Anisotropic Metropolis Adjusted Langevin Algorithm: convergence and utility in Stochastic EM...

Anisotropic Metropolis adjusted Langevin algorithm:Convergence and utility in stochastic EM algorithm.

Stephanie Allassonniere

CMAP, Ecole Polytechnique

BigMC, January 2012

Join work with Estelle Kuhn (INRA, France)

Stephanie Allassonniere (CMAP) AMALA BigMC, January 2012 1 / 42

Introduction

Introduction:

Where does the problem came from?

Image analysis: Compare two observations via the quantification ofthe deformation from one to the other (D’Arcy Thompson, 1917)

Registration

/ Variance

Each element of a population is a smooth deformation of a template

Template estimation

/ Mean

Introduction

Introduction:

Registration

/ Variance

Template estimation

/ Mean

Introduction

Introduction:

Registration

/ Variance

Template estimation

/ Mean

Introduction

Introduction:

Registration

/ Variance

Template estimation / Mean

Introduction

Introduction:

Registration / Variance

Template estimation / Mean

Introduction

Introduction:

Deformable template model: (u = voxel, vu its position)

y(u) = I0(vu −m(vu)) + σε(u) ,

Template I0 and geometry Law(m) estimation

High dimensional setting, Low sample size

Considering the LDDMM framework through the shooting equations

Introduction

Introduction:

y(u) = I0(vu −m(vu)) + σε(u) ,

Introduction

Introduction:

y(u) = I0(vu −m(vu)) + σε(u) ,

Introduction

Introduction:

y(u) = I0(vu −m(vu)) + σε(u) ,

Introduction

Introduction:

y(u) = I0(vu −m(vu)) + σε(u) ,

Introduction

Outline:

1. AMALA: simulation of random variables in high dimension

Anisotropic MALA descriptionConvergence property

2. AMALA within stochastic algorithm for parameter estimation

Maximum likelihood estimation for incomplete datasettingAMALA-SAEMConvergence properties

3. Experiments

BME-Template model: small deformation settingBME-Template model: LDDMM setting

Introduction

Outline:

3. Experiments

Anisotropic Metropolis Adjusted Langevin algorithm (AMALA)

Introduction:

General setting:

Simulation of random variable in high dimension settings: → GibbsSampler not useful

Metropolis Adjusted Langevin Algorithm (MALA)

Target distribution: πAt iteration k of this algorithm, Xk the current valueSimulate Xc w.r.t. N (Xk + δD(Xk), δIdd)where D(x) = b

max(b,|∇ log π(x)|)∇ log π(x).

Update Xk+1 = Xc with probability

α(Xk ,Xc) = min(

1, π(Xc )qMALA(Xc ,Xk )qMALA(Xk ,Xc )π(Xk )

)and Xk+1 = Xk otherwise.

Problem: isotropic covariance matrix = numerically trapped(α(Xk ,Xc) = 0)

Introduction:

General setting:

α(Xk ,Xc) = min(

Introduction:

General setting:

α(Xk ,Xc) = min(

Introduction:

General setting:

Target distribution: π

At iteration k of this algorithm, Xk the current valueSimulate Xc w.r.t. N (Xk + δD(Xk), δIdd)where D(x) = b

α(Xk ,Xc) = min(

Introduction:

General setting:

α(Xk ,Xc) = min(

Introduction:

General setting:

α(Xk ,Xc) = min(

Introduction:

General setting:

α(Xk ,Xc) = min(

→ Anisotropic Metropolis Adjusted Langevin Algorithm (AMALA)

Anisotropic Metropolis Adjusted Langevin algorithm (AMALA) Description of the algorithm

How including anisotropy?

Following the magnitude of the gradient

First approximation: independence of directions

Bounded covariance (same as bounded drift)

Anisotropic Metropolis Adjusted Langevin Algorithm (AMALA)

For all k = 1 : kend Iterates of Markov chain

Sample Xc with respect to

N (Xk + δD(Xk), δΣ(Xk))

with D(Xk) = bmax(b,|∇ log π(Xk )|)∇ log π(Xk) and

Σ(Xk) = Idd + diag(([∇ log π(Xk)]21 ∧ b), ... , ([∇ log π(Xk)]2d ∧ b)

)Compute the acceptance ratio

α(Xk ,Xc) = min

(1,π(Xc)qc(Xc ,Xk)

qc(Xk ,Xc)π(Xk)

)(qc = the pdf of this distribution).

Sample Xk+1 = Xc with probability α(Xk ,Xc) and Xk+1 = Xk withprobability 1− α(Xk ,Xc) = Acceptation/reject

Compute the acceptance ratio

α(Xk ,Xc) = min

(1,π(Xc)qc(Xc ,Xk)

qc(Xk ,Xc)π(Xk)

α(Xk ,Xc) = min

(1,π(Xc)qc(Xc ,Xk)

qc(Xk ,Xc)π(Xk)

α(Xk ,Xc) = min

(1,π(Xc)qc(Xc ,Xk)

qc(Xk ,Xc)π(Xk)

Anisotropic Metropolis Adjusted Langevin algorithm (AMALA) Geometric ergodicity of the chain

Geometric ergodicity of the Markov chain

Condition:

π super-exponential: Smoothness condition on the target distribution

(B1) The density π is positive with continuous first derivative such that:

lim|x|→∞

n(x).∇ log π(x) = −∞ (1)

andlim sup|x|→∞

n(x).m(x) < 0 (2)

where ∇ is the gradient operator in Rd , n(x) = x|x| is the unit vector

pointing in the direction of x and m(x) = ∇π(x)|∇π(x)| is the unit vector in

the direction of the gradient of the stationary distribution at point x .

Result:

Existence of a small set

Π(x ,A) ≥ εν(A)1C(x), ∀x ∈ X and ∀A ∈ B

Drift condition: pulls the chain back into the small set

ΠV (x) ≤ λV (x) + b1C(x) .

Geometric ergodicity

supx∈X

|ΠnV (x)− π(x)|V (x)

≤ Rρn . (3)

Result:

ΠV (x) ≤ λV (x) + b1C(x) .

supx∈X

≤ Rρn . (3)

Result:

ΠV (x) ≤ λV (x) + b1C(x) .

supx∈X

≤ Rρn . (3)

Result:

ΠV (x) ≤ λV (x) + b1C(x) .

supx∈X

≤ Rρn . (3)

Experiments on synthetic data

Target: 10 dimensional Gaussian distribution with zero mean anddiagonal covariance matrix with diagonal coefficients randomly pickedbetween 1 and 2500

Comparison of AMALA and symmetric random walk

500, 000 iterations for each algorithm starting at zero

Mean squared jump distance (MSJD) in stationarity:AMALA 0.1504 - random walk 0.0407.

Experiments on synthetic data

Figure: Autocorrelation functions of the AMALA (red) and the random walk(blue) samplers for four of the ten components of the Gaussian 10 dimensionaldistribution.

Why not using exising MALA-like algorithms?

Optimised MALA-like algorithms are usually adaptive

Good performances in practice

Good theoretical properties

Numerical problem at the first iterations (not yet stationary):convergence time?

Most important: Our goal = parameter estimation

AMALA = one tool inside another algorithm

Adaptive + estimation algorithm = numerical issues: too manydegree of freedom

However

Applying AMALA within SAEM

Outline:

3. Experiments

Applying AMALA within SAEM Maximum likelihood estimation for incomplete data setting

Maximum likelihood estimation for incomplete data setting

y ∈ Rn: observed data

z ∈ Rl : missing data

(y , z) ∈ Rn+l : complete data

P = {f (y , z ; θ), θ ∈ Θ}: family of parametric pdfs on Rn+l

Assumption:∃θ ∈ Θ s.t. the complete data likelihood q(y , z ; θ) = f (y , z ; θ)

Observed likelihood:

g(y ; θ) =

∫f (y , z ; θ)µ(dz). (4)

Given a sample of observations (yi )1≤i≤n = yn1Find: θg in Θ s.t.

θg = arg maxθ∈Θ

g(yn1 ; θ)

g(y ; θ) =

∫f (y , z ; θ)µ(dz). (4)

g(yn1 ; θ)

g(y ; θ) =

∫f (y , z ; θ)µ(dz). (4)

g(yn1 ; θ)

g(y ; θ) =

∫f (y , z ; θ)µ(dz). (4)

g(yn1 ; θ)

g(y ; θ) =

∫f (y , z ; θ)µ(dz). (4)

g(yn1 ; θ)

g(y ; θ) =

∫f (y , z ; θ)µ(dz). (4)

g(yn1 ; θ)

g(y ; θ) =

∫f (y , z ; θ)µ(dz). (4)

Given a sample of observations (yi )1≤i≤n = yn1

Find: θg in Θ s.t.θg = arg max

θ∈Θg(yn1 ; θ)

g(y ; θ) =

∫f (y , z ; θ)µ(dz). (4)

g(yn1 ; θ)

Applying AMALA within SAEM Description of the algorithm

AMALA-SAEM

Incomplete data setting + maximum likelihood estimation = EMalgorithm

General case −→ E step not tractable

Stochastic Approximation EM for convergence properties

with MCMC method for simulation step.

AMALA-SAEM

→ AMALA-SAEM: using AMALA as the MCMC method

Description of the algorithm

Assumption: model in the exponential family = all information carried bysufficient statistics S

For k = 1 : kend Iteration of SAEM

Sample zk through a single AMALA step (simulation andacceptation/reject) using current parameter θk−1

Compute the stochastic approximation

sk = sk−1 + γk (S(zk)− sk−1) ,

where (γk)k is a sequence of positive step sizes.

Update the parameterθk = θ(sk).

sk = sk−1 + γk (S(zk)− sk−1) ,

Can require truncation on random boundaries for convergence purposes

Applying AMALA within SAEM Convergence properties

Convergence properties

Conditions:

Smoothness of the model (classic conditions for convergence ofstochastic approximation and EM)

Condition for AMALA geometric ergodicity (B1)

Results:

Convergence of (sk) a.s. towards critical point of mean field of theproblem

Convergence of estimated parameters (θk) a.s. towards critical pointof observed likelihood

Central limit theorem for (θk) with rate 1/√γk

Conditions:

Results:

Conditions:

Results:

Conditions:

Results:

Conditions for the SA to converge

Define for any V : X → [1,∞] and any g : X → Rm the norm

‖g‖V = supz∈X

‖g(z)‖V (z)

(A1’) S is an open subset of Rm, h : S → Rm is continuous and there existsa continuously differentiable function w : S → [0,∞[ with thefollowing properties.

(i) There exists an M0 > 0 such that

L , {s ∈ S, 〈∇w(s), h(s)〉 = 0} ⊂ {s ∈ S, w(s) < M0} .

Conditions for the SA to converge (2)

(ii) There exists a closed convex set Sa ⊂ S for whichs → s + ρHs(z) ∈ Sa for any ρ ∈ [0, 1] and (z , s) ∈ X × Sa (Sa isabsorbing) and such that for any M1 ∈]M0,∞], the set WM1 ∩ Sa is acompact set of S where WM1 , {s ∈ S, w(s) ≤ M1}.

(iii) For any s ∈ S\L 〈∇w(s), h(s)〉 < 0.(iv) The closure of w(L) has an empty interior.

(A2’) For any s ∈ S, Hs : X → S is measurable and∫‖Hs(z)‖πs(dz) <∞.

(A3”) There exist a function V : X → [1,∞] such that{z ∈ X ,V (z) <∞} 6= ∅, constants a ∈]0, 1], p ≥ 2 , r > 0 andq ≥ 1 such that for any compact subset K ⊂ S,

sups∈K‖Hs‖V <∞ , (5)

sups∈K

(‖gs‖V + ‖Πsgs‖V ) <∞ , (6)

sups,s′∈K

‖s − s ′‖−a{‖gs − gs′‖V q + ‖Πsgs − Πs′gs′‖V q} <∞ , (7)

where for anys ∈ S a solution of the Poisson equationg − Πsg = Hs − πs(Hs) is denoted by gs .

(ii) For any sequence ε = (εk)k≥0 satisfying εk < ε for an ε sufficientlysmall, for any sequence γ = (γk)k≥0, there exist a constant C suchthat and for any z ∈ X ,

sups∈K

supk≥0

Eγz,s[V p(zk)1σ(K)∧ν(ε)≥k

]≤ CV p+r (z) , (8)

where ν(ε) = inf{k ≥ 1, ‖sk − sk−1‖ ≥ εk} andσ(K) = inf{k ≥ 1, sk /∈ K} and the expectation is related to thenon-homogeneous Markov chain ((zk , sk))k≥0 using the step-sizesequence γ = (γk)k≥0.

(A4) The sequences γ = (γk)k≥0 and ε = (εk)k≥0 are non-increasing,

positive and satisfy:∞∑k=0

γk =∞, limk→∞

εk = 0 and

∞∑k=1

{γ2k + γkε

ak + (γkε

−1k )p} <∞, where a and p are defined in (A3”).

Condition for AMALA-SAEM to converge

(M1) The parameter space Θ is an open subset of Rp. The completedata likelihood function is given by:

f (y , z ; θ) = exp {−ψ(θ) + 〈S(z), φ(θ)〉} ,

where S is a Borel function on Rl taking its values in an open subsetS of Rm. Moreover, the convex hull of S(Rl) is included in S, and,for all θ in Θ, ∫

||S(z)||pθ(z)µ(dz) <∞.

(M2) The functions ψ and φ are twice continuously differentiable onΘ.

Condition for AMALA-SAEM to converge (2)

(M3) The function s : Θ→ S defined as

s(θ) ,∫

S(z)pθ(z)µ(dz)

is continuously differentiable on Θ.

(M4) The function l : Θ→ R defined as the observed-datalog-likelihood

l(θ) , log g(y ; θ) = log

∫f (y , z ; θ)µ(dz)

is continuously differentiable on Θ and

∫f (y , z ; θ)µ(dz) =

∫∂θf (y , z ; θ)µ(dz).

(M5) There exists a function θ : S → Θ, such that:

∀s ∈ S, ∀θ ∈ Θ, L(s; θ(s)) ≥ L(s; θ).

Moreover, the function θ is continuously differentiable on S.

(M6) The functions l : Θ→ R and θ : S → Θ are m timesdifferentiable.

(M7)(i) There exists an M0 > 0 such that{

s ∈ S, ∂s l(θ(s)) = 0}⊂ {s ∈ S, −l(θ(s)) < M0} .

(ii) For all M1 > M0, the set ¯Conv(S(Rl)) ∩ {s ∈ S, −l(θ(s)) ≤ M1} is acompact set of S.

(M8) There exists a polynomial function P of degree 2 such that forall z ∈ X

||S(z)|| ≤ |P(z)| .

(B3) For any compact subset K of S, there exists a polynomialfunction Q of the hidden variable such that

sups∈K|∇z log pθ(s)(z)| ≤ |Q(z)|

Application on Bayesian Mixed effect template estimation

Outline:

3. Experiments

Application on Bayesian Mixed effect template estimation Description of the BME Template model

BME Template model with small deformations

y(u) = I0(vu −m(vu)) + σε(u) ,

Parametric template and deformation:

Iα(v) = (Kpα)(v) =kp∑j=1

Kp(v , rp,k)αj and

mz(v) = (Kgz(v) =kg∑j=1

Kg (v , rg ,k)z j .

Generative model:z ∼ ⊗n

i=1N2kg (0, Γg ) | Γg ,

y ∼ ⊗ni=1N|Λ|(mzi Iα, σ

2Id) | z, α, σ2 ,

Bayesian framework → MAP estimator (= penalised MLE)

y(u) = I0(vu −m(vu)) + σε(u) ,

Kp(v , rp,k)αj and

Kg (v , rg ,k)z j .

i=1N2kg (0, Γg ) | Γg ,

2Id) | z, α, σ2 ,

y(u) = I0(vu −m(vu)) + σε(u) ,

Kp(v , rp,k)αj and

Kg (v , rg ,k)z j .

i=1N2kg (0, Γg ) | Γg ,

2Id) | z, α, σ2 ,

y(u) = I0(vu −m(vu)) + σε(u) ,

Kp(v , rp,k)αj and

Kg (v , rg ,k)z j .

i=1N2kg (0, Γg ) | Γg ,

2Id) | z, α, σ2 ,

Application on Bayesian Mixed effect template estimation Results on the template estimation

Training sets

Figure: Left: Training set (inverse video). Right: Noisy training set (inversevideo).

Application on Bayesian Mixed effect template estimation Results on the template estimation

Estimated templates

Algorithm/Noise level

FAM-EM H.G.-SAEM AMALA-SAEM

No Noise

Noisyof Variance 1

Figure: Estimated templates using different algorithms and two level of noise.The training set includes 20 images per digit.

Application on Bayesian Mixed effect template estimation Results on the covariance matrix estimation

Estimated geometric variability

Figure: Synthetic samples generated with respect to the BME template model.Left: No noise. Right: Noisy data.

Application on Bayesian Mixed effect template estimation CLT empirical proof

Empirical proof of the CLT

Figure: Evolution of the estimation of the noise variance along the SAEMiterations. Left: original data. Right: Noisy training set.

Application on Bayesian Mixed effect template estimation CLT empirical proof

Figure: Evolution of the estimation of the noise variance along the SAEMiterations. Test of convergence towards the Gaussian distribution of the estimatedparameters.

Application on Bayesian Mixed effect template estimation Medical image template estimation

Corpus callosum data base

Figure: Medical image template estimation: 10 Corpus callosum and spleniumtraining images among the 47 available.

Figure: Grey level mean. FAM-EM estimated template. Hybrid Gibbs - SAEMestimated template.AMALA-SAEM estimation .

Application on Bayesian Mixed effect template estimation Results on the template estimation using LDDMM shooting

BME Template model with LDDMM

y(u) = I0(φ−1β(0)(vu)) + σε(u) ,

Parametric template: Iα(v) = (Kpα)(v) =kp∑j=1

Kp(v , rp,k)αj and φ

LDDMM solution of shooting with initial momentum β(0).

i=1N2kg (0, Γg ) | Γg ,

y ∼ ⊗ni=1N|Λ|(φβ(0)Iα, σ

2Id) | z, α, σ2 ,

LDDMM: parametric deformation:

Fix some control points: c(t) = (c1(t), ..., cng (t))

Choose a kernel Kg

Start from an initial momentum β(0) = β1(0), ..., βng (0)

Then, Hamiltonian System → Time evolution of both momenta andcontrol points

∂β(c, β) = Kg (c(t))β(t)

dt= −∂H

∂c(c, β) = −1

2∇c(t)K (β(t), β(t))

LDDMM: parametric deformation (2):

Interpolating on any point of the domain:

vt(r) = (Kgβ(t))(r) =

ng∑k=1

Kg (r , ck(t))βk(t) ∀r ∈ D (10)

Deformation = solution of the flow equation: ∂φβ(0)(t)

∂t= vt ◦ φβ(0)(t)

φ0 = Id .

φβ(0) = φβ(0)(1)

Gradient computation

E (ci , βi︸︷︷︸

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸+σ2 Reg(φ1)︸︷︷︸

L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

∇S0E = dS0y(0)T∇y(0)A +∇S0L

∇yk (0)A = 2 (I0(yk(0))− I (yk(1)))∇yk (0)I0

- Momenta decrease image discrepancy

- Control Points attracted by image contours

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸+σ2 Reg(φ1)︸︷︷︸

L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}i

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

∇S0E = dS0y(0)T∇y(0)A +∇S0L

∇yk (0)A = 2 (I0(yk(0))− I (yk(1)))∇yk (0)I0

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸+σ2 Reg(φ1)︸︷︷︸

L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

∇S0E = dS0y(0)T∇y(0)A +∇S0L

∇yk (0)A = 2 (I0(yk(0))− I (yk(1)))∇yk (0)I0

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸+σ2 Reg(φ1)︸︷︷︸

L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

∇S0E = dS0y(0)T∇y(0)A +∇S0L

∇yk (0)A = 2 (I0(yk(0))− I (yk(1)))∇yk (0)I0

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸

A(yk (0))

+σ2 Reg(φ1)︸︷︷︸

L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

∇S0E = dS0y(0)T∇y(0)A +∇S0L

∇yk (0)A = 2 (I0(yk(0))− I (yk(1)))∇yk (0)I0

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸

A(yk (0))

+σ2 Reg(φ1)︸︷︷︸L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

∇S0E = dS0y(0)T∇y(0)A +∇S0L

∇yk (0)A = 2 (I0(yk(0))− I (yk(1)))∇yk (0)I0

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸

A(yk (0))

+σ2 Reg(φ1)︸︷︷︸L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

∇S0E = dS0y(0)T∇y(0)A +∇S0L

∇yk (0)A = 2 (I0(yk(0))− I (yk(1)))∇yk (0)I0

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸

A(yk (0))

+σ2 Reg(φ1)︸︷︷︸L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

∇S0E = dS0y(0)T∇y(0)A +∇S0L

∇yk (0)A = 2 (I0(yk(0))− I (yk(1)))∇yk (0)I0

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸

A(yk (0))

+σ2 Reg(φ1)︸︷︷︸L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

∇S0E = dS0y(0)T∇y(0)A +∇S0L

∇yk (0)A = 2 (I0(yk(0))− I (yk(1)))∇yk (0)I0

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸

A(yk (0))

+σ2 Reg(φ1)︸︷︷︸L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸

A(yk (0))

+σ2 Reg(φ1)︸︷︷︸L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

E (ci , βi︸︷︷︸S0

) =∑

k (I0(φ−11 (yk))− I (yk))2︸︷︷︸

A(yk (0))

+σ2 Reg(φ1)︸︷︷︸L(S0)=β(0)tΓg(q(0),q(0))β(0)

S0 = {(ci , βi )}idS(t)

dt= F (S(t)) S(0) = S0

dt= G (S(t), y(t)) y(1) = y

dη(t)

dt= ∂S(t)G

Tη(t), η(0) = ∇y(0)A

dξ(t)

dt= ∂y(t)G

Tη(t)− dFT ξ(t), ξ(1) = 0

∇S0E = ξ(0) +∇S0L

Using LDDMM deformations via shooting (preliminary results)

AMALA : GH :

Conclusion

Good performances (as accurate as other algorithms)

Reduce computational time

Can handle the movement of control points in practice (theory toconfirm)

Can handle sparsity of the template (' model selection)

Removing control points ? In practice, why not... theory ?

Conclusion

Good performances (as accurate as other algorithms)

Reduce computational time

Can handle the movement of control points in practice (theory toconfirm)

Can handle sparsity of the template (' model selection)

Removing control points ? In practice, why not... theory ?

Thank you !

Conclusion

Anisotropic Metropolis Adjusted Langevin Algorithm: convergence and utility in Stochastic EM...

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