t ss r srt - - PDF document

t s s r s r t s r
SMART_READER_LITE
LIVE PREVIEW

t ss r srt - - PDF document

rs rt s ss t ss r srt


slide-1
SLIDE 1

▼ ❖ ❉ ❆ ✽

❲♦r❦s❤♦♣ ♦♥ ▼♦❞❡❧✲❖r✐❡♥t❡❞ ❉❡s✐❣♥ ❛♥❞ ❆♥❛❧②s✐s

❖♣t✐♠❛❧ ❞❡s✐❣♥s ❢♦r ❞✐s❝r✐♠✐♥❛t✐♥❣ ❛♠♦♥❣ s❡✈❡r❛❧ ♥♦♥✲◆♦r♠❛❧ ♠♦❞❡❧s

❈❤✐❛r❛ ❚♦♠♠❛s✐

❯♥✐✈❡rs✐t② ♦❢ ▼✐❧❛♥♦✱ ■t❛❧② ❏✉♥❡ ✹✕✽✱ ✷✵✵✼

slide-2
SLIDE 2

❖❯❚▲■◆❊

  • ◆♦t❛t✐♦♥ ❛♥❞ ❇❛❝❦❣r♦✉♥❞✳
  • ❑▲✕♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥✿ ❞✐s❝r✐♠✐♥❛t✐♦♥ ❜❡✲

t✇❡❡♥ t✇♦ r✐✈❛❧ ♠♦❞❡❧s✳

  • ●❡♥❡r❛❧✐③❡❞ ❑▲✕♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥✿ ❞✐s✲

❝r✐♠✐♥❛t✐♦♥ ❛♠♦♥❣ ♠♦r❡ t❤❛♥ t✇♦ ♠♦❞✲ ❡❧s✳

  • ❖t❤❡r ♣♦ss✐❜❧❡ ❝r✐t❡r✐❛✳
  • ❆♥ ❡①❛♠♣❧❡✿ ❞✐❝r✐♠✐♥❛t✐♦♥ ❛♠♦♥❣ t❤r❡❡

❧♦❣✐st✐❝ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳

  • ❙♦♠❡ r❡❢❡r❡♥❝❡s✳
slide-3
SLIDE 3

◆❖❚❆❚■❖◆

  • x ∈ X✿ ❡①♣❡r✐♠❡♥t❛❧ ❝♦♥❞✐t✐♦♥ ❝❤♦s❡♥

❜② t❤❡ ❡①♣❡r✐♠❡♥t❡r✳

  • ❆♣♣r♦①✐♠❛t❡ ❞❡s✐❣♥ ξ✿ ❛ ❞✐s❝r❡t❡ ♣r♦❜✲

❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ X ✇❤✐❝❤ ✐s s✉♣♣♦rt❡❞ ♦♥ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts✿ ξ =

  • x1

· · · xk ω1 · · · ωk

  • , 0 ≤ ωi ≤ 1,

k

  • i=1

ωi = 1.

  • y = y(x)✿

♦❜s❡r✈❛❜❧❡ ❡①♣❡r✐♠❡♥t❛❧ r❡✲ s♣♦♥s❡✳

  • fi(y, x, θi)✿ i−t❤ st❛t✐st✐❝❛❧ ♠♦❞❡❧✱ ✇❤❡r❡

θi ∈ Ωi ⊂ I Rmi ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✉♥✲ ❦♥♦✇♥ ♣❛r❛♠❡t❡r ✈❡❝t♦r✳

slide-4
SLIDE 4
  • ❖❆▲

❆✐♠ ♦❢ t❤❡ ❡①♣❡r✐♠❡♥t✿ t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦❢ k r✐✈❛❧ st❛t✐st✐❝❛❧ ♠♦❞❡❧s✱ fi(y, x, θi), i = 1, . . . , k, ✐s t❤❡ ♠♦r❡ ❛❞❡q✉❛t❡✳

slide-5
SLIDE 5

❇❆❈❑●❘❖❯◆❉

  • ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ♠♦❞❡❧s

❆t❦✐♥s♦♥ ✭✶✾✼✷✮✿ t❤❡ ♠♦❞❡❧ ✐s ❡♠❜❡❞❡❞ ✐♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ♦♥❡ ❛♥❞ t❤❡ ❞❡s✐❣♥ ❧♦♦❦s ❢♦r ❡st✐♠❛t✐♥❣ t❤❡ ❛❞❞✐t✐♦♥❛❧ ♣❛✲ r❛♠❡t❡rs ✐♥ t❤❡ ❜❡st ✇❛②✳ ❆t❦✐♥s♦♥ ❛♥❞ ❈♦① ✭✶✾✼✹✮✿ t❤❡ ♣r❡✈✐✉s ❛♣✲ ♣r♦❛❝❤ ✐s ❣❡♥❡r❛❧✐③❡❞ ❢♦r ❝♦♠♣❛r✐♥❣ s❡✈✲ ❡r❛❧ ❧✐♥❡❛r ♠♦❞❡❧s✳

  • ❘❡❣r❡ss✐♦♥ ♠♦❞❡❧s ✭❧✐♥❡❛r ♦r ♥♦t✮

❆t❦✐♥s♦♥ ❛♥❞ ❋❡❞♦r♦✈ ✭✶✾✼✺❛✱✶✾✼✺❜✮✿ ❚✲ ♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥ ❢♦r ❝♦♠♣❛r✐♥❣ t✇♦ ♦r ♠♦r❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳ ❯❝✐➠s❦✐ ❛♥❞ ❇♦❣❛❝❦❛ ✭✷✵✵✹✮✿ ●❡♥❡r❛❧✐③❛✲ t✐♦♥ ♦❢ t❤❡ ❚✲❝r✐t❡r✐♦♥ t♦ t❤❡ ❝❛s❡ ♦❢ ❤❡t✲ ❡r♦s❝❡❞❛st✐❝ r❛♥❞♦♠ ❡rr♦rs✳

  • ●❡♥❡r❛❧✐③❡❞ ❧✐♥❡❛r ♠♦❞❡❧s ✭❣❧♠✮

P♦♥❝❡ ❞❡ ▲❡♦♥ ❛♥❞ ❆t❦✐♥s♦♥ ✭✶✾✾✷✮✿ ❚❤❡ ❚✲❝r✐t❡r✐♦♥ ✐s ❡①t❡♥❞❡❞ t♦ t❤❡ ❝❛s❡ ♦❢ ❣❧♠✳

slide-6
SLIDE 6

❚❍❊ ❑▲✲❖P❚■▼❆▲■❚❨ ❈❘■❚❊❘■❖◆ ❑✉❧❧❜❛❝❦✕▲❡✐❜❧❡r ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ f1(y, x, θ1) ✭t❤❡ ❛ss✉♠❡❞ ✏tr✉❡✑ ♠♦❞❡❧✮ ❛♥❞ f2(y, x, θ2)✿ I [f1(y, x, θ1), f2(y, x, θ2)] =

  • f1(y, x, θ1) log
  • f1(y, x, θ1)

f2(y, x, θ2)

  • dy.

❑▲✕♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥ ✭▲ó♣❡③✕❋✐❞❛❧❣♦✱ ❚♦♠♠❛s✐ ❛♥❞

❚r❛♥❞❛✜r✱ ✷✵✵✼✮✿

I2,1(ξ) = min

θ2∈Ω2

  • X I [f1(y, x, θ1), f2(y, x, θ2)] ξ(dx)
slide-7
SLIDE 7

P❘❖P❊❘❚■❊❙ ✶✳ ❋♦r r❡❣r❡ss✐♦♥ ♠♦❞❡❧s ✇✐t❤ ◆♦r♠❛❧ ❡rr♦r ❞✐str✐❜✉t✐♦♥✱ t❤❡ ❑▲✲ ♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥ ❝♦✐♥❝✐❞❡s ✇✐t❤

  • t❤❡ ❚✲♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥✱ ✐♥ t❤❡ ❤♦♠♦s❝❤❡❞❛st✐❝ ❝❛s❡❀
  • t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❚✲♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥ ♣r♦✈✐❞❡❞

❜② ❯❝✐➠s❦✐ ❛♥❞ ❇♦❣❛❝❦❛ ✭✷✵✵✹✮✱ ✐♥ t❤❡ ❤❡t❡r♦s❝❤❡❞❛st✐❝ ❝❛s❡✳ ✷✳ ❋♦r ❣❡♥❡r❛❧✐③❡❞ ❧✐♥❡❛r ♠♦❞❡❧s✱ t❤❡ ❑▲✲❝r✐t❡r✐♦♥ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❚✲♦♣t✐♠❛❧✐t② ❝r✐t❡r✐♦♥ ♣r♦✲ ✈✐❞❡❞ ❜② P♦♥❝❡ ❞❡ ▲❡♦♥ ❛♥❞ ❆t❦✐♥s♦♥ ✭✶✾✾✷✮✳

slide-8
SLIDE 8

❚❍❊ ●❊◆❊❘❆▲■❩❊❉ ❑▲✲❖P❚■▼❆▲■❚❨ ❈❘■❚❊❘■❖◆

  • fk+1(y, x, θk+1) ✿ ❡①t❡♥❞❡❞ ♠♦❞❡❧ ✇❤✐❝❤

✐♥❝❧✉❞❡s t❤❡ k r✐✈❛❧ ♠♦❞❡❧s ❛s s♣❡❝✐❛❧ ❝❛s❡s✳

  • Ii,k+1(ξ) ✿ ❞❡t❡❝ts ❞❡♣❛rt✉r❡s ❢r♦♠ fi(y, x, θi)

✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ♦t❤❡r ♠♦❞❡❧s✳

  • ❊✣❝✐❡♥❝② ♦❢ ξ ❢♦r ❞❡t❡❝t✐♥❣ ❞❡♣❛rt✉r❡s

❢r♦♠ fi(y, x, θi)✿ Effi,k+1(ξ)= Ii,k+1(ξ) Ii,k+1(ξ∗

i )

ξ∗

i =argmax ξ

Ii,k+1(ξ)

  • ●❡♥❡r❛❧✐③❡❞ ❑▲✲❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✿

Iα(ξ)=k

i=1 αi · Effi,k+1(ξ)

✇❤❡r❡ 0 ≤ αi ≤ 1,

k

  • i=1

αi = 1✳

slide-9
SLIDE 9

P❘❖P❊❘❚■❊❙ ✶✳ ❚❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥ Iα(ξ) ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥✳ ✷✳ ■❢ ξ ✐s ❛ r❡❣✉❧❛r ❞❡s✐❣♥ t❤❡♥ ∂Iα(ξ, ¯ ξ) =

k

  • i=1

αi ∂Ii,k+1(ξ, ¯ ξ) Ii,k+1(ξ∗

i )

=

  • X ψα(x, ξ) ¯

ξ(dx), ✇❤❡r❡ ψα(x, ξ) = ∂Iα(ξ, ξx) ❛♥❞ ξx ✐s ❛ ❞❡s✐❣♥ ✇❤✐❝❤ ♣✉ts t❤❡ ✇❤♦❧❡ ♠❛ss ❛t ♣♦✐♥t x✳ ✸✳ ❊q✉✐✈❛❧❡♥❝❡ ❚❤❡♦r❡♠ ξ∗

α = argmax ξ

Iα(ξ) ⇔ ψα(x, ξ∗

α) ≤ 0, x ∈ X.

slide-10
SLIDE 10

❖❚❍❊❘ P❖❙❙■❇▲❊ ❈❘■❚❊❘■❆

  • ❚❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥ ♦❢ ❡✣❝✐❡♥❝✐❡s✿

IGM

α

(ξ) = k

i=1

  • Effi,k+1(ξ)
  • αi

ξ∗

GM =arg max ξ

IGM

α

(ξ)=arg max

ξ

log IGM

α

(ξ). log IGM

α

(ξ) =

k

  • i=1

αi · log

  • Ii,k+1(ξ)

k

  • i=1

αi · log

  • Ii,k+1(ξ∗

i )

  • .

❚❤✉s✱ ξ∗

GM ♠❛①✐♠✐③❡s t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛♥

♦❢ Ii,k+1(ξ)✱ ✇✐t❤♦✉t t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡✐r ♣♦ss✐❜❧❡ ❞✐✛❡r❡♥t ♠❛❣♥✐t✉❞❡✳ ❋♦r t❤✐s r❡❛s♦♥ t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ Iα(ξ)=

k

  • i=1

αi · Effi,k+1(ξ) ✐s ♣r❡❢❡rr❡❞ ❤❡r❡✳

slide-11
SLIDE 11
  • ❚❤❡ ♠✐♥✐♠✉♠ ❡✣❝✐❡♥❝② ❝r✐t❡r✐♦♥✿

Im(ξ) = min

i=1,...,k Effi,k+1(ξ)

❚❤✐s ❝r✐t❡r✐♦♥ ♥❛t✉r❛❧❧② ❣✐✈❡s ❡q✉❛❧ ❡✣✲ ❝✐❡♥❝✐❡s t♦ t❤❡ ♠♦❞❡❧s t❤❛t ❛r❡ t❤❡ ♠♦st ❞✐✣❝✉❧t t♦ ❞✐s❝r✐♠✐♥❛t❡ t❤✉s✱ ✐t s❡❡♠s ✈❡r② ❛♣♣r♦♣r✐❛t❡ ❢♦r ❞✐s❝r✐♠✐♥❛t✐♥❣ ♣✉r♣♦s❡s✳ ❚❤✐s ❝r✐t❡r✐♦♥ ✇✐❧❧ ❜❡ st✉❞✐❡❞ ✐♥ ❞❡♣t❤ ✐♥ ❢✉t✉r❡✳

slide-12
SLIDE 12

❊❳❆▼P▲❊

  • ❙t❛t✐st✐❝❛❧ ♠♦❞❡❧s✿

P(y = 1, x, θi) = F(ηi) = eηi 1 + eηi, x ∈ [0, 1].

  • ❘✐✈❛❧ ♠♦❞❡❧s ❢♦r t❤❡ ❧♦❣✐t ♦❢ t❤❡ ❡①✲

♣❡❝t❡❞ r❡s♣♦♥s❡✿ η1 = β1x η2 = β0 + β1x η3 = β1x + β2x2.

  • ❈♦♠❜✐♥❡❞ ♠♦❞❡❧ ❢♦r t❤❡ ❧♦❣✐t ♦❢ t❤❡ ❡①✲

♣❡❝t❡❞ r❡s♣♦♥s❡✿ η4 = β0 + β1x + β2x2.

slide-13
SLIDE 13
  • ❊◆❊❘❆▲■❩❊❉ ❑▲✲❈❘■❚❊❘■❖◆ ❋❯◆❈❚■❖◆

Iα(ξ) = α1 I1,4(ξ) 0.1098 + α2 I2,4(ξ) 0.0026 + (1 − α1 − α2) I3,4(ξ) 0.1102.

α1 α2 ξ∗

α

Eff1,4(ξ∗

α)

Eff2,4(ξ∗

α)

Eff3,4(ξ∗

α)

✶ ✵

  • 0.5

1 0.9993 0.0003 0.0003

  • ✵✳✾✾✾✸✹

✵✳✵✵✵✹✵ ✵✳✾✾✾✸✸ ✵ ✶

  • 0.4160

1 0.2201 0.4655 0.3144

  • ✵✳✷✸✽✺✸

✶ ✵✳✷✷✵✵✾ ✵ ✵

  • 0.5

1 0.9993 0.0003 0.0003

  • ✵✳✾✾✾✸✹

✵✳✵✵✵✹✵ ✵✳✾✾✾✸✸ 1/3 1/3

  • 0.4

1 0.9360 0.0400 0.0240

  • ✵✳✾✸✽✸✺

✵✳✶✵✼✶✵ ✵✳✾✸✻✻✷ ✵✳✸ ✵✳✹

  • 0.368

1 0.6162 0.2391 0.1446

  • ✵✳✻✸✵✼✺

✵✳✻✷✶✸✶ ✵✳✻✶✻✻✺

slide-14
SLIDE 14

❆▲▼❖❙❚ ❊◗❯❆▲▲❨ ❊❋❋■❈❊■◆❚ ❖P❚■▼❆▲ ❉❊❙■●◆ ξ∗

(0.3,0.4) =

  • 0.368

1 0.6162 0.2391 0.1446

  • ❑▲✕♦♣t✐♠❛❧ ❞❡s✐❣♥s ❢♦r ❞✐s❝r✐♠✐♥❛t✐♥❣ ❜❡t✇❡❡♥ t✇♦ ♠♦❞❡❧s✿

❈♦♠♣❛r✐s♦♥s ❑▲✕♦♣t✐♠❛❧ ❞❡s✐❣♥ Eff

  • ξ∗

(0.3,0.4)

  • η1 ✈❡rs✉s η2

ξ0 ✵✳✻✻✾✺ η3 ✈❡rs✉s η2 ξ0 ✵✳✻✶✻✻ η2 ✈❡rs✉s η3

  • 0.4538

1 0.2396 0.4667 0.2937

  • ✵✳✻✸✺✼

η1 ✈❡rs✉s η3

  • 0.3683

1 0.6599 0.3401

  • ✵✳✸✽✶✺
slide-15
SLIDE 15

❙❖▼❊ ❘❊❋❊❘❊◆❈❊❙

❆t❦✐♥s♦♥✱ ❆✳ ❈✳✿ P❧❛♥♥✐♥❣ ❡①♣❡r✐♠❡♥ts t♦ ❞❡t❡❝t ✐♥❛❞✲ ❡q✉❛t❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✳ ❇✐♦♠❡tr✐❦❛✱ ✺✾✱ ✷✼✺✕✷✾✸ ✭✶✾✼✷✮✳ ❆t❦✐♥s♦♥✱ ❆✳ ❈✳✱ ❈♦①✱ ❉✳❘✳✿ P❧❛♥♥✐♥❣ ❡①♣❡r✐♠❡♥ts ❢♦r ❞✐s❝r✐♠✐♥❛t✐♥❣ ❜❡t✇❡❡♥ ♠♦❞❡❧s✳ ❏✳❘✳ ❙t❛t✐st✳ ❙♦❝✳ ❇✱ ✸✻✱ ✸✷✶✕✸✹✽ ✭✶✾✼✹✮✳ ❆t❦✐♥s♦♥✱ ❆✳ ❈✳✱ ❋❡❞♦r♦✈✱ ❱✳❱✳✿ ❚❤❡ ❞❡s✐❣♥s ♦❢ ❡①✲ ♣❡r✐♠❡♥ts ❢♦r ❞✐s❝r✐♠✐♥❛t✐♥❣ ❜❡t✇❡❡♥ t✇♦ r✐✈❛❧ ♠♦❞❡❧s✳ ❇✐♦♠❡tr✐❦❛✱ ✻✷✱ ✺✼✕✼✵ ✭✶✾✼✺❛✮✳ ❆t❦✐♥s♦♥✱ ❆✳ ❈✳✱ ❋❡❞♦r♦✈✱ ❱✳❱✳✿ ❖♣t✐♠❛❧ ❞❡s✐❣♥✿ ❡①✲ ♣❡r✐♠❡♥ts ❢♦r ❞✐s❝r✐♠✐♥❛t✐♥❣ ❜❡t✇❡❡♥ s❡✈❡r❛❧ ♠♦❞❡❧s✳ ❇✐♦♠❡tr✐❦❛✱ ✻✷✱ ✷✽✾✕✸✵✸ ✭✶✾✼✺❜✮✳ ▲ó♣❡③✕❋✐❞❛❧❣♦✱ ❏✳✱ ❚♦♠♠❛s✐✱ ❈✳✱ ❚r❛♥❞❛✜r✱ P✳❈✳✿ ❆♥ ♦♣t✐♠❛❧ ❡①♣❡r✐♠❡♥t❛❧ ❞❡s✐❣♥ ❝r✐t❡r✐♦♥ ❢♦r ❞✐s❝r✐♠✐♥❛t✲ ✐♥❣ ❜❡t✇❡❡♥ ♥♦♥✲◆♦r♠❛❧ ♠♦❞❡❧s✳ ❏✳❘✳❙t❛t✐st✳❙♦❝ ❇✱ ✻✾✱ ✷✸✶✲✷✹✷ ✭✷✵✵✼✮✳ P♦♥❝❡ ❞❡ ▲❡♦♥✱ ❆✳❈✱ ❆t❦✐♥s♦♥✱ ❆✳❈✳✿ ❚❤❡ ❞❡s✐❣♥ ♦❢ ❡①♣❡r✐♠❡♥ts t♦ ❞✐s❝r✐♠✐♥❛t❡ ❜❡t✇❡❡♥ t✇♦ r✐✈❛❧ ❣❡♥✲ ❡r❛❧✐③❡❞ ❧✐♥❡❛r ♠♦❞❡❧s✳ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❙t❛t✐st✐❝s ✲ ❆❞✈❛♥❝❡s ✐♥ ●▲▼ ❛♥❞ ❙t❛t✐st✐❝❛❧ ▼♦❞❡❧❧✐♥❣✳ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✭✶✾✾✷✮✳ ❯❝✐➠s❦✐✱ ❉✳✱ ❇♦❣❛❝❦❛✱ ❇✳✿ ❚✲♦♣t✐♠✉♠ ❞❡s✐❣♥s ❢♦r ♠✉❧✲ t✐r❡s♣♦♥s❡ ❞②♥❛♠✐❝ ❤❡t❡r♦s❝❡❞❛st✐❝ ♠♦❞❡❧s✳ ♠❖❉❛ ✼ ✲ ❆❞✈❛♥❝❡s ✐♥ ▼♦❞❡❧✲❖r✐❡♥t❡❞ ❉❡s✐❣♥ ❛♥❞ ❆♥❛❧②s✐s✳ P❤②s✐❝❛✲❱❡r❧❛❣✱ ❍❡✐❞❡❧❜❡r❣✱ ✭✷✵✵✹✮✳

slide-16
SLIDE 16

❆ ❋■❘❙❚ ❖❘❉❊❘ ❆▲●❖❘■❚❍▼ ❋❖❘ ❈❖▼P❯❚■◆● ξ∗

α

❛✮ ❣✐✈❡♥ ξs✱ ✜♥❞ θi,s = arg min

θi∈Ωi

  • X I [fk+1(y, x, θk+1), fi(y, x, θi)]ξs(dx), i = 1, . . . , k

xs = arg max

x∈χ ψα(x, ξs)

❜✮ ❈❤♦♦s❡ 0 ≤ γs ≤ 1 s✉❝❤ t❤❛t lim

s→∞ γs=0, ∞

  • s=1

γs=∞,

  • s=1

γ2

s <∞

✭❤❡r❡ γs = 1/s✮ ❛♥❞ ❝♦♥str✉❝t ξs+1 = (1 − γs) ξs + γs ξxs, ξxs =

  • xs

1

  • .
slide-17
SLIDE 17

❆ ❙❚❖PP■◆● ❘❯▲❊

  • 1 + maxx∈χ ψα(x; ξ)

Iα(ξ)

−1

≤ Iα(ξ) Iα(ξ∗

α) ≤ 1

✇❤❡r❡ ψα(x, ξ) = ∂Iα(ξ, ξx) ❚❤✉s✱ t❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡❞✉r❡ st♦♣s ✇❤❡♥ ξs ✐s s✉❝❤ t❤❛t

  • 1 + maxx∈χ ψα(x; ξs)

Iα(ξs)

−1

> δ, 0 < δ < 1.