hamilton-jacobi approach to contrast functions and...

Hamilton-Jacobi approach to contrast functions and geodetical motion in

information geometry

Juan Manuel Pérez-Pardo

J.M. Pérez-Pardo

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J.M. Pérez-Pardo

F l o r i o M C i a g l i a

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D o m e n i c o F e l i c e

Joint work with

J.M. Pérez-Pardo

Outline

Introduction and motivation

Generating Potentials for arbitrary tensors

Hamilton-Jacobi approach to contrast functions

J.M. Pérez-Pardo

Geometrical Quantum Mechanics

Pure states are a complex projective space

Orthonormal basis ,

h̃ =hd ⌦ d ih | i � h |d i

h | i ⌦ hd | ih | i

⇡ : H ! PH⇡(| i) = | ih |

h | i{|eji} hej |eki = �jk

zj = hej | i z̄j = h |eji

h̃ =dz̄j ⌦ dzj

kzk2 � (z̄jdzj⌦)⌦ (dz̄kzk)

kzk4

J.M. Pérez-Pardo

Geometrical Quantum Mechanics

non-linear coordinateszj =

ppje

i'j

⇣pj � 0 ,

X

j

pj = 1⌘

h̃ =1

4

X

j

1

pjdpj ⌦ dpj +

⇥dpj ⌦ d'k + d'j ⌦ d'k

⇤

h̃ =dz̄j ⌦ dzj

kzk2 � (z̄jdzj⌦)⌦ (dz̄kzk)

kzk4

J.M. Pérez-Pardo

Geometrical Quantum MechanicsH ⌘ L2(�) | i ! hx | i = (x)

h |xi =

⇤(x)

Immersion of a measure space manifold into ⇥ PH (x|✓) =

pp(x|✓)ei↵(x|✓)

d ln (x|✓) = 1

2d ln p(x|✓) + id↵(x|✓)

h = g � iwg = 1

4Ep[(d ln p)⌦ (d ln p)] + Ep[d↵⌦ d↵]� Ep(d↵)⌦ Ep(d↵)

w = Ep[d ln p ^ d↵]

F = 14Ep(d ln p⌦ d ln p) Fisher-Rao metric

Ep(df) :=R�(df)pdx

J.M. Pérez-Pardo

Statistical Manifold:Each point represents a probability density

Fisher-Rao metric, a Riemannian metric:Infinitesimal “distinguishability” of probability densities

Dualistic Structure:

Usually defined by perturbation of the hermitean connection

Information Geometry

Mp 2 M

g

(g,r,r⇤)

X,Y, Z 2 X(M)

�ijk = g�ijk ± Tijk

X(g(Y, Z)) = g(rXY, Z) + g(Y,r⇤XZ)

J.M. Pérez-Pardo

Divergence Function

Statistical manifold:Divergence function:

Oriented distance-like function If is differentiable

is a generating functionGenerates tensors intrinsically

M

S : M⇥M ! R

S(x, y) 6= S(y, x)

S

@S

@x

i

��x=y

=@S

@y

i

��x=y

= 0

S(x,y) � 0

S(x,y) = 0 x = y,

S@

2S

@x

i@x

j= gij

J.M. Pérez-Pardo

Outline




J.M. Pérez-Pardo

Some examplesHessian Riemannian Structures[1]:

Kähler Potential

Generating Potentials for arbitrary Tensors

! =@2f

@z@z̄

f : U ! RConvex function:

Local coordinates on U ⇢ Mx = {xi}

gij(x) :=@

2f

@x

i@x

j

This is not a tensor!

Quote from Wikipedia: “There is no comparable way of describing a general Riemannian metric in terms of a single function.”

[1]: J.J. Duistermaat. On Hessian Riemannian Structures. Asian J. Math, 5 (1) (2001), pp. 79-92

J.M. Pérez-Pardo

Two copies of the manifold to define a two-point function

Provides several canonical structures:

Contrast Functions as Generating Potentials I

S : M⇥M ! R

⇡R : M⇥M ! M⇡L : M⇥M ! M

X 2 X(M)

Left-lift:Right-lift:

XL 2 X(M⇥M)

XR 2 X(M⇥M)

Diagonal embedding:d : M ,! M⇥M

p 7! (p, p)

Restriction to the diagonal:S|d := d⇤S

J.M. Pérez-Pardo

Consider the contrast function

It is clearly f-linear inLet us show that it is also f-linear in

Contrast Functions as Generating Potentials II

S : M⇥M ! R

(LXLLYLS)��d:= g(X,Y )

X

Y

LXLL(fY )LS = LXL(fLYLS)

= fLXLLYLS + LXLf · LYLS

J.M. Pérez-Pardo




S : M⇥M ! R


X

Y



( )��d ( )

��d

( )��d

( )��d

0

= fg(X,Y )

J.M. Pérez-Pardo




S : M⇥M ! R


X

Y



( )��d ( )

��d

( )��d

( )��d

0

= fg(X,Y )

One can define more rank 2 tensors:

(LXRLYRS)��d= (LXLLYLS)

��d= �(LXRLYLS)

��d= g(X,Y )

J.M. Pérez-Pardo

Contrast Functions as Generating Potentials III

This procedure can be used for tensors of order 3

One needs to find suitable linear combinations to cancel the non-tensorial terms

There are 8 possible combinations that lead to tensorial quantities.

They define the same symmetric tensor up to a signIn information geometry this is used to define pairs of dual connections

(LXLLYRLZRS � LXRLYLLZLS)��d= T (X,Y, Z)

or

(LXLLYLLZRS � LXRLYRLZLS)��d= T (X,Y, Z)

J.M. Pérez-Pardo

Can one go to higher order tensors?For instance the Riemann tensor

Can one generate non-symmetric tensors?

J.M. Pérez-Pardo

Outline




J.M. Pérez-Pardo

Find a canonical contrast functionGivenConstruct a two point function with vanishing derivatives

Hamilton’s Principal function

Solution of the equations of motion

Hamilton-Jacobi Approach to Contrast Functions

g T

�

�(0) = x �(1) = y

S[x,y] =

Z 1

0L(�(t), �̇(t))dt

S[�] =

Z tfin

tin

L(�(t), �̇(t))dt

)x = y

pi(x,y)|x=y

= 0�(t) = x

dS = pi(x,y)dxi � Pi(x,y)dy

i

Pi(x,y)|x=y

= 0

J.M. Pérez-Pardo

Hamilton-Jacobi Approach to Contrast Functions II

Choose the Lagrangian

Compute derivatives

We need the relation between andTaylor expansion

L(x, v) =1

2gijv

iv

j +1

6Tijkv

iv

jv

j

@S

@yj= gjk(y)v

kfin +

1

2Tjkl(y)v

kfinv

lfin

x,yvfin

�(t) = �(1) + vfin(1� t) +1

2v̇fin(1� t)2 +

1

6v̈fin(1� t)3

vfin = y � x+1

2v̇fin � 1

6v̈fin

Eq. of motion)v̇l = �T l

jkvkv̇j � �l

jkvjvk � 1

6glrArjksv

jvkvs

J.M. Pérez-Pardo

Assuming that is analytic in

Hamilton-Jacobi Approach to Contrast Functions III

v̇ v

�k := y

k � x

k

@S

@yj= g(y)jk�

k � 1

2�jkl(y)�

k�l +1

2Tjkl(y)�

k�l

+1

6�jkr(y)�

rls(y)�

k�l�s � 1

12Ajkls(y)�

k�l�s

@

2S

@x

i@y

j

��d= �gij(x)

@

3S

@x

l@x

k@y

j

��d= ��jkl + Tjkl

@

3S

@y

l@y

k@x

j

��d= ��jkl � TjklBy similar calculations:

J.M. Pérez-Pardo

Hamilton-Jacobi Approach to Contrast Functions IV

Find a combination of 4th order derivatives that is f-linear

Two possible linear combinations:LXLLYRLZLLWRS

��d=: LRLR

Q1(X,Y, Z,W ) =(LRRL�RLLR) + (LRRR�RLLL)

+ (LRLL�RLRR) + (LRLR�RLRL)

Q2(X,Y, Z,W ) =(LLLL�RRRR) + (LLLR�RRRL)

+ (LLRL�RRLR) + (LLRR�RRLL)

J.M. Pérez-Pardo

Hamilton-Jacobi Approach to Contrast Functions IV

Find a combination of 4th order derivatives that is f-linear

Two possible linear combinations:LXLLYRLZLLWRS

��d=: LRLR

Q1(X,Y, Z,W ) =(LRRL�RLLR) + (LRRR�RLLL)

+ (LRLL�RLRR) + (LRLR�RLRL)

Q2(X,Y, Z,W ) =(LLLL�RRRR) + (LLLR�RRRL)

+ (LLRL�RRLR) + (LLRR�RRLL)

Q1 = Q2 = 0

After a lengthy calculation:

J.M. Pérez-Pardo

Hamilton-Jacobi Approach to Contrast Functions V

Choose a lagrangian

L(x, v) =1

2gijv

iv

j +1

6Tijkv

iv

jv

j +1

24Cijklv

iv

jv

jv

l

J.M. Pérez-Pardo


Choose a lagrangian

L(x, v) =1

2gijv

iv

j +1

6Tijkv

iv

jv

j +1

24Cijklv

iv

jv

jv

l

Q1 = Q2 = 0


J.M. Pérez-Pardo


Choose a lagrangian

L(x, v) =1

2gijv

iv

j +1

6Tijkv

iv

jv

j +1

24Cijklv

iv

jv

jv

l

Q1 = Q2 = 0


What about non-symmetric tensors?

J.M. Pérez-Pardo

References

F.M. Ciaglia, F. di Cosmo, D. Felice, S. Mancini, G. Marmo and J.M. Pérez-Pardo. Hamilton-Jacobi approach to Potential Functions in Information Geometry. J. Math. Phys. 58, 063506. (2017).

P. Facchi, R. Kulkarni, V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Ventriglia. Classical and quantum Fisher information in the geometrical formulation of quantum mechanics. Phys. Lett. A 374, (2010), 4801-4803

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