x3.4 - x3.5
@liang4673
2013/3/13()
2013/3/13() 1 / 14
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1 RiemannRiemann
2 Fisher-3
Legendre
2013/3/13() 2 / 14
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abstract
1 - - - -
2
2013/3/13() 3 / 14
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review
-
n S = fpg
()ij;k := E
@i@jl +1 2
@il@jl
(@kl)
< r()@i @j ; @k >= ()ij;k
r(0) : Fisher Riemannr(1) : (exponential family)r(e)r(1) : (mixture family)r(m)
2013/3/13() 4 / 14
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review
(S; g;r;r) X;Y; Z 2 T (S)
Z < X;Y >=< rZX;Y > + < X;rZY >
r r g (dual)@kgij = ki;j + kj;ir()r() Fishier
2013/3/13() 5 / 14
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alpha-ane manifold
-:
S = fp(xj )j 2 g Rp(xj )dx = constant
L()(u) :=
(2
1u(1)=2 ( 6= 1)
log u ( = 1)
l()(xj ) := L()(p(xj ))
2013/3/13() 6 / 14
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alpha-ane manifold
-:
Fisher
gij() =Z@il
()(xj ) @jl()(xj )dx
()ij;k =Z@i @j l
()(xj ) @k l()(xj )dx
2013/3/13() 7 / 14
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alpha-ane manifold
-:
S = fp(xj )j 2 g -[i]
@i @j l()(xj ) = 0
x fC;Fig
l()(xj ) = C(x) + iFi(x)
S
2013/3/13() 8 / 14
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alpha-ane manifold
-:
!?(S; g;r();r())
i :=ZFi(x)l()(xj )dx
[i] i
() =
(2
1+
Rp(xj )dx ( 6= 1)R
p(xj )(log p(xj ) 1)dx ( = 1)
2013/3/13() 9 / 14
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alpha-divergence
-:
- D()(pjjq)8p;q 2 P;8 2 R
D()(pjjq) := 41 2
Z 1 2
p+1 + 2
q
p(1)=2q(1+)=2dx ( 6= 1)
D(1)(pjjq) = D(1)(qjjp) :=Z
q p+ p log pq
dx
2013/3/13() 10 / 14
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alpha-divergence
-:,
D(1)(jj) : Kullback-Leibler divergence (ex.EM-algorithm : mixture model)
D(0)(jj) : Hellinger
2013/3/13() 11 / 14
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alpha-family
-:
: S = fp(xj)j 2 g (3.35)
S := fp(xj )j 2 ; > 0g
- : S - S = fp(xj)j 2 g
p(xj ) =(
nXi=0
ci()Fi(x)
)2=(1)
2013/3/13() 12 / 14
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alpha-family
-:
(1)- : (1)-(1)-1- : 1-(1)- S () M
M S e- () M - S M ( 3.11)
2013/3/13() 13 / 14
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foliation
: ( 3.4)
2013/3/13() 14 / 14
previous contentstoday's contentsabstractreviewalpha-affine manifoldalpha-divergencealpha-familyfoliation
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