differential geometry

Element of differential geometry Riemannian geometry No Riemannian geometry

Element of differential geometry and applications

to probability and statistics

Jerome Lapuyade-Lahorgue

Jerome Lapuyade-Lahorgue 1/ 49

Element of differential geometry and applications to probability and statistics


Plan

1 Element of differential geometry

2 Riemannian geometryDefinitionsDistance and geodesic curvesInformation geometryGradient, Laplacian and Brownian motions on manifolds

3 No Riemannian geometryNotion of connexionThe Riemannian case and Levi-Civita connexionSecond-order derivative, torsion and curvature




Main definitions

Let M a C1-differentiable manifold.The parametrisation:

θ ∈ Θ ⊂ Rn → ϕ(θ) ∈ M; (1)




Main definitions


θ ∈ Θ ⊂ Rn → ϕ(θ) ∈ M; (1)

The directional derivative of f : M → R at P along a curveγ : R → M:

limt→0

f (γ(t)) − f (γ(0))

t, (2)

where γ(0) = P.




Main definitions


θ ∈ Θ ⊂ Rn → ϕ(θ) ∈ M; (1)

The directional derivative of f : M → R at P along a curveγ : R → M:

limt→0

f (γ(t)) − f (γ(0))

t, (2)

where γ(0) = P.A tangent vector of M at P: An application which associates to f

a directional derivative.The set of tangent vectors at P is denoted TP(M) and (v .f )(P)the directional derivative at P for γ(0) = v .




Main definitions

Vector field: An application which associates at θ a tangent vectorat P = ϕ(θ). The set of vector field, called tangent space, isdenoted T (M)The derivative V .f of f along the vector map V is an applicationwhich associates at θ the derivative (V (θ).f )(P);k-differential form: An application which associates at θ ak-multilinear, antisymmetric form from TP(M) × . . . × TP(M) toR. The set of k-differential forms is

∧k(M).




Properties of the tangent space

We have: dimT (M) = n, and a base is given by:

∂

∂θi, (3)

the application which associates the derivative along the curvet → ϕ(θ1, . . . , θi−1, t, θi+1, . . . , θn).




Operations on differential forms

We have: dim∧k(M) = C k

n and:

α =∑

1≤i1<...<ik≤n

αi1,...,ikωi1,...,ik , (4)

where ωi1,...,ik is the base of∧k(M).

A possible base is such that ωi1,...,ik

(

∂∂θi1

, . . . , ∂∂θik

)

= 1. For k = 1

and such a base, ωj is denoted dθj .





Exterior product

The exterior product is the application:

∧ :∧k ×∧l → ∧k+l ,

such that:

α→ α ∧ β is linear;

α ∧ β = (−1)klβ ∧ α.

We have ωi1,...,ik = dθi1 ∧ . . . ∧ dθik , where (dθ1, . . . , dθn) is the

dual base of(

∂∂θ1, . . . , ∂

∂θn

)

.





Integral of differential form

For the form ω = fdθ1 ∧ . . . ∧ dθn:∫

Vω = ε

∫

ϕ−1(V)f (θ)dθ1 . . . dθn, (5)

where V ⊂ M has a no empty interior, and ε = 1 or −1 dependingon the orientation. If ε doesn’t depend on f , the manifold isorientable.Measurable manifolds: The form fdθ1 ∧ . . . ∧ dθn such that|f |dθ1 . . . dθn is the Lebesgue measure on M is called volume form.





Restriction of differential form

Let S ⊂ M a k-dimensional sub-manifold and

α =∑

1≤i1<...<ik

fi1,...,ikdθi1 ∧ . . . ∧ dθin ∈k∧

(M). Its restriction

α|S ∈ ∧k(S) is:

α|S = ε∑

fi1,...,ik ((ϕ−1ψ)(u1, . . . , uk))

∣

∣

∣

∣

D(θi1, . . . , θik )

D(u1, . . . , uk)

∣

∣

∣

∣

du1∧. . .∧duk ,

(6)where ε is the orientation of S,ψ : (u1, . . . , uk) → ψ(u1, . . . , uk) ∈ S a parametrization of S and∣

∣

∣

∣

D(θi1 , . . . , θik )

D(u1, . . . , uk)

∣

∣

∣

∣

the Jacobian of ϕ−1 ψ.





Integral of k-form

The integral of α ∈ ∧k(M) is:

∫

Sα =

∫

Sα|S , (7)

where S is a k-dimensional sub-manifold.





Border of a manifold and exterior derivative

Definition

Let f and g two functions from E to F , f and g are k-homotopicif there exists an application:

H : [0, 1]k × E → F , with [0, 1]0 = 0, 1 . (8)

such (u1, . . . , uk) → H(u1, . . . , uk , x) continuous,H(0, . . . , 0, x) = f (x) and H(1, . . . , 1, x) = g(x).

Two topological spaces E and F has the same k-homotopy if thereexists two functions f : E → F and g : F → E such g f and f g

are k-homotopic to the respective identity functions.






Example

Two sets with different number of connected componants cannot be k-homotopic;

The sets R and x are 1-homotopic;

R and the circle S1 are not 1-homotopic but 0-homotopic;

The sphere S2 and the torus T 2 are not 1-homotopic but2-homotopic;

Two homeomorph sets have the same k-homotopy.






Definition

The border application is an application δ which associates to an-dimensional connexe manifold M:

If the manifold has the same n-homotopy than Sn, δM = ∅;If the manifold is homeomorph to the hypercube or anhalf-space of R

n, δM is the image of the set of faces or thedelimitation of the half-space by the homeomorphism;

In other cases, the manifold has a hole, the border is theunion of the border of the manifold completed without thehole and the border of the hole.






We have the following properties:

δM is either empty or dim δM = dimM− 1;

δδM = ∅;If M is a star-shaped open set, then δM = ∅ implies that Mis a border;

The set Kerδ/Imδ is called De Rham homology.






The exterior derivative is an application d such

d(

∧k(M))

⊂ ∧k+1(M) such that for any S ⊂ M,

k + 1-dimensional sub-manifold and for any α ∈ ∧k(M):

∫

Sdα =

∫

δSα(Green-Stockes formula) (9)

There exists a unique linear d such (9) holds and:

df =

n∑

j=1

∂f

∂θjdθj ;

d(f α) = df ∧ α+ fdα (Leibnitz rule).





Orientation of the border

Let V = ϕ([0, 1]n) a n-dimensional sub-manifold such δV 6= ∅ andconsider:

α = fdθ1 ∧ . . . ∧ θj−1 ∧ θj+1 ∧ . . . ∧ θn (10)

We have:∫

δVα = εj

∫

[0,1]n−1

[f (θj = 1) − f (θj = 0)] dθ1 . . . dθj−1dθj+1 . . . dθn

= εj

∫

[0,1]n

∂f

∂θjdθ1 . . . dθn

= εjε(−1)j−1

∫

Vdα. (11)

So εjε = (−1)j−1.





Properties of the exterior derivative

d is R-linear;

df =

n∑

j=1

∂f

∂θjdθj ;

d d = 0;

d(α ∧ β) = dα ∧ β + (−1)|α|α ∧ dβ (Leibnitz rule).

Remark: The set Ker d/Im d is called the De Rham cohomology.





Interior derivative

Let V a vector field, the interior derivative ιV is the unique linearapplication from

∧k to∧k−1 such that:

ιV f = 0;

ιV (df ) = V .f ;

ιV (α ∧ β) = ιVα ∧ β + (−1)|α|α ∧ ιVβ (Leibnitz rule).




Lie derivative

Integral curves and flows

Let V be a vector field. An integral curve t → θ(t) with origineP = ϕ(θ(0)) is a solution of:

V (θ(t)) = θ(t), (12)

with:

θ =

n∑

j=1

θj∂

∂θj. (13)

The flow (φt)t∈R of V is a set of applications φt which associatesto P ∈ M the point Q = ϕ(θ(t)), where θ is the integral curvewith origine P.




Lie derivative

Properties of the flow

φs φt = φt φs = φs+t ;

φ0 = IdM.




Lie derivative

Properties of the flow

φs φt = φt φs = φs+t ;

φ0 = IdM.

We define the pullback φ∗t , for P = ϕ(θ) and Q = φt(P):

If W (θ) ∈ TP(M) then (φ∗t W )(θ) ∈ TQ(M);

If α(θ) defined on TP(M) × . . . × TP(M), then (φ∗tα)(θ)defined on TQ(M) × . . .× TQ(M).




Lie derivative

The Lie derivative

The pullbacks are respectively defined by:

φ∗t f = f φt ;

(φ∗t W )(θ)(f ) = W ((ϕ−1 φt ϕ)(θ))(φ∗t f );

(φ∗tα)(θ)(φ∗t W1(θ), . . . , φ∗t Wk(θ)) =

α((ϕ−1 φt ϕ)(θ))(W1(θ), . . . ,Wk(θ)).

The Lie derivative is then defined as:

LV (T ) = limt→0

φ∗t T − φ∗0T

t, (14)

where T = f , W or α.




Lie derivative

Properties of the Lie derivative

The Lie derivative satisfies:

T → LV T is linear;

LV (f ) = V .f : extends the directional derivative;

LV (fW ) = (V .f )W + f LV (W ): Leibnitz rule for vector fields;

LV (W .f ) = LV (W ).f + W .LV (f ): composition ofdirectional derivative;

LV (α ∧ β) = LV (α) ∧ β + α ∧ LV (β): Leibnitz rule fordifferential form.




Lie derivative

Lie derivative of vector field: LV (W ) is the unique vector fieldsuch that:

LV (W ).f = V .(W .f ) − W .(V .f ). (15)

Proof.

V .(W .f ) = LV (W .f ) because W .f is a function,

= LV (W ).f + W .LV (f ),

= LV (W ).f + W .(V .f ). (16)




Lie derivative

Lie derivative of differential form: LVα is given by theGreen-Ostrogradski formula:

LVα = (d ιV )α+ (ιV d)α. (17)

Example

If ω ∈ ∧n(M), LVω = divVω. We have the classicalGreen-Ostrogradski formula:

∫

VdivVω =

∫

δVιVω. (18)




Definitions

In the first part, we were studying topology and measure theory ona manifold.From now, we will study geometry.A first mean to provide a manifold with a geometry: to define it asa Riemannian manifold.

A Riemannian manifold is a manifold such that each tangent spaceTP(M) is provided with an inner product. We denote:

gi ,j(θ) =

⟨

∂

∂θi(θ),

∂

∂θj(θ)

⟩

, (19)

and ei = ∂∂θi

.In Riemannian manifolds, the volume form isω =

√det Gdθ1 ∧ . . . ∧ dθn.




Definitions

Example (The sphere S2)

A parametrisation and the associated base are:

x = sin(θ) cos(ϕ)y = sin(θ) sin(ϕ)

z = cos(θ)

eθ = cos(θ) cos(ϕ)ex + cos(θ) sin(ϕ)ey

− sin(θ)ez

eϕ = − sin(θ) sin(ϕ)ex + sin(θ) cos(ϕ)ey

If the inner product is inheritated from R3 then:

G =

(

1 00 sin2 θ

)

, (20)

and the volume form is ω = sin θdθ ∧ dϕ.




Distance and geodesic curves

Riemannian geometry and geodesic curves

The distance between two points P = ϕ(θ(1)) and Q = ϕ(θ(2)) isthe minimum of:

∫ t(2)

t(1)

√

∑

i ,j

gi ,j(θ(t))θi (t)θj(t)dt, (21)

where t → θ(t) describes the set of curves of M.Let us denote L(θ, θ) =

∑

i ,j gi ,j(θ)θi θj , the minimum is reachedfor t → θ(t), called geodesic curve, solution of:

∂L

∂θk(θ(t), θ(t)) − d

dt

(

∂L

∂θk(θ(t), θ(t))

)

= 0. (22)




Distance and geodesic curves

Riemannian geometry and geodesic curves

The Euler-Lagrange equation can be expressed as:

∑

i

gi ,k(θ(t))θi(t)+∑

i ,j

θi (t)θj(t)

[

∂gi ,k(θ(t))

∂θj− 1

2

∂gi ,j(θ(t))

∂θk

]

= 0.




Information geometry

Geometry of the set of probability densities

We study parameterized set of probability distribution.At each θ ∈ Θ ⊂ R

k , we associate a probability densityy → p(y ; θ):

∫

p(y ; θ)dy = 1 and Pθ(Y ∈ [a, b]) =

∫ b

a

p(y ; θ)dy ;

For each fixed y , θ → p(y ; θ) is differentiable.

The Riemannian metric is chosen in order to the volume form isthe prior distribution such that an infinite sample of p(y ; θ)provides the maximum of information.






θ ∈ Θ a parameter that we want to estimate, y → p(y ; θ) thecorresponding density and y1:n = (y1, . . . , yn) a sample of p(y ; θ).The Bayesian inference consists in:

1 A prior knowledge on θ: p(θ);

2 y1:n brings posterior knowledge: p(θ|y1:n) =p(y1:n; θ)p(θ)

p(y1:n),

where:

p(y1:n) =

∫

p(y1:n; θ)p(θ)dθ. (23)






Statistical inference is like observing the parameter through a noisychannel:

H(Θ) H(Y1:n)

Noisy channel

H(Y1:n|Θ)

H(Θ) = −∫

log(p(θ))p(θ)dθ: prior information at input;

H(Y1:n) = −∫

log(p(y1:n))p(y1:n)dy1:n: total information

from observation at output;

H(Y1:n|Θ) = −∫

log(p(y1:n; θ))p(y1:n; θ)p(θ)dθdy1:n:

information added from noise of the channel (randomness);





Jeffreys’ priors and Fisher metric

Finally: H(Y1:n) − H(Y1:n|Θ) is the remaining information on θ.Asymptotically: When N big enough, this quantity is maximal for:

p(θ) ∝√

det IY (θ) Jeffreys prior. (24)

where:

IY (θ)i ,j = E

[

∂

∂θilog p(Y ; θ) × ∂

∂θjlog p(Y ; θ)

]

Fisher information.

(25)One can show that the Jeffreys prior corresponds to the volumeform, consequently G = IY (θ).





Distance between probability distributions

Let D(θ1, θ2) the distance between two distributions p(y ; θ1) andp(y ; θ2) and θN

ML = argmax p(y1, . . . , yN ; θ) the maximumlikelihood estimator. We show that:

limN→+∞

D(θNML, θ0) =

∣

∣

∣

∣

N(

0,1

N

)∣

∣

∣

∣

in distribution, (26)

where θ0 is the true parameter.Utility: In interval estimation, the confident interval

θ : D(θ, θNML) < ε

doesn’t depend on the true parameter.




Gradient, Laplacian and Brownian motions on manifolds

The gradient and Laplace-Beltrami operator

Let f be a function from M to R, there exists a unique vectorfield, called gradient of f , denoted gradf such that:

〈gradf ,V 〉 = df (V ), (27)

for all vector fields V .

The Laplace-Beltrami operator of f is defined as:

∆f = divgradf . (28)





Denote g i ,j the coefficients of G−1, we have:

gradf = G−1

[

∂f

∂θ1e1 + . . .+

∂f

∂θ1en

]

;

divV =

n∑

j=1

∂Vj

∂θj+

1

2Vj

n∑

i ,k=1

g i ,k ∂gk,i

∂θj

;

∆f =

n∑

j=1

[

n∑

i=1

(

g i ,j ∂2f

∂θi∂θj+∂g i ,j

∂θj

∂f

∂θi

)

+1

2

(

n∑

l=1

g l ,j ∂f

∂θl

)

n∑

i ,k=1

g i ,k ∂gk,i

∂θj

.





Martingals, local martingals and semi-martingals

Let (Ω,A,P) a probability space.

A continuous time process (Mt) is a martingal ifE [Mt |σ((Mu)u≤s)] = Ms for any s ≤ t;

A continuous time process (Mt) is a local martingal if thereexists an increasing sequence of stopping times (Tn) such thatthe processes (Mt∧Tn) are martingals;

A predictible process (At) is a process such that for anyω ∈ Ω, there exists a Radon measure µ(ω) such thatAt(ω) − As(ω) = µ(ω) (]s, t]);

A semi-martingal is the sum of a local martingal and apredictible process.





Stochastic integral and differential forms

If M and N are two semi-martingals, we define the Ito integral:

∫ t

0f (Ms)dNs =L2

limn→+∞

n−1∑

k=0

f (Mtk )(Ntk+1− Ntk );

(29)

Pt =∫ t

0 f (Ms)dNs will be denoted dPt = f (Ms)dNs .





Ito formula

Let M and N two local martingals, there exists a unique predictibleprocess denoted 〈M,N〉 such that MN − 〈M,N〉 is a localmartingal.If M and N are semi-martingals, 〈M,N〉 is the predictible processassociated to their respective local martingal parts.Let M = (M(1), . . . ,M(n)) ∈ R

n a semi-martingal and f a functionfrom R

n to R, then:

df (Mt) =n∑

j=1

∂f

∂mj

(Mt)dM(j)t +

1

2

n∑

i ,j=1

∂2f

∂mi∂mj

(Mt)d⟨

M(i),M(j)⟩

t.





Brownian motion in a manifold

A Brownian motion in Rn is the martingal Gaussian process

(B(1), . . . ,B(n)) such that⟨

B(i),B(i)⟩

= t and⟨

B(i),B(j)⟩

= 0.A function f from M to R is solution of the Heat equation if:

1

2∆f +

∂f

∂t= 0. (30)

The Brownian motion M = ϕ(Θ) on M is such that, if f is

solution of the Heat equation and(

∂∂θ1, . . . , ∂

∂θn

)

orthonormal

base:

df (Mt , t) =n∑

j=1

∂f

∂θjdΘj

t.





Expression of the Brownian motion

If ∆ =

n∑

i=1

ai∂

∂θi+∑

i ,j

bi ,j∂2

∂θi∂θj, then:

dΘ(i)t =

ai

2dt +

k∑

l=1

α(i)l dB

(l)t , (31)

where B(l) is Brownian motion on R and:

α(1)1 . . . α

(1)k

......

...

α(k)1 . . . α

(k)k

×

α(1)1 . . . α

(k)1

......

...

α(1)k . . . α

(k)k

=

b1,1 . . . b1,k...

......

bk,1 . . . bk,k

.




Notion of connexion

Notion of connexion

If (e1, . . . , en) is a base of the tangent space Lei(ej ) = 0, however

the tangent space is not constant.The change of the tangent spaces defines the geometry of themanifold. For this, we define another mean to derive vector fields:the connexion:

∇V is linear;

∇V (fW ) = (V .f )W + f ∇V (W ) (Leibnitz rule);

∇fV (W ) = f ∇V (W ).




Notion of connexion

The Christoffel coefficients

The Christoffel coefficients Γki ,j determines the connexion:

∇ei(ej ) =

n∑

k=1

Γki ,jek ,

so we have:

∇V (W ) =n∑

k=1

n∑

i=1

Vi

∂Wk

∂θi+

n∑

j=1

WjΓki ,j

ek .




The Riemannian case and Levi-Civita connexion

The Levi-Civita connexion

The geodesics are the integral curves of the vector field V suchthat:

∇V (V ) = 0. (32)

In Riemannian case: A connexion is Riemannian if the previousequation is equivalent to the Euler-Lagrange equation, we show:

∑

k

gi ,kΓil ,j =

∂gl ,k

∂θj− 1

2

∂gl ,j

∂θk. (33)

A connexion without torsion is a connexion such that Γki ,j = Γk

j ,i .The Levi-Civita connexion is the unique Riemannian withouttorsion connexion.




The Riemannian case and Levi-Civita connexion

If ∇ is the Levi-Civita connexion, we have:

∑

k

gi ,kΓil ,j =

1

2

[

∂gl ,k

∂θj+∂gj ,k

∂θl− ∂gl ,j

∂θk

]

.

The Levi-Civita is the unique without torsion connexion such that:

ek . 〈ei , ej 〉 = 〈∇ekei , ej 〉 + 〈ei ,∇ek

ej 〉 .




Second-order derivative, torsion and curvature

Second-order derivative

(v .f )(P) is correctly defined, because depends only on v = γ(0).However v .(V .f ) depends also on γ(0), where γ integral curve ofV .We define the second order derivative of f as:

(∇2v f )(P) = v .(V .f )(P),





Second-order derivative

(v .f )(P) is correctly defined, because depends only on v = γ(0).However v .(V .f ) depends also on γ(0), where γ integral curve ofV .We define the second order derivative of f as:

(∇2v f )(P) = v .(V .f )(P),

where V is the vector field associated to geodesic such P = γ(0)and v = γ(0).Remark: The second derivative depends on the geometry.






We define the second-order derivative:

∇2V f : θ → (∇2

V (θ)f )(ϕ(θ)).

We have:∇2

V f = V .(V .f ) − (∇V V ).f

Remark: It coincides with the classical second-order derivative if V

is geodesic vector field.





We define:

∇2V ,W f = V .(W .f ) − (∇V W ).f ;

∇2V ,WZ = ∇V (∇W Z ) −∇∇V W Z .

(34)

The torsion of a connexion is defined as:

T (V ,W ) = ∇V W −∇W V − LV W .

We have:∇2

V ,W f −∇2W ,V f = −T (V ,W ).f , (35)

the torsion says that the second-order derivatives don’t commutefor functions.





The curvature of a without torsion connexion is defined as:

R(V ,W ,Z ) = ∇V (∇W Z ) −∇W (∇V Z ) −∇LV W Z .

We have:∇2

V ,W Z −∇2W ,V Z = R(V ,W ,Z ), (36)

the curvature says that the second-order derivatives don’tcommute for vector fields.In a three dimensional space:

R(V ,W ,Z ).U = −KVol(V ,W )Vol(Z ,U), (37)

K is called Gauss curvature.





No riemannian set of probability distributions

The study of no Riemannian or information geometry with torsionhas been by S. I. Amari and H. Nagaoka in Methods of InformationGeometry



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