A gradient-like Variational Bayesian approach for inverse scattering problems

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Submitted on 6 Feb 2014

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A gradient-like Variational Bayesian approach for

inverse scattering problems

Leila Gharsalli

To cite this version:

Leila Gharsalli. A gradient-like Variational Bayesian approach for inverse scattering problems. 2014.

�hal-00942863�

❆ ❣r❛❞✐❡♥t✲❧✐❦❡ ❱❛r✐❛t✐♦♥❛❧ ❇❛②❡s✐❛♥ ❛♣♣r♦❛❝❤ ❢♦r ✐♥✈❡rs❡ s❝❛tt❡r✐♥❣

♣r♦❜❧❡♠s

▲❡✐❧❛ ●❤❛rs❛❧❧✐

▲❛❜♦r❛t♦✐r❡ ❞❡s ❙✐❣♥❛✉① ❡t ❙②stè♠❡s ✭▲✷❙✮

❯▼❘✽✺✵✻✿ ❈◆❘❙✲❙❯P❊▲❊❈✲❯♥✐✈ P❛r✐s✲❙✉❞

✸ r✉❡ ❏♦❧✐♦t✲❈✉r✐❡✱ ✾✶✶✾✵ ●✐❢✲s✉r✲❨✈❡tt❡✱ ❋r❛♥❝❡

❊✲♠❛✐❧✿ ❧❡✐❧❛✳❣❤❛rs❛❧❧✐❅❧ss✳s✉♣❡❧❡❝✳❢r

❆❜str❛❝t

■♥ t❤✐s ❞♦❝✉♠❡♥t✱ ✇❡ ♣r❡s❡♥t ❝♦♠♣✉t❛t✐♦♥s ♦❢ ✉♣❞❛t✐♥❣ s❤❛♣✐♥❣ ♣❛r❛♠❡t❡rs ❢♦r ❛ ♥❡✇ ♠❡t❤♦❞ ❜❛s❡❞ ♦♥ t❤❡

✈❛r✐❛t✐♦♥❛❧ ❇❛②❡s✐❛♥ ❛♣♣r♦❛❝❤ ✭❱❇❆✮ ❛❧❧♦✇✐♥❣ t♦ s♦❧✈❡ ❛ ♥♦♥❧✐♥❡❛r ✐♥✈❡rs❡ s❝❛tt❡r✐♥❣ ♣r♦❜❧❡♠✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ❞❡t❡❝t ❛♥ ✉♥❦♥♦✇♥ ♦❜❥❡❝t ❢r♦♠ ♠❡❛s✉r❡♠❡♥ts ♦❢ t❤❡ s❝❛tt❡r❡❞ ✜❡❧❞ ❛t ❞✐✛❡r❡♥t ❢r❡q✉❡♥❝✐❡s ❛♥❞ ❢♦r s❡✈❡r❛❧

✐❧❧✉♠✐♥❛t✐♦♥s✳ ❚❤✐s ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ✐s ❦♥♦✇♥ t♦ ❜❡ ♥♦♥✲❧✐♥❡❛r ❛♥❞ ✐❧❧✲♣♦s❡❞✳ ❙♦ ✐t ♥❡❡❞s t♦ ❜❡ r❡❣✉❧❛r✐③❡❞ ❜②

✐♥tr♦❞✉❝✐♥❣ ❛ ♣r✐♦r✐ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤✐s ✐s t❛❝❦❧❡❞ ✐♥ ❛ ❇❛②❡s✐❛♥ ❢r❛♠❡✇♦r❦ ✇❤❡r❡ t❤❡ ♣❛rt✐❝✉❧❛r ♣r✐♦r ✐♥❢♦r♠❛t✐♦♥

✇❡ ❛❝❝♦✉♥t ❢♦r ✐s t❤❛t t❤❡ ♦❜❥❡❝t ✐s ❝♦♠♣♦s❡❞ ♦❢ ❛ ✜♥✐t❡ ❦♥♦✇♥ ♥✉♠❜❡r ♦❢ ❞✐✛❡r❡♥t ♠❛t❡r✐❛❧s ❞✐str✐❜✉t❡❞ ✐♥ ❝♦♠♣❛❝t r❡❣✐♦♥s✳ ❚❤❡♥ ✇❡ ♣r♦♣♦s❡ t❤❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ tr✉❡ ❥♦✐♥t ♣♦st❡r✐♦r ❜② ❛ s❡♣❛r❛❜❧❡ ❧❛✇ ❜② ♠❡❛♥ ♦❢ ❛ ❣r❛❞✐❡♥t✲❧✐❦❡

❱❛r✐❛t✐♦♥❛❧ ❇❛②❡s✐❛♥ t❡❝❤♥✐q✉❡✳ ❚❤✐s ❧❛tt❡r ✐s ❛♣♣❧✐❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ♣♦st❡r✐♦r ❡st✐♠❛t♦rs ❜② ❛❧❧♦✇✐♥❣ ❛ ❥♦✐♥t

✉♣❞❛t❡ ♦❢ t❤❡ s❤❛♣❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♥❣ ♠❛r❣✐♥❛❧s ❛♥❞ r❡❝♦♥str✉❝t t❤❡ s♦✉❣❤t ♦❜❥❡❝t✳ ❚❤❡ ♠❛✐♥ ✇♦r❦

✐s ❣✐✈❡♥ ✐♥ ❬✹❪✱ ✇❤✐❧❡ t❡❝❤♥✐❝❛❧ ❞❡t❛✐❧s ♦❢ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ❝❛❧❝✉❧❛t✐♦♥s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❝✉rr❡♥t ♣❛♣❡r✳

✶ ■♥tr♦❞✉❝t✐♦♥

❚❤❡ ❣r❛❞✐❡♥t✲❧✐❦❡ ❱❛r✐❛t✐♦♥❛❧ ❇❛②❡s✐❛♥ ❛♣♣r♦❛❝❤ ❢r♦♠ ♥♦✇ ♦♥ ❞❡♥♦t❡❞ ❛s ●❱❇❆ ✐s ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❝❧❛ss✐❝❛❧ ✈❛r✐❛t✐♦♥❛❧

❇❛②❡s✐❛♥ ❛♣♣r♦❛❝❤ ✭❱❇❆✱ ❬✶❪✮ t❤❛t ❛✐♠s ❛t ❛♣♣r♦①✐♠❛t✐♥❣ ❛ ❥♦✐♥t ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ p(x|y) ❜② ❛ s❡♣❛r❛❜❧❡ ❧❛✇

q(x) =Q

iqi(xi)✇❤✐❝❤ ✐s ❛s ❝❧♦s❡ t♦ t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ❛s ♣♦ss✐❜❧❡ ✐♥ t❡r♠s ♦❢ t❤❡ ❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r ❞✐✈❡r❣❡♥❝❡✳

■t ❝❛♥ ❜❡ ♥♦t❡❞ t❤❛t ♠✐♥✐♠✐③✐♥❣ t❤❡ ❑▲ ❞✐✈❡r❣❡♥❝❡ ✐s ❡q✉✐✈❛❧❡♥t t♦ ♠❛①✐♠✐③✐♥❣ t❤❡ ❢r❡❡ ♥❡❣❛t✐✈❡ ❡♥❡r❣② ❞❡r✐✈❡❞

❢r♦♠ st❛t✐st✐❝❛❧ ♣❤②s✐❝s F(q) =R

q(x) ln (p(y, x)/q(x))❞x✳ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❛❧t❡r♥❛t❡

♦♣t✐♠✐③❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❡❛❝❤ qi(xi)❛♥❞ ✐s ❣✐✈❡♥ ❜②✿

qi(xi) ∝ expD

ln(p(x, y))E

Q

j6=iqj(xj)



. ✭✶✮

❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ qi r❡q✉✐r❡s t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ ❛❧❧ qj✱ j 6= i✳ ❍♦✇❡✈❡r✱ r❡❝❡♥t❧②✱ ♦t❤❡r ✇❛②s t❤❛♥ t❤✐s ❝❧❛ss✐❝❛❧

❛❧t❡r♥❛t❡ ♦♣t✐♠✐③❛t✐♦♥ ❤❛✈❡ ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞ ❬✺❪✳ ■♥ ❢❛❝t✱ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ✐♥✈♦❧✈❡❞ ✐♥ ❱❇❆ ✐s ❛♥ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧

❝♦♥❝❛✈❡ ♣r♦❜❧❡♠✳ ❍❡♥❝❡✱ ❛♣♣r♦①✐♠❛t✐♥❣ ❞❡♥s✐t✐❡s qi(xi)❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❛❞❛♣t✐♥❣ ❛ ❝❧❛ss✐❝❛❧ ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠✱

s✉❝❤ ❛s ❛ ❣r❛❞✐❡♥t ♠❡t❤♦❞✱ t♦ ❱❇❆✳ ❯s✐♥❣ t❤❡ ♥♦t✐♦♥ ♦❢ ♦♣t✐♠❛❧ st❡♣✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♥❣ ♠❛r❣✐♥❛❧s ❤❛✈❡ ❛♥ ✐t❡r❛t✐✈❡

❢✉♥❝t✐♦♥❛❧ ❢♦r♠✳ ❆t ✐t❡r❛t✐♦♥ n✱ t❤❡② r❡❛❞✿

e

q⁽ⁿ⁾_i (xi) ∝ eq_i⁽ⁿ⁻¹⁾(xi)(1−α)

× exp



αD

ln(p(x, y))E

Q

i6=j eq(n−1) j (xj )



 ✭✷✮

✇❤❡r❡ α ≥ 0 ✐s ❛ ❞❡s❝❡♥t st❡♣ t❤❛t ♠✐♥✐♠✐③❡s t❤❡ ❢r❡❡ ♥❡❣❛t✐✈❡ ❡♥❡r❣② ❛t ❡❛❝❤ ✐t❡r❛t✐♦♥✳

✶

✷ ❇❛②❡s✐❛♥ ❝♦♠♣✉t❛t✐♦♥s

▲❡t ✉s r❡❝❛❧❧ ♥♦✇ ❛❧❧ t❤❡ ❡①♣r❡ss✐♦♥s ♦❢ ♣r✐♦rs ❛♥❞ ❧✐❦❡❧✐❤♦♦❞s ✉s❡❞ ✐♥ t❤❡ ❇❛②❡s✐❛♥ ❢r❛♠❡✇♦r❦ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛ ♥♦♥✲❧✐♥❡❛r

✐♥✈❡rs❡ s❝❛tt❡r✐♥❣ ♣r♦❜❧❡♠ ❬✸❪✳ ❚❤❡♥ ✇❡ ❣✐✈❡ t❤❡ ❢♦r♠ ♦❢ s❡♣❛r❛t✐♦♥ ❛♥❞ ❡①♣r❡ss✐♦♥s ♦❢ ❛♣♣r♦①✐♠❛t✐♥❣ ❧❛✇s ❢♦r ❞✐✛❡r❡♥t

♣❛r❛♠❡t❡rs ✭✇❡ ❝❛♥ ❝❤❡❝❦ s❡✈❡r❛❧ r❡❢❡r❡♥❝❡s ❬✷✱ ✸✱ ✻❪ ❢♦r t❤❡ ❢♦r✇❛r❞ ♠♦❞❡❧❧✐♥❣ ♦❢ t❤❡ ♣r♦❜❧❡♠ ❛♥❞ ✐ts ❢♦r♠✉❧❛t✐♦♥✮✳

❲❡ ❤❛✈❡✿

q(y|w, vǫ) = N (G^ow, vǫ■), q(w|χ, vξ) = N (mw, Vw), q(χ|z, m, v) = N (mχ, Vχ), q(z) = exp

( λX

r X

r^′

δ(z(r) − z(r^′)) )

, p(mk) = N (mk|µ0, τ0), p(vk) = IG(vk|η0, φ0),

p(vǫ) = IG(vǫ|ηǫ, φǫ), p(vξ) = IG(vξ|ηξ, φξ),

✭✸✮

❲❡ ❞❡♥♦t❡ ψ = {m, v, vǫ, vξ} t❤❡ s❡t ♦❢ t❤❡ ❤②♣❡r✲♣❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦❞❡❧✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥

♦❢ t❤❡ ✉♥❦♥♦✇♥s r❡❛❞s✿

p (χ, w, z, ψ|y) ∝ p (y|w, vǫ) p (w|χ, vξ) p (χ|z, m, v) p (z|λ) p (m|µ0, τ0)

× p (v|η0, φ0) p (vǫ|ηǫ, φǫ) p (vξ|ηξ, φξ)

∝ η^K(φ₀ ⁰⁻¹⁾η^(φǫ ^ǫ−1)η^K(φ_ξ ^ξ−1)

A exp



−||y − G^ow||²₂ 2vǫ



× exp



−||w − XE^inc− XG^cw||²₂ 2vξ



× exp



−(χ − mχ)^TVχ−1(χ − mχ) 2



× exp



λX i

X j

δ(z(i) − z(j)) − T (λ)



v^−φ_ǫ ^ǫ−1

× exp



−ηǫ

vǫ



vǫ^−φξ−1exp



−ηξ

vξ

YK

k=1

v_k^−φ0−1

× exp



−η0

vk

 exp



−|mk− µ0|² 2τ0



∝ exp {L}, ✭✹✮

✇❤❡r❡ A = (2π)^M+N (NP +1)+K

2 (vǫ)^M2 (vξ)^N×NP2 |Vχ|^(−1/2)(τ0)^(K/2)Γ(φ0)^KΓ(φǫ)Γ(φξ)✇❤❡r❡ M ✐s t❤❡ ♥✉♠❜❡r ♦❢ s♦✉r❝❡s✱

N t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ ❡❧❡♠❡♥t❛r② sq✉❛r❡ ♣✐①❡❧s✱ NP t❤❡ ♥✉♠❜❡r ♦❢ ♣♦❧❛r✐③❛t✐♦♥ ✭NP = 1❉✱ 2❉ ♦r 3❉✮ ❛♥❞ L r❡❛❞s✿

L = −

M 2



log(vǫ) −

N NP

2

 log(vξ)

− 1

2vǫ

||y − G^ow||²₂− 1 2vξ

||w − XE^inc− XG^cw||²₂

− P

kNklog (vk)

2 −X

k

X r

|χ(r) − mk(r)|² 2vk

+ λX

r X

r^′

δ(z(r) − z(r^′)) − P

k|mk− µ0|² 2τ0

− X

k

 η0

vk

+ (φ0+ 1) log(vk)

−ηǫ

vǫ

− (φǫ+ 1) log (vǫ)

− ηξ

vξ

− (φξ+ 1) log (vξ). ✭✺✮

✷

❲❡ ♠❛② ♥♦t❡ t❤❛t ❛♣♣❧②✐♥❣ t❤❡ ❥♦✐♥t ♠❛①✐♠✉♠ ❛ ♣♦st❡r✐♦r✐ ✭❏▼❆P✮ ♦r t❤❡ ♣♦st❡r✐♦r ♠❡❛♥ ✭P▼✮ t♦ ❝♦♠♣✉t❡ t❤❡

❥♦✐♥t ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ✭❡q✉❛t✐♦♥ ✭✹✮✮ ②✐❡❧❞s ✐♥tr❛❝t❛❜❧❡ ❢♦r♠ ❛♥❞ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ♥❡❡❞❡❞ t♦ ♦❜t❛✐♥ ❛ ♣r❛❝t✐❝❛❧

s♦❧✉t✐♦♥✳ ❚❤✐s ✐s ❞♦♥❡ ❜② ♠❡❛♥s ♦❢ t❤❡ ●❱❇❆✳ ❋✐rst✱ ❛ str♦♥❣ s❡♣❛r❛t✐♦♥ ✐s ❝❤♦s❡♥✿

q(x) = q(vǫ)q(vξ) ×Y

i

q(χi)q(wi)q(zi)Y

k

q(mk)q(vk). ✭✻✮

❚❤❡♥✱ ✉s✐♥❣ ❡q✉❛t✐♦♥ ✭✷✮✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♥❣ ♠❛r❣✐♥❛❧ ❢♦r ❡❛❝❤ ✉♥❦♥♦✇♥ ✈❛r✐❛❜❧❡ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ♠❡❛♥s ♦❢

❢✉♥❝t✐♦♥❛❧ ♦♣t✐♠✐③❛t✐♦♥✳ ❯♣❞❛t✐♥❣ t❤❡ ❛♣♣r♦①✐♠❛t❡ ♣♦st❡r✐♦r r❡q✉✐r❡s 7 ❞✐✛❡r❡♥t ❣r❛❞✐❡♥t st❡♣s t❤❛t ✇❡ ❞❡♥♦t❡ ❜② αw✱ αχ✱ αz✱ αvǫ✱αvξ✱ αvk ❛♥❞ αmk✳

✷✳✶ ❯♣❞❛t❡ s❤❛♣✐♥❣ ♣❛r❛♠❡t❡rs

❆t t❤✐s ♣♦✐♥t✱ ✐t ❝❛♥ ❜❡ ♠❡♥t✐♦♥❡❞ t❤❛t t❛❦✐♥❣ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ✐♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧② ❛♥❞ ✉s✐♥❣ ❝♦♥❥✉❣❛t❡ ♣r✐♦rs ✇✐❧❧

r❡s✉❧t ✐♥ ❥♦✐♥t ♣♦st❡r✐♦rs ❛♥❞ ♠❛r❣✐♥❛❧s r❛♥❣✐♥❣ ✐♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②✳ ❖♣t✐♠✐③❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ qi(xi)t❤❡♥

r❡s✉❧ts ✐♥ ♦♣t✐♠✐③✐♥❣ t❤❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡s❡ ❧❛✇s✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❡ s✉♠♠❛r✐③❡ t❤❡ r❡s✉❧ts ❢♦r t❤❡ ●❛✉ss✐❛♥ ✲ ✐♥✈❡rs❡

❣❛♠♠❛ ❝❛s❡✿

q(w) = N (mfw, eVw), q(χ) = N (mfχ, eVχ), q(mk) = N (eµk, eτk), q(vk) = IG(eηk, eφk),

q(vǫ) = IG(eηǫ, eφǫ), q(vξ) = IG(eηξ, eφξ),

q(z) = eζ_k∝ exp



λX r∈D

X r^′∈V (r)

ζ(re ^′)



, ✭✼✮

✇❤❡r❡ t✐❧❞❡❞ ♣❛r❛♠❡t❡rs ❛r❡ ❣✐✈❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❚❤❡ ❦❡② ✉s❡❞ t♦ ♦❜t❛✐♥ t❤❡s❡ ❢♦r♠s ✐s ❡q✉❛t✐♦♥ ✭✷✮✳

✷✳✶✳✶ ❈♦♥tr❛st s♦✉r❝❡ w

log (eqⁿ(w(i))) ∝ (1 − αw) log (eq(w(i))) + αw hlog (p (χ, w, z, ψ, y))i_q(/w(˜ i⁾⁾

∝ (1 − αw)h e

v_w(i⁾w(i)²− 2fm_w(i⁾w(i)i

+ αw hlog (p(y|w, vǫ)) + log (p(w|χ, vξ))i_q(/w(˜ i))

∝ (1 − αw)h e

v_w(i)w(i)²− 2fm_w(i)w(i)i

− αw

2

||y − G^ow||²₂ vǫ

+ ||w − XE^inc− XG^cw||²₂ vξ

˜

q(/w(i⁾⁾ ✭✽✮

✇❤❡r❡ q(/w(i)) =Q

j6=iq(w(j))q(χ)q(z)q(ψ)✳ ❖r ||y − G^ow||²₂

˜

q(/w(i)) ∝ X l

|G^o(l, i)|²|w(i)|²

+ 2 ℜe



X k

G^o∗(k, i)



y(k) −X j⁶⁼i

G^o(k, j) emw(j)



 w^∗(i)



 ,

❛♥❞

✸

||w − XE^inc− XG^cw||²₂

˜

q(/w(i⁾⁾ ∝ |w(i)|²− G^c∗(i, i) ¯χ^∗(i) |w(i)|²− G^c(i, i) ¯χ(i) |w(i)|²

+ X

j

|G^c(j, i)| |χ(i)|²|w(i)|²− 2 ℜe E^inc(i) ¯χ(i)w^∗(i)

+ 2 ℜe



X j

|G^c∗(j, i)| |χ(j)|²|w^∗(j)|²





− 2 ℜe



X

j6=i

¯

χ(i) G^c(i, j) ¯w(j) w^∗(i)





− 2 ℜe



X

j6=i

¯

χ^∗(j) G^c∗(j, i) ¯w(j) w^∗(i)





+ 2 ℜe



X j

G^c∗(j, i)|χ(j)|² X k6=i

G^c(j, k) ¯w(k) w^∗(i)





+ cte.

✇❤❡r❡^∗ ❞❡♥♦t❡s t❤❡ ❝♦♥❥✉❣❛t❡ ❝♦♠♣❧❡①✳

❍❡♥❝❡✱ ❜② ❝♦♠❜✐♥✐♥❣ ❛❧❧ t❤❡ t❡r♠s✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♥❣ ❧❛✇✱ ❛t t❤❡ ✐t❡r❛t✐♦♥ n✱ ❜❡❝♦♠❡s qⁿ(w) = N (fmⁿ_w, eVⁿ_w)

✇❤❡r❡✿

Ve_wⁿ(i) =

"

(1 − αw) eVw(i) + αw vǫ⁻¹

X j

|G^o(j, i)|²

+ v_ξ⁻¹



1 − 2 ℜe G^c(i, i) emχ(i)

+X

j

|G^c(j, i)|² | emχ(i)|²+ eVχ(i)
!#⁻¹ ,

e mⁿ_w(i)

Ve_wⁿ(i) = (1 − αw) emw(i) + αw

"

vǫ⁻¹

X k

G^o∗(k, i)



y(k) −X

j6=i

G^o(k, j) emw(j)





+ v⁻¹_ξ E^inc(i) emχ(i) −X j

G^c∗(j, i)



| emχ(j)|²+ eVχ(j)

 E^inc(j)

+ X

j6=i e

mχ(i)G^c(i, j) emw(j) +X j6=i

e

m^∗_χ(i)G^c∗(j, i) emw(j)

− X

j

G^c∗(j, i)



| emχ(i)|²+ eVχ(i) X k6=j

G^c(j, k)w(k)

!#

,

❇② ❛❞❞✐♥❣ t❤❡ ♠✐ss✐♥❣ emw t♦ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❡q✉❛t✐♦♥ ❛♥❞ ♣✉tt✐♥❣ ✐t ✐♥ ❛ ✈❡❝t♦r✐❛❧ ❢♦r♠✱ ✇❡

♦❜t❛✐♥ t❤❡ ✜♥❛❧ ❢♦r♠✿





























Veⁿ_w=h

(1 − αw) eV⁻¹_w + αw ❉✐❛❣

v⁻¹ǫ Γ^o+ v⁻¹_ξ Γ^xci−1

f

mⁿ_w=fmw+ αwVeⁿ_w

""

v⁻¹ǫ G^oH(y − G^ofmw) + v_ξ⁻¹

f

mχE^inc− G^cH f m²_χ+ eVχ

 E^inc

− fmw+mfχG^cmfw+ G^cHfm^∗_χmfw− G^cH f m²_χ+ eVχ

 G^cfmw

##

✭✾✮

✇❤❡r❡ ♦✈❡r❧✐♥❡ ❞❡♥♦t❡s t❤❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ ✈❛r✐❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ q ✭✐✳❡✳ u = ❊(u)q✮✱ s✉♣❡rs❝r✐♣t^H ✐♥❞✐❝❛t❡s t❤❡

❝♦♥❥✉❣❛t❡ tr❛♥s♣♦s❡ ❛♥❞ Γ^o❛♥❞ Γ^xc ❛r❡ ❣✐✈❡♥ ❛s✿

✹

Γ^o(i) = X j

|G^o(j, i)|²,

Γ^xc(i) = 1 − 2ℜe(G^c(i, i) emχ(i)) + ( em²_χ(i) + eVχ(i))X

j

|G^c(j, i)|²

✷✳✶✳✷ ❈♦♥tr❛st χ

log (eqⁿ(χ(i))) ∝ (1 − α) log (eq(χ(i))) + αχ hlog (p (χ, w, z, ψ, y))i_q(/χ(˜ i))

∝ (1 − αχ)h e

v_χ(i)χ(i)²− 2fm_χ(i)χ(i)i

+ αχ hlog (p(w|χ, vǫ)) + log (p(χ|z, v, m))i_q(/χ(˜ i))

∝ (1 − αχ)h e

v_χ(i⁾χ(i)²− 2fm_χ(i⁾χ(i)i + αχ

||w − XE||²₂ vξ

+ (χ − mχ)^TV⁻¹_χ (χ − mχ)

˜ q(/χ(i))

, ✭✶✵✮

✇❤❡r❡ q(/χ(i)) =Q

j6=iq(χ(j))q(w)q(z)q(ψ)✱ ♦r ||w − XE||²₂

˜

q(/χ(i)) ∝ −2ℜe

w^∗(i)E(i)χ(i)

+ E²(i)|χ(i)|²

+

*

||w||²₂− 2ℜe



X

j⁶⁼i

w^∗(j)E(j)χ(j)



 +X j⁶⁼i

|E(j)χ(j)|² +

˜ q(/χ(i⁾⁾

,

❛♥❞

D

(χ − mχ)^TV⁻¹_χ (χ − mχ)E

˜

q(/χ(i⁾⁾ ∝ X j

||χ(j) − mk||²₂ vk

˜ q(/χ(i))

∝ v_k⁻¹ |χ(i)|²− 2ℜe (fmkχ^∗(i))
,

❇② ✐❞❡♥t✐✜❝❛t✐♦♥✱ ✇❡ ✜♥❞

Ve_χⁿ(i) ∝ h

(1 − αχ) eVχ(i)⁻¹+ αχ



v_ξ⁻¹E²(i) + ev^χ−1(i)i−1

,

❛♥❞

e mⁿ_χ(i)

Ve_χⁿ(i) ∝ (1 − αχ) emχ(i) + αχ

"

e mχ(i)

Veχ(i) + v_ξ⁻¹w(i)E^∗(i)

# ,

✇❤❡r❡ wE^∗ ✐s t❤❡ ♠❡❛♥ ♦❢ t❤❡ ✈❡❝t♦r wE^∗s✉❝❤ t❤❛t✿

wE^∗(i) = X

NfNvN

E^inc∗(i) emw(i) + emw(i)X

jb

G^c∗(i, i^′) em^∗_w(i^′)

+ G^c∗(i, i) eVw(i),

❛♥❞ E²✐s ❛ ❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡ ✇❤♦s❡ ❡❧❡♠❡♥ts ❛r❡ ✇r✐tt❡♥ s✉❝❤ t❤❛t✿

E²(i) = X

N_fN_vN_P

|E^inc(i)|²+ 2ℜe E^inc∗(i)G^cmew(i)

+ |X i^′

G^c(i, i^′) emw(i)|²+X i^′

|G^c(i, i^′)|²Vew(i),

✺

❚❤❡♥ t❤❡ ✈❡❝t♦r✐❛❧ ❢♦r♠ ✐s ❣✐✈❡♥ ❜②✿





Veⁿ_χ=h

(1 − αχ) eV⁻¹_χ + αχ

❉✐❛❣

v_ξ⁻¹E²+ V⁻¹_χ i−1

f

mⁿ_χ = αχ Veⁿ_χhP

kv⁻¹_k eζ_k◦ fmk+ v⁻¹_ξ w◦ E^∗i ✭✶✶✮

✇❤❡r❡ Vχ⁻¹(i, i) =P

keζ_k(i)v⁻¹_k .

✷✳✶✳✸ ❚❤❡ ❤✐❞❞❡♥ ✜❡❧❞ z

log (eqⁿ(z(i))) ∝ (1 − αz) log (eq(z(i))) + α hlog (p (χ, w, z, ψ, y))i_q(/z(˜ i))

∝ (1 − αz) log (eq(z(i))) + αz

*

−1

2log(vk) +||χ(i) − mχ(i)||²₂ 2vk

− λ X

l∈V(i)

δ (z(l) − k) +

˜ q(/z(i))

∝ (1 − α) log ζek(i)

−αz

2

"

log(vk) + v⁻¹_k h|χ(i) − mχ(i)|i_q(/z(˜ i))

− 2λ X

j∈V(i)

ζek(j)

#

∝ (1 − αz) log ζek(i)

−αz

2

"

Ψ(˜ηk) − log ˜φk+ v_k⁻¹h

| emχ(i)|²

+ me^†_k− 2ℜe em^†_χme^∗_χ(i)
i

+ λ X

j∈V(i)

ζek(j)

# ,

✇❤❡r❡ q(/z(i)) =Q

j6=iq(z(j))q(χ)q(w)q(ψ)❛♥❞ em^†2_χ = em²_k+ eτk✳

❇② ✐❞❡♥t✐✜❝❛t✐♦♥✱ ✐t ❜❡❝♦♠❡s✿

ζe_kⁿ = ζe_k^(1−αz)exp (

−αz

2 Ψ(eηk) + log( eφk) + v_k⁻¹

( emχ(i) − eµk)²+ eτk

+ Veχ(r)

− λ X

i^′∈V(r⁾ eζ_k(i^′)

!)

, ✭✶✷✮

✇❤❡r❡ Ψ(x) =_∂x^∂ log Γ(x)t❤❡ ❞✐❣❛♠♠❛ ❢✉♥❝t✐♦♥ ✇✐t❤ Γ(x) ✐s t❤❡ ❣❛♠♠❛ ❢✉♥❝t✐♦♥✳

✷✳✶✳✹ ❚❤❡ ♦❜s❡r✈❛t✐♦♥ ✈❛r✐❛♥❝❡ vǫ

log (eqⁿ(vǫ)) ∝ (1 − αvǫ) log (eq(vǫ)) + αvǫ hlog (p (χ, w, z, ψ, y))i_{q(/v˜ ǫ)}

∝ (1 − αvǫ)



−vǫ

˜

ηǫ + ( ˜φǫlog(vǫ))



+ αvǫ hp(y|w, vǫ)p(vǫ|ηǫ, φǫ)i_{q(/v˜ ǫ)}, ✭✶✸✮

✇❤❡r❡ q(/vǫ)) = q(χ)q(w))q(z)Q

l6=vǫq(ψl)✱ ✇✐t❤✿

hp(y|w, vǫ)p(vǫ|ηǫ, φǫ)i_{q(/v˜ ǫ)} ∝

−M

2 log(vǫ) −||y − G^ow||²₂ 2vǫ

−ηǫ

vǫ

− (φǫ− 1) log(vǫ)

˜ q(/vǫ)

∝ −(1 + φǫ+M

2 ) log(vǫ) − v⁻¹_ǫ

ηǫ+||y − G^ow||²₂ 2

˜ q(/vǫ)

,

✇❤❡r❡

✻

||y − G^ow||²₂

˜

q(/v_ǫ) ∝ X i

|y(i)|²+X j

X k

G^o(i, k) emw(j) em^∗_w(k) +X j

|G^o(i, j)|evw(j)

− 2ℜe y^∗X

j

G^o(i, j) em^∗_w(j)! ,

❇② ✐❞❡♥t✐✜❝❛t✐♦♥✱ ✇❡ ♦❜t❛✐♥✿







φeⁿ_ǫ = (1 − αvǫ) eφǫ+ αvǫ φǫ+^M₂
e

ηⁿ_ǫ =^1−α_eη^vǫ

ǫ + αvǫ ηǫ+¹₂

||y||²₂+ ||G^ofmw||²₂− 2ℜe(y^HG^ofmw) + ||G^o2evw||1

!

✭✶✹✮

✷✳✶✳✺ ❈♦✉♣❧✐♥❣ ♥♦✐s❡ vξ

log (eqⁿ(vξ)) ∝ 1 − αvξ

log (eq(vξ)) + αξ hlog (p (χ, w, z, ψ, y))i_{q(/v˜ ξ)}

∝ 1 − αv_ξ


−vξ

˜ ηξ

+ ( ˜φξ− log(vξ))



+ αvξ hp(w|χ, vξ)p(vξ|ηξ, φξ)i_{q(/v˜ ξ)}, ✭✶✺✮

✇❤❡r❡ q(/vξ)) = q(χ)q(w))q(z)Q

l6=vξq(ψl)✇✐t❤

hp(w|χ, vξ)p(vξ|ηξ, φξ)i_q(/v˜

ξ) ∝

log v_ξ^{−NP N2} −||w − XE||²₂ 2vξ

−ηξ

vξ

− (φξ+ 1) log(vξ)

˜ q(/vξ)

∝



ηξ+NPN 2



log (vξ) − v⁻¹_ξ

ηξ+||w − XE||²₂ 2

˜ q(/vξ)

,

❛♥❞

||w − XE||²₂

˜

q(/vξ) = X i

| emw(i)|²+ ev^w(i) +

| emχ(i)|²+ ev^w(i) E²(i)

− 2ℜe e

mχ(i)w^∗(i)E(i)! ,

❇② ✐❞❡♥t✐✜❝❛t✐♦♥ ✇❡ ✜♥❞✿

















φeⁿ_ξ = 1 − αv_ξ
eφξ+ αv_ξ φξ+^NP₂^N

αv_ξ dχ+ 1 − αv_ξ
f mχ
e

η_ξⁿ = ^1−α_eη^vξ

ξ + αvξ ηξ+¹₂

||fmw||²₂+ || eVw||²₂+ || f m²_χ+ eVχ

 E²||1

− 2ℜe f

m^H_χw◦ E^∗ ! ✭✶✻✮

✇❤❡r❡ dχ=P

kv_k⁻¹eζ_k◦ fm^†_χ+ v⁻¹_ξ w◦ E^∗✳

✷✳✶✳✻ ❱❛r✐❛♥❝❡ ♦❢ t❤❡ ❝❧❛ss❡s vk

∀κ ∈ {1, ..., K}✇❡ ❤❛✈❡✿

✼

log (eqⁿ(vk)) ∝ (1 − α) log (eq(vk)) + αv_k hlog (p (χ, w, z, ψ, y))i_{q(/v˜ k)}

∝ (1 − αvk)



−vk

˜ ηk

+ ( ˜φk− 1) log(vk)

 + αvk

*

−1 2

X i

δ(z(i) = κ)

× 

log(vk) +|χ(i) − mk|² vk

 +η0

vk

+ (φ0+ 1) log(vk) +

˜ q(/vk)

∝ (1 − αvk)



−vk

˜ ηk

+ ( ˜φk− log(vk))

 + αvk

"

− φ0+ 1 + P

i eζk(i) 2

!

− v_k⁻¹

*

η0+|χ(i) − mk|² 2

+

˜ q(/vk)

#

, ✭✶✼✮

q(/vk)) = q(χ)q(w))q(z)Q

l6=v_kq(ψl)✳

❇② ✐❞❡♥t✐✜❝❛t✐♦♥ ✇❡ ✜♥❞✿











φeⁿ_k = (1 − αvk) eφk+ αvk

 φ0+

P

i^ζek(i⁾

2



e

η_kⁿ= ^1−α_ηek^vk + αv_k η0+¹₂P i eζk(i)

| emχ(i)|²+ eVχ(i) + m²_k+ ˜τ_k²! ✭✶✽✮

✷✳✶✳✼ ▼❡❛♥s ♦❢ ❝❧❛ss❡s mk

log (eqⁿ(mk)) ∝ (1 − αmk) log (eq(mk)) + αmk hlog (p (χ, w, z, ψ, y))iq(/m˜ k)

∝ (1 − αmk) e

τkm²_k− 2eµkmk

+ αmk −

*1 2

X i

δ (z(i) = κ)

×

|χ(i) − mχ(i)|² vk



+|mk− µ0| τ0

+

˜ q(/mk)

!

, ✭✶✾✮

✇✐t❤ q(/mk)) = q(χ)q(w))q(z)Q

l6=mkq(ψl)

❇② ✐❞❡♥t✐✜❝❛t✐♦♥✱ ✇❡ ♦❜t❛✐♥✿







 e τ_kⁿ=

"

(1 − αmk) eτk+ αmk

τ₀⁻¹+ v⁻¹_k P

i eζk(i)#−1

e

µⁿ_k = eτ_kⁿh

(1 − αmk) eµk+ αmk

τek

µ0

τ0 + v_k⁻¹P

i eζk(i) emχ(i)i ✭✷✵✮

✸ ❚❤❡ ❢r❡❡ ♥❡❣❛t✐✈❡ ❡♥❡r❣②

❚❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❢r❡❡ ♥❡❣❛t✐✈❡ ❡♥❡r❣② ❞✉r✐♥❣ t❤❡ ✐t❡r❛t✐♦♥ ♣r♦❝❡ss ❛❧❧♦✇s t♦ ❤❛✈❡ ❛♥ ✐♥❞✐❝❛t♦r ♦♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡

♦❢ t❤❡ ❛❧❣♦r✐t❤♠✳ ■♥❞❡❡❞✱ ✐ts ✈❛❧✉❡ ❛t t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❛❧❧♦✇s ❢♦r ❛♥ ❡st✐♠❛t❡ ♦❢ t❤❡ ❡✈✐❞❡♥❝❡ t❤❛t t❤❡ ♠♦❞❡❧ ✐s ✉s❡❢✉❧

❢♦r t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ❧❛tt❡r✳ ■♥ t❤❡ ♥♦♥❧✐♥❡❛r ❝❛s❡✱ t❤❡ ❢r❡❡ ♥❡❣❛t✐✈❡ ❡♥❡r❣② ✐s ❣✐✈❡♥ ❜②✿

F(q) = hp(y, χ, w, z, ψ|M)iq + H(q(χ, w, z, ψ)) ✭✷✶✮

✇❤❡r❡ H ✐s t❤❡ ❡♥tr♦♣② ♦❢ q ❛♥❞ ✐s ❣✐✈❡♥ ❜②✿

✽

H(q(χ, w, z, ψ)) = X

Nv

X

Nf

X

NP

X i

X

k

logp

2πeev^w(i)

+X

i X

k

logq

2πeev^χ(i)



+ X

i X

k

ζ˜κ(i) λ X j^∈V(i⁾

ζ˜k(j) − log X

k

exp



λ X j^∈V(i⁾

ζ˜k(j)





!

+ X

κ

logp 2πeeτκ



+X

k

e

ηk+ log

φekΓ(eηk)

− (1 + eηk)Ψ(eηk)

!

+ ηeǫ+ log

φeǫΓ(eηǫ)

− (1 + eηǫ)Ψ(eηǫ) + eηξ+ log

φeξΓ(eηξ)

− (1 + eηξ)Ψ(eηξ), ✭✷✷✮

❍❡♥❝❡✿

hp(y, χ, w, z, ψ)iq = −M + N (1 + NP) + K

2 log(2π) + (φǫ− 1) log (ηǫ) − Γ(φǫ)

− M + 2φǫ+ 2 2



Ψ(eηǫ) − log( eφǫ)

− ηǫv⁻¹ǫ + (φξ− 1) log (ηξ) − Γ(φξ)

− N NvNPNf+ 2φξ+ 2 2

Ψ(eηξ) − log( eφξ)

− ηξv_ξ⁻¹+ log(η^K(φ₀ ⁰⁻¹⁾)

− X

k



2φ0+ 1 +P

i eζk(i)

Ψ(eηk) − log(fφk)

− 2η0v⁻¹_k

2 − K Γ(φ0)

− X

k

e

m²_k+ eτk+ µ²₀− 2 emkµ0

2τ0

+ λX i

X

k

ζek(i) X j^∈V(i⁾

ζek(j) −vǫ⁻¹

2

× ||y||²₂+ ||G^ofmw||²₂− 2y^tG^omfw+ || eVwΓ^o||1

!

− v_ξ⁻¹

2 ||fmw||²₂+ || eVw||1+    

f mχ+ eVχ

 E²

1− 2fm^H_χw◦ E^∗

!

− X

i X

k

v_k⁻¹ζek(i)

| emχ(i)|²₂+ evχ(i) + em^†2_χ(i) − 2 emχ(i) em^†_χ(i)

2 , ✭✷✸✮

❲❡ ♠❛② ♥♦t❡ t❤❛t t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❢r❡❡ ♥❡❣❛t✐✈❡ ❡♥❡r❣② ❞❡♣❡♥❞s ♠❛✐♥❧② ✐♥ t❡r♠s ✉s❡❞ t♦ ✉♣❞❛t❡ t❤❡ ✈❛❧✉❡s

♦❢ t❤❡ ♣❛r❛♠❡t❡rs ❛♥❞ ✐ts ❡✈❛❧✉❛t✐♦♥ ❞♦❡s ♥♦t r❡q✉✐r❡ ❛ ❝♦st ♦❢ ❛❞❞✐t✐♦♥❛❧ ❝♦♠♣✉t❛t✐♦♥✳

✹ ❖♣t✐♠❛❧ st❡♣ ✈❛❧✉❡s ❝♦♠♣✉t❛t✐♦♥

❇② ❡①❛♠✐♥✐♥❣ ❡q✉❛t✐♦♥s ✭✾✮✱ ✭✶✶✮✱ ✭✶✷✮✱ ✭✶✹✮✱ ✭✶✻✮✱ ✭✶✽✮✱ ✭✷✵✮✱ ✇❡ r❡♠❛r❦ t❤❛t t❤❡r❡ ✐s ♥♦ ❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ ❡❧❡♠❡♥ts

♦❢ t❤❡ s❛♠❡ ❣r♦✉♣ ❢♦r χ✱ z✱ vǫ✱ vξ✱ vk ❛♥❞ mk✱ ✇❤❡r❡❛s t❤❡ ✉♣❞❛t❡ ♦❢ t❤❡ ♠❡❛♥ ✈❛❧✉❡ ♦❢ t❤❡ ❝♦♥tr❛st s♦✉r❝❡ fmⁿ_w❛t t❤❡

✐t❡r❛t✐♦♥ n ❞❡♣❡♥❞s ♦♥ t❤❡ ♠❡❛♥ ✈❛❧✉❡ fmw✳ ❚❤❡♥ t❤❡ ❣r❛❞✐❡♥t st❡♣s αρ✱ ρ = χ, z, vǫ, vξ, mk, vk ❝❛♥ ❜❡ s❡t t♦ ✶ ✐♥

♦r❞❡r t♦ ❛❝❝❡❧❡r❛t❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ ♦♥❧② t❤❡ ❝♦♥tr❛st s♦✉r❝❡ ✉♣❞❛t✐♥❣ st❡♣ αw✐s ❝♦♠♣✉t❡❞✳ ❚❤✐s st❡♣ ✐s ❝♦♠♣✉t❡❞

✐♥ ❛♥ ♦♣t✐♠❛❧ ✇❛② ✐♥ ♦r❞❡r t♦ ❡♥s✉r❡ ❛ ❢❛st ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❤❛♣✐♥❣ ♣❛r❛♠❡t❡r t♦✇❛r❞s t❤❡✐r ✜♥❛❧ ✈❛❧✉❡s✳ ❚❤✐s ✐s

❞♦♥❡ ❢r♦♠ t❤❡ ♥❡❣❛t✐✈❡ ❢r❡❡✲❡♥❡r❣② ❢✉♥❝t✐♦♥ ❜② ♠❡❛♥s ♦❢ ❛ ◆❡✇t♦♥✬s ♠❡t❤♦❞✳ ❚❤❡ ♦♣t✐♠❛❧ st❡♣ t❤❡♥ r❡❛❞s✿

α^opt_w = ∆F(αw)/∆²F(αw)   αw=0

, ✭✷✹✮

✇✐t❤ ∆F = ∂F/∂αw ❡t ∆²F = ∂²F/∂²αw

❇❡❢♦r❡ ❛♣♣❧②✐♥❣ t❤❡ ◆❡✇t♦♥ ♠❡t❤♦❞✱ ✇❡ r❡✇r✐t❡ t❤❡ ✉♣❞❛t❡ ❡q✉❛t✐♦♥s ❢♦r w ✇✐t❤ ♠♦r❡ ❝♦♥✈❡♥✐❡♥t ♥♦t❛t✐♦♥s✿

✾







 Veⁿ_w=

"

(1 − αw) eV⁻¹_w + α Rw

#−1

f

mⁿ_w=fmw+ αw Veⁿ_wdw

✭✷✺✮

✇❤❡r❡ Rw=❉✐❛❣

vǫ⁻¹Γ^o+ v⁻¹_ξ Γ^xc

❛♥❞ dw=

"

v⁻¹ǫ G^oH(y − G^ofmw) + v_ξ⁻¹ f

mχE^inc− G^cH f m²_χ+ eVχ

 E^inc

− fmw+fmχG^cfmw+ G^cHmf^∗_χfmw− G^cH f m²_χ+ eVχ

 G^cfmw

#

✳

❚❤❡♥✱ ✇❡ r❡✇r✐t❡ t❤❡ ♥❡❣❛t✐✈❡ ❢r❡❡ ❡♥❡r❣② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ αw ✐♥ ♦r❞❡r t♦ ✉s❡ ✐t t♦ ♦❜t❛✐♥ t❤❡ ♦♣t✐♠❛❧ ✈❛❧✉❡✳ ■t r❡❛❞s✿

F(αw) = −vǫ⁻¹

2 ||y||²₂+ ||G^omfw||²₂− 2y^tG^ofmw+ || eVwΓ^o||1

!

−v_ξ⁻¹

2 ||fmw||²₂+ || eVw||1

+    

f mχ+ eVχ

 E²

1− 2fm^H_χw◦ E^∗

!

+X

i

logp

2πeev^w(i) ,

✭✷✻✮

❚❤❡♥✱ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ✜rst ❛♥❞ t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s t❤❛t ②✐❡❧❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿

∂F(αw)

∂αw

= mf^′_w d^t_w+X

i

ev^′w(i)

2 ev^w(i)− ev^′w(i)rw(i), ✭✷✼✮

∂²F(αw)

∂²αw

= fm^′′_w d^t_w+fm^′_w d^t_w^′ +X

i

evw^′′(i)evw(i) − (evw^′ (i))²

2 (evw(i))² − evw^′′(i)rw(i). ✭✷✽✮

✇❤❡r❡ vw(i)❛♥❞ rw(i)st❛♥❞ r❡s♣❡❝t✐✈❡❧② ❢♦r ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts i ♦❢ ♠❛tr✐① Vw ❛♥❞ Rw✱ ❛♥❞

















Ve^′n_w =

Veⁿ_w2

Ve⁻¹_w − Rw



Ve^′′n_w = Veⁿ_w3

Ve⁻¹_w − Rw

2

f

m^′n_w = eVⁿ_wdw+ α eV^′n_wdw

f

m^′′n_w = 2 eV^′n_wdw+ α eV^′′n_w dw

✭✷✾✮

t❤❡♥ t❤❡ ♦♣t✐♠❛❧ ✈❛❧✉❡ ✐♥ t❤✐s ❝❛s❡ ❜❡❝♦♠❡s

α^opt_w = d^t_wmf^′_w0+¹₂P

is²_wi f

m^′′_w0 d^t_w+fm^′_w0 d^t_w^′ +¹₂P

is³_wi ✭✸✵✮

✇❤❡r❡ 





















∂ fmw

∂αw

  αw=0

=fm^′_w0= eVw0dw

∂²mfw

∂²αw

  αw=0

=fm^′′_w0= 2 eV^′_w0dw

∂ fVw

∂αw

  αw=0

= eV^′_w0= − eVw0Sw

✭✸✶✮

❛♥❞

d^t_w^′ = mf^′_w^t v⁻¹ǫ G^oG^H− v_ξ⁻¹

1 −mfχG^c− G^cHfm^∗_χ− G^cH(fmχ+ eVχ)G^c!t

,

swi = rwievw0i− 1. ✭✸✷✮

✶✵

❘❡❢❡r❡♥❝❡s

❬✶❪ ❱✳ ❙♠í❞❧ ❛♥❞ ❆✳ ◗✉✐♥♥✱ ❚❤❡ ❱❛r✐❛t✐♦♥❛❧ ❇❛②❡s ▼❡t❤♦❞ ✐♥ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣✱ ❙♣r✐♥❣❡r ❱❡r❧❛❣✱ ❇❡r❧✐♥✱ ✷✵✵✻✳

❬✷❪ ❍✳ ❆②❛ss♦✱ ❇✳ ❉✉❝❤ê♥❡✱ ❛♥❞ ❆✳ ▼♦❤❛♠♠❛❞✲❉❥❛❢❛r✐✱ ✏❖♣t✐❝❛❧ ❞✐✛r❛❝t✐♦♥ t♦♠♦❣r❛♣❤② ✇✐t❤✐♥ ❛ ✈❛r✐❛t✐♦♥❛❧ ❇❛②❡s✐❛♥

❢r❛♠❡✇♦r❦✱✑ ■♥✈❡rs❡ Pr♦❜❧❡♠s ✐♥ ❙❝✐❡♥❝❡ ❛♥❞ ❊♥❣✐♥❡❡r✐♥❣✱ ✈♦❧✳ ✷✵✱ ♥♦✳ ✶✱ ♣♣✳ ✺✾✕✼✸✱ ✷✵✶✷✳

❬✸❪ ▲✳ ●❤❛rs❛❧❧✐✱ ❍✳ ❆②❛ss♦✱ ❇✳ ❉✉❝❤ê♥❡✱ ❛♥❞ ❆✳ ▼♦❤❛♠♠❛❞✲❉❥❛❢❛r✐✱ ✏▼✐❝r♦✇❛✈❡ t♦♠♦❣r❛♣❤② ❢♦r ❜r❡❛st ❝❛♥❝❡r

❞❡t❡❝t✐♦♥ ✇✐t❤✐♥ ❛ ✈❛r✐❛t✐♦♥❛❧ ❇❛②❡s✐❛♥ ❛♣♣r♦❛❝❤✱✑ ✐♥ ■❊❊❊ ❊✉r♦♣❡❛♥ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣ ❈♦♥❢❡r❡♥❝❡ ✭❊❯❙■P❈❖✮✱

▼❛rr❛❦❡❝❤✱ ▼❛r♦❝❝♦✱ ✷✵✶✸✳

❬✹❪ ▲✳ ●❤❛rs❛❧❧✐✱ ❍✳ ❆②❛ss♦✱ ❇✳ ❉✉❝❤ê♥❡✱ ❛♥❞ ❆✳ ▼♦❤❛♠♠❛❞✲❉❥❛❢❛r✐✱ ✏❆ ❣r❛❞✐❡♥t✲❧✐❦❡ ✈❛r✐❛t✐♦♥❛❧ ❇❛②❡s✐❛♥ ❛♣♣r♦❛❝❤✿

❛♣♣❧✐❝❛t✐♦♥ t♦ ♠✐❝r♦✇❛✈❡ ✐♠❛❣✐♥❣ ❢♦r ❜r❡❛st t✉♠♦r ❞❡t❡❝t✐♦♥✱✑ s✉❜♠✐tt❡❞ ✐♥ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥

■♠❛❣❡ Pr♦❝❡ss✐♥❣ ✭■❈■P✮✱ ❋r❛♥❝❡✱ P❛r✐s✱ ✷✵✶✹✳

❬✺❪ ❆✳ ❋r❛②ss❡ ❛♥❞ ❚✳ ❘♦❞❡t✱ ✏❆ ♠❡❛s✉r❡✲t❤❡♦r❡t✐❝ ✈❛r✐❛t✐♦♥❛❧ ❜❛②❡s✐❛♥ ❛❧❣♦r✐t❤♠ ❢♦r ❧❛r❣❡ ❞✐♠❡♥s✐♦♥❛❧ ♣r♦❜❧❡♠s✱✑

❚❡❝❤✳ ❘❡♣✳ ❤❛❧✲✵✵✼✵✷✷✺✾✱ ❤tt♣✿✴✴❤❛❧✳❛r❝❤✐✈❡s✲♦✉✈❡rt❡s✳❢r✴❞♦❝s✴✵✵✴✼✵✴✷✷✴✺✾✴P❉❋✴✈❛r❴❜❛②❱✽✳♣❞❢✱ ✷✵✶✷✳

❬✻❪ ❲✳ ❈✳ ●✐❜s♦♥✱ ❚❤❡ ▼❡t❤♦❞ ♦❢ ▼♦♠❡♥ts ✐♥ ❊❧❡❝tr♦♠❛❣♥❡t✐❝s✱ ❈❤❛♣♠❛♥ ✫ ❍❛❧❧✴❈❘❈✱ ❇♦❝❛ ❘❛t♦♥✱ ✷✵✵✼✳

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