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WAGOS
Conformal Changes of Divergence and Information Geometry
Shun-ichi AmariRIKEN Brain Science Institute
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Information GeometryInformation Geometry
Systems Theory Information Theory
Statistics Neural Networks
Combinatorics PhysicsInformation Sciences
Riemannian ManifoldDual Affine Connections
Manifold of Probability Distributions
Math. AI
Vision, Shape
optimization
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22
1; , ; , exp
22
xS p x p x
Information Geometry ?Information Geometry ?
p x
;S p x θ
Riemannian metricDual affine connections
( , ) θ
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Manifold of Probability DistributionsManifold of Probability Distributions
1 2 3 1 2 3
1,2,3 { ( )}
, , 1
x p x
p p p p p p
1p
p
3p
2p
;S p x
0 1
1 0 1
1
{ | ( , ,..., ); 1; 0}
{ | ( , ,..., ); 0}
n n i i
n n i
n n
S p p p p p
S p p p p
S S
p
p
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Riemannian Structure
2 ( )
( )
( ) ( )
Euclidean
i jij
T
ij
ds g d d
d G d
G g
G E
Fisher information
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Affine Connectioncovariant derivative
,
0, X=X(t)
( )
X c
X
i jij
Y X Y
X
s g d d
minimal
geodesic
distance
straight line
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DualityDuality
, , , i jijX Y X Y X Y g X Y
Riemannian geometry:
X
Y
X
Y
**, , ,X XX Y Z Y Z Y Z
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Dual Affine Connections
e-geodesic
m-geodesic
log , log 1 logr x t t p x t q x c t
, 1r x t tp x t q x
,
q x
p x
*( , )
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Divergence :D Z Y
: 0
: 0, iff
: ij i j
D
D
D d g dz dz
z y
z y z y
z z z
positive-definite
Z
Y
M
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Metric and Connections Induced by Divergence(Eguchi)
'
' '
:
:
:
ij i j
ijk i j k
ijk i j k
g D
D
D
y z
y z
y z
z z y
z z y
z z y
', i ii iz y
Riemannian metric
affine connections
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Duality:
, ,
k ij kij kji
ijk ijk ijk
g
T
M g T
*, , ,X XX Y Z Y Z Y Z
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Two Types of Divergence Invariant divergence (Chentsov, Csiszar)
f-divergence: Fisher- structure
Flat divergence (Bregman)
KL-divergence belongs to both classes
q(x)
D[p : q] = p(x)f{ }dxp(x)
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Invariant divergence (manifold of probability distributions; )
: one-to-one
sufficient statistics
y k x
: :X X Y YD p x q x D p y q y
{ ( , )}S p x
ChentsovAmari -Nagaoka
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Csiszar f-divergence
: ,if i
i
qD p f
p
p q
: convex, 1 0,f u f
: :cf fD cDp q p q
1f u f u c u
' ''1 1 0 ; 1 1f f f 1
( )f u
u
Ali-SilveyMorimoto
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Invariant geometrical structurealpha-geometry (derived from invariant divergence)
,S p x
ij i j
ijk i j k
g E l l
T E l l l
log , ; i i
l p x
-connection
, ;ijk ijki j k T
: dually coupled
, , ,X XX Y Z Y Z Y Z
α
Fisher information
Levi-civita:
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: Dually Flat Structure
1 2 1 2 1 2
affine coordinates
dual affine coordinates
potential ,
,
:D
nS
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Dually flat manifold:Manifold with Convex Function
S coordinates 1 2, , , n L
: convex function
negative entropy log ,p p x p x dx energy
Euclidean 21
2i
mathematical programming, control systems, physics, engineering, economics
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Riemannian metric and flatness
Bregman divergence
, grad D
1,
2i j
ijD d g d d
, ij i j i ig
: geodesic (notLevi-Civita)Flatness (affine)
{ , ( ), }S
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Legendre Transformation
i i
one-to-one
0ii
,i i i
i
,D
( ) max { ( )}ii
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Two flat coordinate systems ,
: geodesic (e-geodesic)
: dual geodesic (m-geodesic)
“dually orthogonal”
,
,
j ji i
ii i
i
*, , ,X XX Y Z Y Z Y Z
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Geometry
ijG GRiemannian metric
1: ,
2TD d G p p p p p
1
G
G G
p p
p p
Straightness (affine connection)
: -geodesic
: -geodesic
t t
t t
p a b
p a b
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Pythagorean Theorem (dually flat manifold)
: : :D P Q D Q R D P R
Euclidean space: self-dual
21
2 i
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Projection Theorem min :
Q MD P Q
Q = m-geodesic projection of P to M
min :Q M
D Q P
Q’ = e-geodesic projection of P to M
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dually flat space
convex functionsBregman
divergence
invariance
invariant divergence Flat divergence
KL-divergenceF-divergenceFisher inf metricAlpha connection
: space of probability distributions }{pS
logp(x)
D[p : q] = p(x) { }dxq(x)
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, 0 : ( 1 not n holds)i iS p p p
Space of positive measures : vectors, matrices, arrays
f-divergence
α-divergence
Bregman divergence
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divergence
1 1
2 21 1
[ : ] { }2 2i i i iD p q p q p q
[ : ] { log }ii i i
i
pD p q p p q
q
KL-divergence
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α-representation (Amari-Nagaoka, Zhang)
1
2 , 1
log , 1i
i
i
prp
iU r r
typical case: u-representation,
2
1 , 1
, 1z
zU ze
i iU r p
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Divergence over α-representation
: i i i iD U r V r rr p q
1
2
1
2
: -geodesic
2: -geodesic1
i i
i i
r p
r p
log 1i i
i i
r p
r p
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β-divergence (Eguchi)
11
1 , 01
U z z
0 expU z z
0 : :D KLp q p q
-div -div
-divKL
( 1) ( 1)
1[ : ] { ( ) ( )
( 1
-structure
dually flat: -divergenc
)
( 1) ( ) ( ( ) ( )}
e
D p q p x q x
q x p x q x dx
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( , ) divergence
, [ : ] { }i i i iD p q p q p q
: divergence
1: -divergence
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Tsallis q-Entropy--
1
111
11 , 1
1ln
log , 1
exp ln 1 (1 ) } , ex
1ln 1
1
p
1 q
T
q
q
q
qq q
u qqu
u q
u u q u
H E p x dxp q
u
x
Shannon entropy1
[log ] ( ) log ( )( )
H E p x p x dxp x
Generalized log
α structure
2 1
1
2
q
q
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1 11 : log
1 1T R q
q
q
q
H h H hq
h
q
p p x dx
: convex: ( ) { ( )}q qh p f h p p
1 11 ln
1q q qq
H E p xq h
p
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1
: ln
11
1
p q
q q
r xD p x r x E
p x
p x r x dxq
: -divergence ; -structure
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q -exponential family cf Pistone exponential
conve
escort distrib
, exp
:
( ) ( )ˆ ( ) :
1 11 ln
1
ution
x
q i i
q q
p p
i q i
q qq
p x x
p x p xdx p x
h h
E x
E p xq h
x
x x
p
κ
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0
0
1 10
1
1
1
1
n
n ii
n
i ii
i q qi
qi i
q
S p
p x p x
p pq
ph
p
q-Geometry derived from : dually flat
( ) and ( )
0
0 01
log ( ) log ( )
(log log ) ( ) log
n
q q i ii
n
q i q i qi
p x p x
p p x p
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Dually flat structure of q-escort
esc
ˆˆ: log
ˆ
ˆ log log q
q
pD p r p dx
rh rp x
q p x dxr x h p
esc ˆ ˆlog logij q i jg E p p
geodesic: exponential family
dual geodesic: q-family
1 2log , log 1 logp x t t p x t p x c t
1 2ˆ ˆ ˆ, 1p x t tp x t p t
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q-escort probability distribution
1ˆ ( ) ( )
( )q
q
p x p xh p
Escort geometry
ˆ ˆ ˆ( ) [ log ( , ) log ( , )]ij q i jg E p x p x
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q -escort geometry
1ˆ q
q
p x p xh
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Dually flat structure of q-escort
esc
ˆˆ: log
ˆ
ˆ log log q
q
pD p r p dx
rh rp x
q p x dxr x h p
esc ˆ ˆlog logij q i jg E p p
geodesic: exponential family
dual geodesic: q-family
1 2log , log 1 logp x t t p x t p x c t
1 2ˆ ˆ ˆ, 1p x t tp x t p t
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Pythagorean theoremq
: : :q qD p q D q r D p r
q - geodesic :
1 1 1
1 2, 1q q q
p x t tp x t p x c t
dual- -geodesic:q
1 2, 1q q q
p x t tp x t p x c t
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Projection theorem
arg min :qr M
D p r p
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Max-entropy theorem
constraint
max
s: q k k
q
E a x c
h
p
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q -Cramer Rao theorem : ,p x
1
ˆ-unbiased estima
ˆ
tor
ˆ ˆ
q
q i i j j ij
E
q
E g
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q -maximum likelihood estimator
1ˆ ˆarg max , , ;
1arg max ;
q N
q
iN
q
p x x
p xh
/ Bayes MAP with
N q
qh
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q -super-robust estimator (Eguchi)
1
1
1 1
0 1
,ˆmax , max
bias-corrected -estimating function
ˆ, , log
1log
1
1ˆ, 0 max ,
q
q i q
q q
N
q i ii q
p xp x
h
q
s x p x p c
c hq
s x p xh
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Conformal change of divergence
: :D p q p D p q
ij ijg p g
( )
logijk ijk k ij j ik i jk
i i
T T s g s g s g
s
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q -Fisher information
( ) ( )( )
q Fij ij
q
qg p g p
h p
conformal transformation
11[ ( ) : ( )] (1 ( ) ( ) )
(1 ) ( )q q
q divergence
D p x r x p x r x dxq h p
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Total Bregman divergence (Vemuri)
2
( ) ( ) ( ) ( )TBD( : )
1 | ( ) |
p q q p qp q
q
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Total Bregman Divergence and its Applications to Shape
Retrieval
•Baba C. Vemuri, Meizhu Liu, Shun-ichi Amari, Frank Nielsen
IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2010
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Total Bregman Divergence
2
::
1
DTD
f
x yx y
•rotational invariance
•conformal geometry
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TBD examples
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Clustering : t-center
1, , mE x x
arg min , ii
TD x x x
E
y
T-center of E x
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t-center
2
1
1
i i
i
i
i
w ff
w
wf
xx
x
x
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t-center is robust
1, , ;
1; ,
nE
n
x x y
x x z x y
influence fun ;ction z x y
robust as : c z y
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1,
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How good is Total Bregman Divergence
•vision
•signal processing
•geometry (conformal)
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TBD application-shape retrieval
• Using MPEG7 database;• 70 classes, with 20 shapes each class (Meizhu Liu)
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First clustering then retrieval
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Advantages
• Accurate;• Easy to access (shape representation);• Space and time efficient (only need to store the
closed form t-centers, clustering can be done offline, hierarchical tree storage).
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Shape retrieval framework
• Shape-->• Extract boundary points & align them-->• Represent using mixture of Gaussians-->• Clustering & use k-tree to store the clustering
results;• Query on the tree.
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MPEG7 database• Great intraclass variability, and small
interclass dissimilarity.
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Shape representation
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Experimental results
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Other TBD applications
Diffusion tensor imaging (DTI) analysis [Vemuri]
• Interpolation• Segmentation
Baba C. Vemuri, Meizhu Liu, Shun-ichi Amari and Frank Nielsen, Total Bregman Divergence and its Applications to DTI Analysis, IEEE TMI, to appear