 2
)p u
X HX
NKM
(Xi) = k(·, Xi)
: X ! HX
f(x) = hf, k(·, x)iHX
, 8f 2 HX . )

5. u |
b
Y
C 9D
r
• p u
pA u r
• × p t
Z
k(·, y) x(y) = k(·, x)
X ⇠ pX k(x1, x2) = h (x1), (x2)iHX
µX :=
Z
k(·, x)dpX (x) =
Z
(x)dpX(x)
pX

6. u
• × u p
wA oJA x
p G u p
u
• × ʼ ~
-
- u ×
k(x, x0
) = exx0
µX(t) = EX⇠P [k(t, X)] =
Z
etx
dP(x)
=
Z
1 + tx +
t2
x2
2
+
t3
x3
3!
+ · · · dP(x)
= 1 + tEX⇠P [X] +
t2
2
EX⇠P [X2
] +
t3
3!
EX⇠P [X3
] + · · ·
ˆµX :=
1
n
nX
i=1
k(·, xi)

7. !
• w8 O YX O K ?:AD .x
- u w x
-
•
- 5 WKaSW W WOKX NS M OZKXMb!r
w x
- A0B 5
• u HY RS K K O K ?:AD !
- u w pFS SZONSKä
x 5
X1, ..., X` ⇠ P Y1, ..., Yn ⇠ Q
H0 : P = Q vs H1 : P 6= Q
C 9D
rBSS K
y
f |
u fz
M2
(P, Q) := ||µP µQ||2
Hk
µP
µQ
,

8. •
- C 9D
•
r
• u
ä Z
u pS O
hg, CY X fiHY
= E[f(X)g(Y )] 8f 2 HX , g 2 HY
CY X : HX ! HY
7 YW DYXQ O K (
k((x1, y1), (x2, y2)) = kX(x1, x2)kY (y1, y2)
HX ⌦ HY
X, Y ⇠ p HX and (x), HY and (y).
CY X = E[ (Y ) ⌦ (X)] 2 HY ⌦ HX
u
-

9. • a~ H A HeG0a! u
r
|
wERW x
- u r
× pDYXQ O K (
• r
7 YW DYXQ O K (
µY |X=x = E[ (Y )|X = x]
µY |X=x = CY X C 1
XX (x)
ˆµY |X=x = ˆCY X ( ˆCXX + I) 1
(x)
E[g(Y )|X = x] ⇡ hg, ˆµY |X=xiHY
,
8 g 2 HY .
.

10. u
• 3KbO d O
- ~ p p
-
• ~ r 4 4 D 4 234 O Mv
• ×
~
• u u
- u
p × |
t
p⇡
(Y |X = x) / ⇡(Y )p(X = x|Y )
ö

11. O XO 3KbO C O7 WSc O K ?:AD !
• ö
• 8YK
• p ö ~
u ~
• r
• u
ʼ
p
Y ⇡(Y ) p(X = x|Y )
x ⇡(Y )
p⇡
(Y |X = x)
µ⇡
Y (X = x)
µ⇡
Y (X = x) = C⇡
Y X C⇡ 1
XX (x)
3 C⇡
Y X = Ep⇡(Y,X)[ Y (Y ) ⌦ X (X)]
p⇡
(Y, X) = ⇡(Y )p(X|Y ) 6= p(Y, X).
p | ~p(Y, X)

12. • r
- ä u
• EROY OW A YYP SX 2ZZOXNSa! p
~ w u | x
• u | p
~
~ p p
- ×
ˆµ⇡
Y (X = x) = ˆC⇡
Y X ([ ˆC⇡
XX]2
+ I) 1 ˆC⇡
XX (x)
ˆC⇡
XX (B + I) 1
z
(B2
+ I) 1
Bz
C⇡
Y X = C(Y X)Y C 1
Y Y ⇡Y , C⇡
XX = C(XX)Y C 1
Y Y ⇡Y

13. u
• L O _K SYX
• u
8 lXO j NO O K :4 = !
•
COQ3KbO IR O K =C )!
u |s
H6D
|s
ER O RY NSXQ COQ K ScK SYX COQ3KbO
(

14. u
• u
A YYP SX 2ZZOXNSa!
• COQ3KbO IR O K =C )!
• p
min
p(Y |X=x)
KL(p(Y |X = x)||⇡(Y ))
Z
log p(X = x|Y )dp(Y |X = x)
s.t. p(Y |X = x) 2 Pprob
min
p(Y |X=x),⇠
KL(p(Y |X = x)||⇡(Y ))
Z
log p(X = x|Y )dp(Y |X = x) + U(⇠)
s.t. p(Y |X = x) 2 Pprob(⇠)
u
)

15. |
r u p
p
A YZ
ER O RY NSXQ
OQ K ScK SYX
COQ3KbO
ä
"[µ]
"[µ]
"S[µ]
ˆ"S[µ]
ˆ"+
S [µ]
ˆ" ,n[µ]

16. |
r ~ u
A YZ
ER O RY NSXQ
OQ K ScK SYX
COQ3KbO
ä
"[µ]
"[µ]
"S[µ]
ˆ"S[µ]
ˆ"+
S [µ]
ˆ" ,n[µ]
~ ERW )!
ERW !
ERW (!
ERW !

17. • p
p
• p p
- – p p
r u | p
ä A YYP SX 2ZZOXNSa!
• EROY OW )
p
→ ~
u
p⇡
(Y |X = x) µ⇡
Y (X = x) 2 HY
h 2 HY
hh, µ⇡
Y (X = x)iHY
= Ep⇡(Y |X=x)[h(Y )]
"[µ] := sup
||h||HY
1
EX [(EY [h(Y )|X]] hh, µ(X)iHY
)2
]
EX [·] p⇡
(X) EY [·|X] p⇡
(Y |X)
"S[µ] = E(X,Y )⇠p⇡(X,Y )[|| (Y ) µ(X)||2
HY
]
µ⇤
:= argminµ2HK
"[µ] = argminµ2HK
"s[µ] pX a.s.
EROY OW ) Y O NO KS SX 2ZZOXNSXa!
(xi, yi)n
i=1
"[µ] µ : X ! HY
,
"[µ] "S[µ]

18. s
• r
|
•
- r
→
ö u r
QS_OX
→ w pERW x
RO O
"S[µ]
f(x, y) := || (y) µ(x)||2
HY
2 HX ⌦ HY
"s[µ] = E(X,Y )[|| (Y ) µ(X)||2
HY
] = hf, µ(X,Y )iHX ⌦HY
ˆµ(X,Y ) = ˆC(X,Y )Y ( ˆCY Y + I) 1
ˆ⇡Y
ˆ"s[µ] = hˆµ(X,Y ), fiHX ⌦HY
p⇡
(X, Y )
A YZ A YYP SX 2ZZOXNSa!
ˆ⇡Y =
P`
i=1 ˜↵i (˜yi)
"S[µ]
-
ˆ✏s[µ] =
nX
i=1
i|| (yi) µ(xi)||2
HY
,
where. = (GY + n I) 1 ˜GY ˜↵, (GY )ij = kY (yi, yj), ( ˜GY )ij = kY (yi, ˜yj)

19. ER O RY NSXQ OQ K ScK SYX
• pA YZ ~ p
• EROY OW ( | p
R O RY NSXQ OQ K ScK SYX p?S RSbKWK O K K GS_
ü A 5A ! u
| p EROY OW (
MYX S OXMb
i
i
+
i := max(0, i)
ER O RY NSXQ OQ K ScK SYX
✏+
s [µ]✏s[µ]r
|ˆ✏+
s [µ] ✏s[µ]|
p
! 0.
.

20. SXSWScSXQ KK
• p
• EROY OW
- × u p
ˆ✏+
s [µ]
ˆ✏ ,n[µ] =
nX
i=1
+
i || (yi) µ(xi)||2
HY
+ ||µ||2
HK
A YZY S SYX A YYP SX 2ZZOXNSXa!
ˆµ ,n = argminµ ˆ" ,n[µ]
ˆµ ,n(x) = (KX + n⇤+
) 1
K:x
where. = ( (y1), ..., (yn)), (KX )ij = kX (xi, xj),
⇤+
= diag(1/ +
1 , ..., 1/ +
n ), K:x = (kX (x, x1), ..., kX (x, xn))T
.
"S[ˆµ n,n]
P
! min
µ
"S[µ]. u

21. COQ3KbO
• ER O RY NSXQ OQ K ScK SYX p
ʼ
• COQLKbO
- r
- A YZ p
- s
• –p ü m n – 11
• u ö t
L :=
mX
i=1
+
i || (yi) µ(xi)||2
HY
+ ||µ||2
HK
+
nX
i=m+1
||µ(xi) (ti)||2
HY
.
A YZY S SYX (
ˆµreg(x) = (KX + ⇤+
) 1
K:x
where. = ( (y1), ... (yn)), (KX )ij = kX (xi, xj),
⇤+
= diag(1/ +
1 , ..., 1/ +
m, 1/ , ..., 1/ ), K:x = (kX (x, x1), ..., kX (x, xn))T
.

22. 6aZO SWOX
• 4KWO K ZY S SYX OMY_O b !
• r × p
p
r i × !
• r
w x
u u | p u ×
| p × !
{(xt, yt)}m
t=1.
✓t+1 = ✓t + 0.2 + N (0, 4e 1), rt+1 = max(R2, min(R1, rt + N (0, 1))
xt+1 = rt+1 cos ✓t+1, yt+1 = rt+1 sin ✓t+1
{(xt, yt)}m
t=1.
×

23. • u u
u
• ö COQ3KbO ~
→ ä | D ZO _S SYX NK K p
COQ3KbO
• u 3C!p
7!
ER O RY NSXQ OQ K ScK SYX Z 3C!
COQ3KbO D6
- COQ3KbO ä
u | ö ~ p
~
ä ʼ
~
(
R1 = 0, R2 = 10
R1 = 5, R2 = 7
0 ä
~ ʼ
ö
It It+1
(xt, yt) (xt+1, yt+1)
A ONSM SYX
ö !
w x
p((xt+1, yt+1)|I1, ..., IT )
p((xt+1, yt+1)|I1, ..., IT , It+1)

24. COPO OXMO
• 2 R 8 O YX K OX 3Y Q K N K O CK MR 3O XRK N DMRk YZP KXN 2 DWY K 2
O XO WO RYN PY RO Y KWZ O Z YL OW
ZKQO (g ,
• =O DYXQ YXK RKX 9 KXQ 2 Oa DWY K KXN OXTS 7 WSc 9S LO ZKMO OWLONNSXQ YP
MYXNS SYXK NS SL SYX S R KZZ SMK SYX Y NbXKWSMK b OW
ZKQO . g. - 24 .
• =O DYXQ OXTS 7 WSc KXN 2 R 8 O YX O XO OWLONNSXQ YP MYXNS SYXK
NS SL SYX 2 XSPSON O XO P KWO Y PY XYXZK KWO SM SXPO OXMO SX Q KZRSMK WYNO
( ) .- (
• OXTS 7 WSc =O DYXQ KXN 2 R 8 O YX O XO LKbO d O
ZKQO ,(,g ,)
• HKXQ DYXQ X IR KXN HYXQ COX O XO 3KbO SKX :XPO OXMO S R AY O SY
COQ K ScK SYX ZKQO ), ( ),,
• H bK HY RS K K EYWYRK : K K 9S Y RS DK KNK KXN EK O RS HKWKNK 4 Y NYWKSX
WK MRSXQ PY LKQ YP Y N NK K _SK O XO OWLONNSXQ YP K OX NS SL SYX
ZKQO ) ) (
• H ?S RSbKWK 2LNO KW 3Y K SK 2 R 8 O YX KXN OXTS 7 WSc 9S LO ZKMO
OWLONNSXQ YP ZYWNZ K GS_ Z OZ SX K GS_ )--,
• D OPPOX 8 lXO j NO 8 b =O_O = MK 3K NK K O DKW AK O YX 2 R 8 O YX KXN
K SWS SKXY AYX S 4YXNS SYXK WOKX OWLONNSXQ K OQ O Y
ZKQO - (g -(
• X IR ?SXQ 4ROX KXN 6 SM A GSXQ 3KbO SKX SXPO OXMO S R ZY O SY OQ K ScK SYX KXN
KZZ SMK SYX Y SXPSXS O K OX _W
! ,..g -), )
)

25. Kernel Bayesian Inference with Posterior Regularization (Appendix)
Yuchi Matsuoka
2017 3 18
1 Preliminaries
(X, BX ) pX HX k(·, ·) RKHS pX
µX = EpX
[φ(X)] ∈ HX φ(X) = k(X, ·). 1
f ∈ H EpX
[f(X)] = EpX
[⟨f, φ(X)⟩] =
⟨f, µX⟩ universal kernel RKHS H sup norm CX
2 (X, BX ), (Y, BY) φ(x), ψ(y) RKHS HX ,HY p X ×Y
(X, Y ) CXY CXY = Ep[φ(X)⊗ψ(Y )] k((x1, y1), (x2, y2)) = kX (x1, x2)kY(y1, y2)
RKHS HX ⊗ HY µ(XY )
Theorem 1 CXX µX ∈ R(CXX) g ∈ HY E[g(Y )|X = ·] ∈ HX
µY = CY XC−1
XXµX, µY |X=x = E[ψ(Y )|X = x] = CY XC−1
XXφ(x).
2
µX pX {xi}N
i=1 ˆµX = 1
N
N
i=1 φ(xi), CXY
ˆCXY = 1
N
N
i=1 φ(xi) ⊗ ψ(yi)
RKHS Op(N−1/2
)
1
,i.e. supx kX (x, x) < ∞
1

26. 3 Theorem 1
Theorem 1 ((Song et al., 2009, Equation 6)) mΠ mQy HX Π HY QY CXX
mΠ ∈ R(CXX) g ∈ HY E[g(Y )|X = ·] ∈ HX
mQy = CY XC−1
XXmΠ.
C−1
XXmΠ CXX mΠ
Proof CXXf = mΠ f ∈ HX g ∈ HY
⟨CY Xf, g⟩ = ⟨f, CXY g⟩ = ⟨f, CXXE[g(Y )|X = ·]⟩
= ⟨CXXf, E[g(Y )|X = ·]⟩ = ⟨mΠ, E[g(Y )|X = ·]⟩ = ⟨mQy , g⟩.
⟨mΠ, E[g(Y )|X = ·]⟩ = ⟨mQy , g⟩
⟨f, mX⟩ = E[f(X)]
(X, Y ) ∼ p(x, y). U ∼ π(u). (Z, W) ∼ q(x, y) = π(x)p(y|x), qY(y) = q(x, y)dx
⟨mQY
, g⟩HY
= E[g(W)] = g(w)qY(w)dw
⟨mΠ, E[g(Y )|X = ·]⟩ = EU [EY [g(Y )|U]] =
X
(
Y
g(y)p(y|u)dx)π(u)du
=
Y
(
X
g(y)q(u, y)du)dx =
Y
g(y)qY(y)dy.
mQy = CY XC−1
XXmΠ mΠ kX (·, x) E[kY(·, Y )|X = x] = CY XC−1
XXkX (·, x) ✷
4
π(Y ) Y p(X = x|Y ) pπ
(Y |X = x) π(Y ) x pπ
(X, Y ) = π(Y )p(X|Y )
πY CXY µπ
Y (X = x)
pπ
(Y |X = x) ∝ π(Y )p(X = x|Y ).
2

27. p(X|Y ) X × Y p CXY Thm.
1 Cπ
Y X pπ
Cπ
XX pπ
X
µπ
Y (X = x) = Cπ
Y XCπ −1
XX φ(x).
Cπ
Y X HY ⊗ HX µ(Y X) Thm 1.
µ(Y X) = C(Y X)Y C−1
Y Y πY , where. C(Y X)Y := E[ψ(Y ) ⊗ φ(X) ⊗ ψ(Y )].
Cπ
XX
µ(XX) = C(XX)Y C−1
Y Y πY
5
Regularized Bayesian inference (RegBayes) Pprob
minp(Y |X=x) KL(p(Y |X = x)||π(Y )) − log p(X = x|Y )dp(Y |X = x)
s.t. p(Y |X = x) ∈ Pprob
Proof
KL(p(Y |X = x)||π(Y )) − log p(X = x|Y )dp(Y |X = x) = log
p(Y |X = x)
π(Y )
dp(Y |X = x) − log p(X = x|Y )dp(Y |X = x)
= log
p(Y |X = x)
π(Y )p(X = x|Y )
dp(Y |X = x)
= log
p(Y |X = x)
π(Y )p(X=x|Y )
pπ(X=x)
dp(Y |X = x) + log pπ
(X = x)dp(Y |X = x)
= KL p(Y |X = x)||
π(Y )p(X = x|Y )
pπ(X = x)
+ log pπ
(X = x).
arg minp(Y |X=x) KL(p(Y |X = x)||π(Y )) − log p(X = x|Y )dp(Y |X = x) = π(Y )p(X=x|Y )
pπ(X=x)
. ✷
3

28. 6 Vector-valued regression
(RKHS )
E(f) :=
n
i=1
||yj − f(xj)||2
HY
+ λ||f||2
HK
,
where. yj ∈ HY, f : X → HY.
f RKHS HY f RKHS HK
6.0.1 Vector-values regression and RKHSs
{(xi, vi)}i≤m X × V i.i.d X (V, ⟨·, ·⟩V
E(X,V )[||f(X) − V ||2
V]
f : X → V vector-valued regression problem
[Definition] h : X → V (H, ⟨·, ·⟩Γ) x ∈ X, v ∈ V h → ⟨v, h(x)⟩V
RKHS HΓ
Riesz 2
x ∈ X, v ∈ V V HΓ Γx (Γxv ∈ HΓ ) h ∈ HΓ
⟨v, h(x)⟩V = ⟨h, Γxv⟩Γ
HΓ RKHS
Γx L(V) V V
Γ(x, x′
) ∈ L(V)
Γ(x, x′
)v ∈ (Γx′ v)(x) ∈ V
2
(Riesz ) H H R H∗
H φ ∈ H∗
yφ ∈ H x ∈ H φ(x) = ⟨x, yφ⟩
4

29. [Proposition 2.1] Γ : X × X → L(V)
(1) Γ(x, x′
) = Γ(x′
, x)∗
.
(2) n ∈ N, {(xi, vi)}i≤n ⊂ X × V i,j≤n⟨vi, Γ(xi, xj)vj⟩V ≥ 0.
E(X,V )[||f(X) − V ||2
V] n
i=1 ||vi − f(xi)||2
V f RKHS HΓ
HΓ
ˆϵλ(f) :=
n
i=1
||vi − f(xi)||2
V + λ||f||2
Γ.
Γxi
Theorem 2.2.(Adapted from G. Lever and S. Gr¨unew¨alder+ 2012) f∗
ˆϵλ HΓ
f∗
=
n
i=1
Γxi
ci
{ci}, ci ∈ V
i≤n
(Γ(xj, xi) + λδji)ci = vj, 1 ≤ j ≤ n.
ˆϵλ(f)
5

30. 7 ϵs[µ] ε[µ]
Proof
ε[µ] := sup
||h||HY
≤1
EX[(EY [h(Y )|X]] − ⟨h, µ(X)⟩HY
)2
]
= sup
||h||HY ≤1
EX[(EY [⟨h, ψ(Y )⟩HY
|X] − ⟨, h, µ(X)⟩HY
)2
]
≤ sup
||h||HY ≤1
EX,Y [⟨h, ψ(Y ) − µ(X)⟩2
HY
]
≤ sup
||h||HY ≤1
||h||2
HY
EX,Y [||ψ(Y ) − µ(X)||2
HY
]
= EX,Y [||ψ(Y ) − µ(X)||2
HY
] = ϵs[µ].
✷
8 Proposition 1
Proposition 1 (X, Y ) X × Y Y prior π(Y ) p(X|Y ) HX kX φ(x)
RKHS HY kY ψ(y) RKHS φ(x, y) HX ⊗ HY
ˆπY = ℓ
i=1 ˜αiψ(˜yi) πY {(xi, yi)}n
i=1 p(X|Y )
f(x, y) = ||ψ(y) − µ(x)||2
HY
f ∈ HX ⊗ HY
ˆϵs[µ] =
n
i=1
βi||ψ(yi) − µ(xi)||2
HY
,
β = (β1, ..., βn)T
β = (GY + nλI)−1 ˜GY ˜α (GY )ij = kY(yi, yj), ( ˜GY )ij = kY(yi, ˜yj), ˜α = (˜α1, ...˜αℓ)T
.
Proof K. Fukumizu 2016. Kernel Bayes Rule. Proposition 4
6

31. ΦX,Y = (φ(x1, y1), ..., φ(xn, yn)) ˆµ(X,Y ) = ΦX,Y β = ΦX,Y (GY + nλI)−1 ˜GY ˜α
HX ⊗ HY
ϵs[µ] = ⟨ˆµ(X,Y ), f⟩HX ⊗HY
= ⟨ΦX,Y (GY + nλI)−1 ˜GY ˜α, f⟩HX ⊗HY
= ⟨ΦX,Y β, f⟩HX ⊗HY
=
n
i=1
βi||ψ(yi) − µ(xi)||2
HY
.
ˆµ(X,Y ) = ˆC(X,Y )Y ( ˆCY Y + λI)−1
ˆπY h = ( ˆCY Y + λI)−1
ˆπY
h =
n
i=1
aiψ(yi) + h⊥
h⊥ h span(ψ(y1), ..., ψ(yn)} ( ˆCY Y + λI)h = ˆπY
1
n i,j≤n
aikY(yi, yj)ψ(yj) + λ
i≤n
aiψ(yi) + h⊥ =
i≤ℓ
˜αiψ(˜yi)
ψ(yk)|n
k=1
1
n
G2
Y a + λGY a = ˜GY ˜α ⇔
1
n
(GY + nλI)GY a = ˜GY ˜α ⇔
1
n
GY a = (GY + nλI)−1 ˜GY ˜α
ˆµ(X,Y )
ˆµ(X,Y ) =
1
n i≤n
φ(xi, yi) ⊗ ψ(yi) h =
1
n
ΦX,Y GY a = ΦX,Y (GY + nλI)−1 ˜GY ˜α
✷
7

32. 9 Proposition 2
Proposition 2 i β+
i ̸= 0 µ ∈ HK HK K(xi, xj) = kX (xi, xj)I
I : HK → HK
ˆµλ,n(x) = Ψ(KX + λnΛ+
)−1
K:x
Ψ = (ψ(y1), ..., ψ(yn)) (KX)ij = kX (xi, xj) Λ+
= diag(1/β+
1 , ..., 1/β+
n ) K:x = (kX (x, x1), ..., kX (x, xn))T
λn
Proof β+
i = 0 (xi, yi) i β+
i ̸= 0
µ = µ0 + g µ0 = n
i=1 Kxi
ci ˆϵλ,n[µ]
ˆϵλ,n[µ] =
n
i=1
β+
i ||ψ(yi) − µ(xi)||2
HY
+ λn||µ||2
HK
=
n
i=1
β+
i ||ψ(yi) − (µ0(xi) + g(xi))||2
HY
+ λn||µ0 + g||2
HK
=
n
i=1
β+
i ||ψ(yi) − µ0(xi)||2
+ λn||µ0||2
+
n
i=1
β+
i ||g(xi)||2
+ λn||g||2
+ 2λn⟨µ0, g⟩ − 2
n
i=1
β+
i ⟨g(xi), ψ(yi) − µ0(xi)⟩.
i ψ(yi) − n
j=1 kX (xi, xj)cj = λn
β+
i
ci ˆϵλ,n[µ]
λn⟨µ0, g⟩ −
n
i=1
β+
i ⟨g(xi), ψ(yi) − µ0(xi)⟩ = 0
ˆϵλ,n[µ] = ˆϵλ,n[µ0] +
n
i=1
β+
i ||g(xi)||2
+ λn||g||2
≥ ˆϵλ,n[µ0]
ψ(yi) − n
j=1 kX (xi, xj)cj = λn
β+
i
ci ci µ0 = n
i=1 Kxi
ci
(KX + λnΛ+
)c = Ψ
µ0(x) =
n
i=1
kX (x, xi)ci = Ψ(KX + λnΛ+
)−1
K:x
✷
8