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tt tr - - PowerPoint PPT Presentation

Oct 03, 2022 104 likes •1.34k views

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slide-1
SLIDE 1

◆♦♥✲❝♦♥✈❡① ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ◆❡t✇♦r❦ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r②

❈❤❛♥❞r❛ ◆❛✐r

❚❤❡ ❈❤✐♥❡s❡ ❯♥✐✈❡rs✐t② ♦❢ ❍♦♥❣ ❑♦♥❣ ✸r❞ ❏❛♥✉❛r②✱ ✷✵✶✾

slide-2
SLIDE 2

◆♦♥✲❝♦♥✈❡① ♣r♦❜❧❡♠s ❛♥❞ ♥❡t✇♦r❦ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

  • ■♥tr♦❞✉❝t✐♦♥

⋆ ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s ⋆ ❍♦✇ t♦ t❡st t❤❡ ♦♣t✐♠❛❧✐t② ♦❢ ❝♦❞✐♥❣ s❝❤❡♠❡s ⋆ ❲❤❡r❡ ❞♦ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ❛r✐s❡❄

  • ❚✇♦ ♣r♦❜❧❡♠s t♦ ✐❧❧✉str❛t❡ s♦♠❡ ✐❞❡❛s
  • ❖❜s❡r✈❛t✐♦♥s ❛♥❞ ♣♦t❡♥t✐❛❧ ❢✉t✉r❡ ❞✐r❡❝t✐♦♥s

✷

slide-3
SLIDE 3

P♦✐♥t✲t♦✲♣♦✐♥t ❝♦♠♠✉♥✐❝❛t✐♦♥

M ❊♥❝♦❞❡r Xn W ⊗n Y n ❉❡❝♦❞❡r ˆ M ❆ r❛t❡ R ✐s ❛❝❤✐❡✈❛❜❧❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ ❡♥❝♦❞✐♥❣✴❞❡❝♦❞✐♥❣ ♠❛♣s s♦ t❤❛t P(M = ˆ M) → 0 ❛s n → ∞✳ ❈❛♣❛❝✐t②✱ C(W) := sup{R : R ✐s ❛❝❤✐❡✈❛❜❧❡ }✳

❙❤❛♥♥♦♥

❘❛♥❞♦♠ ❝♦❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ ✇❤❡r❡ ✿ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❜❡t✇❡❡♥ ❛♥❞ ◗✉❡st✐♦♥✿ ■s ❄ ✭❨❊❙✮ ✭❙❤❛♥♥♦♥ ✬✹✽✮

✸

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SLIDE 4

P♦✐♥t✲t♦✲♣♦✐♥t ❝♦♠♠✉♥✐❝❛t✐♦♥

M ❊♥❝♦❞❡r Xn W ⊗n Y n ❉❡❝♦❞❡r ˆ M ❆ r❛t❡ R ✐s ❛❝❤✐❡✈❛❜❧❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ ❡♥❝♦❞✐♥❣✴❞❡❝♦❞✐♥❣ ♠❛♣s s♦ t❤❛t P(M = ˆ M) → 0 ❛s n → ∞✳ ❈❛♣❛❝✐t②✱ C(W) := sup{R : R ✐s ❛❝❤✐❡✈❛❜❧❡ }✳

❙❤❛♥♥♦♥

❘❛♥❞♦♠ ❝♦❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ R(W) = sup

µ(x)

I(X; Y ) ✇❤❡r❡ I(X; Y ) :=

  • x,y

µX,Y (x, y) log µX,Y (x, y) µX(x)µY (y)

  • I(X; Y )✿ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❜❡t✇❡❡♥ X ❛♥❞ Y

◗✉❡st✐♦♥✿ ■s ❄ ✭❨❊❙✮ ✭❙❤❛♥♥♦♥ ✬✹✽✮

✸

slide-5
SLIDE 5

P♦✐♥t✲t♦✲♣♦✐♥t ❝♦♠♠✉♥✐❝❛t✐♦♥

M ❊♥❝♦❞❡r Xn W ⊗n Y n ❉❡❝♦❞❡r ˆ M ❆ r❛t❡ R ✐s ❛❝❤✐❡✈❛❜❧❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ ❡♥❝♦❞✐♥❣✴❞❡❝♦❞✐♥❣ ♠❛♣s s♦ t❤❛t P(M = ˆ M) → 0 ❛s n → ∞✳ ❈❛♣❛❝✐t②✱ C(W) := sup{R : R ✐s ❛❝❤✐❡✈❛❜❧❡ }✳

❙❤❛♥♥♦♥

❘❛♥❞♦♠ ❝♦❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ R(W) = sup

µ(x)

I(X; Y ) ✇❤❡r❡ I(X; Y ) :=

  • x,y

µX,Y (x, y) log µX,Y (x, y) µX(x)µY (y)

  • I(X; Y )✿ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❜❡t✇❡❡♥ X ❛♥❞ Y

◗✉❡st✐♦♥✿ ■s R(W) = C(W)❄ ✭❨❊❙✮ ✭❙❤❛♥♥♦♥ ✬✹✽✮

✸

slide-6
SLIDE 6

P♦✐♥t✲t♦✲♣♦✐♥t ❝♦♠♠✉♥✐❝❛t✐♦♥

M ❊♥❝♦❞❡r Xn W ⊗n Y n ❉❡❝♦❞❡r ˆ M ❆ r❛t❡ R ✐s ❛❝❤✐❡✈❛❜❧❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ ❡♥❝♦❞✐♥❣✴❞❡❝♦❞✐♥❣ ♠❛♣s s♦ t❤❛t P(M = ˆ M) → 0 ❛s n → ∞✳ ❈❛♣❛❝✐t②✱ C(W) := sup{R : R ✐s ❛❝❤✐❡✈❛❜❧❡ }✳

❙❤❛♥♥♦♥

❘❛♥❞♦♠ ❝♦❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ R(W) = sup

µ(x)

I(X; Y ) ✇❤❡r❡ I(X; Y ) :=

  • x,y

µX,Y (x, y) log µX,Y (x, y) µX(x)µY (y)

  • I(X; Y )✿ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❜❡t✇❡❡♥ X ❛♥❞ Y

◗✉❡st✐♦♥✿ ■s R(W) = C(W)❄ ✭❨❊❙✮ ✭❙❤❛♥♥♦♥ ✬✹✽✮

✸

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SLIDE 7

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

■t ✐s ❡❛s② ✭✇❤②❄✮ t♦ s❡❡ t❤❛t R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ R(W ⊗ W) = 2R(W) ∀ W. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ ✐s r❡❧❛t✐✈❡❧② ❡❛s②✱ s✐♥❝❡ ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ ✳

✹

slide-8
SLIDE 8

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

■t ✐s ❡❛s② ✭✇❤②❄✮ t♦ s❡❡ t❤❛t R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ R(W ⊗ W) = 2R(W) ∀ W. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ ✐s r❡❧❛t✐✈❡❧② ❡❛s②✱ s✐♥❝❡ ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ ✳

✹

■❢ ∃W s✉❝❤ t❤❛t 1

2R(W ⊗ W) > R(W) t❤❡♥

C(W) ≥ 1

2R(W ⊗ W) > R(W)

✭❍❡♥❝❡ ❡q✉❛❧✐t② ✐s ♥❡❝❡ss❛r②✮

slide-9
SLIDE 9

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

■t ✐s ❡❛s② ✭✇❤②❄✮ t♦ s❡❡ t❤❛t R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ R(W ⊗ W) = 2R(W) ∀ W. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ ✐s r❡❧❛t✐✈❡❧② ❡❛s②✱ s✐♥❝❡ ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ ✳

✹

  • ✐✈❡♥ ǫ > 0 t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦❞❡s s✉❝❤ t❤❛t

1 nI(Xn; Y n) ≥ C(W) − ǫ, ∀n > n0

  • ❋❛♥♦✬s ✐♥❡q✉❛❧✐t②
  • ❉❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t②
slide-10
SLIDE 10

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

■t ✐s ❡❛s② ✭✇❤②❄✮ t♦ s❡❡ t❤❛t R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ R(W ⊗ W) = 2R(W) ∀ W. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ ✐s r❡❧❛t✐✈❡❧② ❡❛s②✱ s✐♥❝❡ ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ ✳

✹

  • ✐✈❡♥ ǫ > 0 t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦❞❡s s✉❝❤ t❤❛t

1 nI(Xn; Y n) ≥ C(W) − ǫ, ∀n > n0 ❍❡♥❝❡✱ ❢♦r k s✉❝❤ t❤❛t N = 2k > n0 ✇❡ ❤❛✈❡ R(W) ✐♥❞❝ = 1 N R(W ⊗ · · · ⊗ W

  • N

) = 1 N I(XN; Y N) ≥ C(W) − ǫ.

slide-11
SLIDE 11

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

■t ✐s ❡❛s② ✭✇❤②❄✮ t♦ s❡❡ t❤❛t R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ R(W ⊗ W) = 2R(W) ∀ W. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ I(X1, X2; Y1, Y2) ≤ I(X1; Y1) + I(X2; Y2). ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ ✐s r❡❧❛t✐✈❡❧② ❡❛s②✱ s✐♥❝❡ ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ ✳

✹

slide-12
SLIDE 12

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

■t ✐s ❡❛s② ✭✇❤②❄✮ t♦ s❡❡ t❤❛t R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ R(W ⊗ W) = 2R(W) ∀ W. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ I(X1, X2; Y1, Y2) ≤ I(X1; Y1) + I(X2; Y2). ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ ✐s r❡❧❛t✐✈❡❧② ❡❛s②✱ s✐♥❝❡ ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ ✳

✹

❙✉❜✲❛❞❞✐t✐✈✐t② ❆ ❢✉♥❝t✐♦♥❛❧ ❞❡✜♥❡❞ ♦✈❡r ❛ ♣r♦❜❛❜✐❧✐t② s✐♠♣❧❡① ✐s s❛✐❞ t♦ ❜❡ s✉❜✲❛❞❞✐t✐✈❡ ✐❢ F12(µX1,X2) ≤ F1(µX1) + F2(µX2) ∀ µX1,X2. ■♥ ❛❜♦✈❡✱ s✐♥❝❡ W ✐s ✜①❡❞✱ I(X; Y ) ✐s ❛ ❢✉♥❝t✐♦♥❛❧ ♦✈❡r µX✱ t❤❡ s♣❛❝❡ ♦❢ ✐♥♣✉t ❞✐str✐❜✉t✐♦♥s✳

slide-13
SLIDE 13

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

■t ✐s ❡❛s② ✭✇❤②❄✮ t♦ s❡❡ t❤❛t R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ R(W ⊗ W) = 2R(W) ∀ W. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ I(X1, X2; Y1, Y2) ≤ I(X1; Y1) + I(X2; Y2). I(X1, X2; Y1, Y2) = I(X1, X2; Y1) + I(X1, X2; Y2|Y1) = I(X1, X2; Y1) + I(Y1, X1, X2; Y2) − I(Y1; Y2) = I(X1; Y1) + I(X2; Y2) − I(Y1; Y2) ≤ I(X1; Y1) + I(X2; Y2). ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ ✐s r❡❧❛t✐✈❡❧② ❡❛s②✱ s✐♥❝❡ ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ ✳

✹

slide-14
SLIDE 14

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

■t ✐s ❡❛s② ✭✇❤②❄✮ t♦ s❡❡ t❤❛t R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ R(W ⊗ W) = 2R(W) ∀ W. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ I(X1, X2; Y1, Y2) ≤ I(X1; Y1) + I(X2; Y2). I(X1, X2; Y1, Y2) = I(X1, X2; Y1) + I(X1, X2; Y2|Y1) = I(X1, X2; Y1) + I(Y1, X1, X2; Y2) − I(Y1; Y2) = I(X1; Y1) + I(X2; Y2) − I(Y1; Y2) ≤ I(X1; Y1) + I(X2; Y2). ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ R(W) = supp(x) I(X; Y ) ✐s r❡❧❛t✐✈❡❧② ❡❛s②✱ s✐♥❝❡ I(X; Y ) ✐s ❛ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ p(x)✳

✹

slide-15
SLIDE 15

❙✉❝❝❡ss❡s

❚❤❡ ✈❛r✐♦✉s ✐❞❡❛s ✐♥tr♦❞✉❝❡❞ ❜② ❙❤❛♥♥♦♥ ❤❛✈❡ ❧❡❞ t♦ ❛♥ ✐♥❢♦r♠❛t✐♦♥ r❡✈♦❧✉t✐♦♥ ❘❛♥❞♦♠ ❝♦❞✐♥❣ ❛♥❞ ✐ts ♦♣t✐♠❛❧✐t② ❤❛✈❡ ❞✐r❡❝t❧② ✐♥s♣✐r❡❞

  • ▲♦✇ ❞❡♥s✐t② ♣❛r✐t② ❝❤❡❝❦ ❝♦❞❡s ✭▲❉P❈✮
  • P♦❧❛r ❝♦❞❡s

⋆ ♣r♦♦❢ ♦❢ s✉❜✲❛❞❞✐t✐✈✐t② ❲❡ ❛r❡ ♥♦✇ ✭❢✉❧❧② ✐♠♠❡rs❡❞✮ ✐♥ ❛ ✇✐r❡❧❡ss ✇♦r❧❞ ◆❡t✇♦r❦ ♦❢ ✉s❡rs s❤❛r✐♥❣ s❛♠❡ ♠❡❞✐✉♠ ❈❧❡❛r ♥❡❡❞ t♦ ♠❛①✐♠❛❧❧② ✉t✐❧✐③❡ t❤❡ ❧✐♠✐t❡❞ r❡s♦✉r❝❡s ✭♣♦✇❡r✱ ❜❛♥❞✇✐❞t❤✱ ❡♥❡r❣②✮ ❉❡✈❡❧♦♣ ❛ s✐♠✐❧❛r ✉♥❞❡rst❛♥❞✐♥❣ ✐♥ ♥❡t✇♦r❦ s❡tt✐♥❣s ❇✉t ✇❡ ✜rst ♥❡❡❞ t♦ ❢✉❧❧② ✉♥❞❡rst❛♥❞ t❤❡ ❜❛s✐❝ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s

✺

slide-16
SLIDE 16

❙✉❝❝❡ss❡s

❚❤❡ ✈❛r✐♦✉s ✐❞❡❛s ✐♥tr♦❞✉❝❡❞ ❜② ❙❤❛♥♥♦♥ ❤❛✈❡ ❧❡❞ t♦ ❛♥ ✐♥❢♦r♠❛t✐♦♥ r❡✈♦❧✉t✐♦♥ ❘❛♥❞♦♠ ❝♦❞✐♥❣ ❛♥❞ ✐ts ♦♣t✐♠❛❧✐t② ❤❛✈❡ ❞✐r❡❝t❧② ✐♥s♣✐r❡❞

  • ▲♦✇ ❞❡♥s✐t② ♣❛r✐t② ❝❤❡❝❦ ❝♦❞❡s ✭▲❉P❈✮
  • P♦❧❛r ❝♦❞❡s

⋆ ♣r♦♦❢ ♦❢ s✉❜✲❛❞❞✐t✐✈✐t② ❲❡ ❛r❡ ♥♦✇ ✭❢✉❧❧② ✐♠♠❡rs❡❞✮ ✐♥ ❛ ✇✐r❡❧❡ss ✇♦r❧❞

  • ◆❡t✇♦r❦ ♦❢ ✉s❡rs s❤❛r✐♥❣ s❛♠❡ ♠❡❞✐✉♠
  • ❈❧❡❛r ♥❡❡❞ t♦ ♠❛①✐♠❛❧❧② ✉t✐❧✐③❡ t❤❡ ❧✐♠✐t❡❞ r❡s♦✉r❝❡s ✭♣♦✇❡r✱ ❜❛♥❞✇✐❞t❤✱ ❡♥❡r❣②✮
  • ❉❡✈❡❧♦♣ ❛ s✐♠✐❧❛r ✉♥❞❡rst❛♥❞✐♥❣ ✐♥ ♥❡t✇♦r❦ s❡tt✐♥❣s

⋆ ❇✉t ✇❡ ✜rst ♥❡❡❞ t♦ ❢✉❧❧② ✉♥❞❡rst❛♥❞ t❤❡ ❜❛s✐❝ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s

✺

slide-17
SLIDE 17

✶✳ ▼✉❧t✐♣❧❡ ❆❝❝❡ss ❈❤❛♥♥❡❧ ✭✉♣❧✐♥❦✮ ✭❙❤❛♥♥♦♥ ✬✻✶✮

M1 M2 ❊♥❝♦❞❡r ✶ ❊♥❝♦❞❡r ✷ Xn

2

Xn

1

W(y|x1, x2) Y n ❉❡❝♦❞❡r ( ˆ M1, ˆ M2)

r❢✇✐r❡❧❡ss✲✇♦r❧❞

❆❤❧s✇❡❞❡

❘❛♥❞♦♠ ❝♦❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ ♣❛✐rs t❤❛t s❛t✐s❢② ❢♦r s♦♠❡ ❀ ✐t s✉✣❝❡s t♦ ❝♦♥s✐❞❡r ✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ ✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② ✭♦♣t✐♠❛❧✮ r❡❣✐♦♥❄ ✭❨❊❙✮ ✭❆❤❧s✇❡❞❡ ✬✼✷✮

✻

slide-18
SLIDE 18

✶✳ ▼✉❧t✐♣❧❡ ❆❝❝❡ss ❈❤❛♥♥❡❧ ✭✉♣❧✐♥❦✮ ✭❙❤❛♥♥♦♥ ✬✻✶✮

M1 M2 ❊♥❝♦❞❡r ✶ ❊♥❝♦❞❡r ✷ Xn

2

Xn

1

W(y|x1, x2) Y n ❉❡❝♦❞❡r ( ˆ M1, ˆ M2)

r❢✇✐r❡❧❡ss✲✇♦r❧❞

❆❤❧s✇❡❞❡

❘❛♥❞♦♠ ❝♦❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ ♣❛✐rs (R1, R2) t❤❛t s❛t✐s❢② R1 ≤ I(X1; Y |X2, Q) R2 ≤ I(X2; Y |X1, Q) R1 + R2 ≤ I(X1, X2; Y |Q) ❢♦r s♦♠❡ p(q)p(x1|q)p(x2|q)❀ ✐t s✉✣❝❡s t♦ ❝♦♥s✐❞❡r |Q| ≤ 2✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ R(W)✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② ✭♦♣t✐♠❛❧✮ r❡❣✐♦♥❄ ✭❨❊❙✮ ✭❆❤❧s✇❡❞❡ ✬✼✷✮

✻

slide-19
SLIDE 19

✶✳ ▼✉❧t✐♣❧❡ ❆❝❝❡ss ❈❤❛♥♥❡❧ ✭✉♣❧✐♥❦✮ ✭❙❤❛♥♥♦♥ ✬✻✶✮

M1 M2 ❊♥❝♦❞❡r ✶ ❊♥❝♦❞❡r ✷ Xn

2

Xn

1

W(y|x1, x2) Y n ❉❡❝♦❞❡r ( ˆ M1, ˆ M2)

r❢✇✐r❡❧❡ss✲✇♦r❧❞

❆❤❧s✇❡❞❡

❘❛♥❞♦♠ ❝♦❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ ♣❛✐rs (R1, R2) t❤❛t s❛t✐s❢② R1 ≤ I(X1; Y |X2, Q) R2 ≤ I(X2; Y |X1, Q) R1 + R2 ≤ I(X1, X2; Y |Q) ❢♦r s♦♠❡ p(q)p(x1|q)p(x2|q)❀ ✐t s✉✣❝❡s t♦ ❝♦♥s✐❞❡r |Q| ≤ 2✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ R(W)✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② ✭♦♣t✐♠❛❧✮ r❡❣✐♦♥❄ ✭❨❊❙✮ ✭❆❤❧s✇❡❞❡ ✬✼✷✮

✻

slide-20
SLIDE 20

✶✳ ▼✉❧t✐♣❧❡ ❆❝❝❡ss ❈❤❛♥♥❡❧ ✭✉♣❧✐♥❦✮ ✭❙❤❛♥♥♦♥ ✬✻✶✮

M1 M2 ❊♥❝♦❞❡r ✶ ❊♥❝♦❞❡r ✷ Xn

2

Xn

1

W(y|x1, x2) Y n ❉❡❝♦❞❡r ( ˆ M1, ˆ M2)

r❢✇✐r❡❧❡ss✲✇♦r❧❞

❆❤❧s✇❡❞❡

❘❛♥❞♦♠ ❝♦❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ ♣❛✐rs (R1, R2) t❤❛t s❛t✐s❢② R1 ≤ I(X1; Y |X2, Q) R2 ≤ I(X2; Y |X1, Q) R1 + R2 ≤ I(X1, X2; Y |Q) ❢♦r s♦♠❡ p(q)p(x1|q)p(x2|q)❀ ✐t s✉✣❝❡s t♦ ❝♦♥s✐❞❡r |Q| ≤ 2✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ R(W)✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② ✭♦♣t✐♠❛❧✮ r❡❣✐♦♥❄ ✭❨❊❙✮ ✭❆❤❧s✇❡❞❡ ✬✼✷✮

✻

slide-21
SLIDE 21

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

❉❡✜♥❡✱ ❢♦r λ ≥ 1✱ Sλ(W) = max

(R1,R2)∈R(W)

  • λR1 + R2}

= max

p1(x1)p2(x2)

  • (λ − 1)I(X1; Y |X2) + I(X1, X2; Y )
  • ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿

✼

slide-22
SLIDE 22

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

❉❡✜♥❡✱ ❢♦r λ ≥ 1✱ Sλ(W) = max

(R1,R2)∈R(W)

  • λR1 + R2}

= max

p1(x1)p2(x2)

  • (λ − 1)I(X1; Y |X2) + I(X1, X2; Y )
  • ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿

✼

R1 + R2 2R1 + R2 3R1 + R2 R2 R1

❙✉♣♣♦rt✐♥❣ ❤②♣❡r♣❧❛♥❡s

slide-23
SLIDE 23

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

❉❡✜♥❡✱ ❢♦r λ ≥ 1✱ Sλ(W) = max

(R1,R2)∈R(W)

  • λR1 + R2}

= max

p1(x1)p2(x2)

  • (λ − 1)I(X1; Y |X2) + I(X1, X2; Y )
  • ❆s ❜❡❢♦r❡✱ R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢

Sλ(W ⊗ W) = 2Sλ(W) ∀ W, λ ≥ 1. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ (λ − 1)I(X11, X12; Y1, Y2|X21, X22) + I(X11, X12, X21, X22; Y1, Y2) ≤ (λ − 1)I(X11; Y1|X21) + I(X11, X21; Y1) + (λ − 1)I(X12; Y2|X22) + I(X12, X22; Y2)

✼

slide-24
SLIDE 24

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

❉❡✜♥❡✱ ❢♦r λ ≥ 1✱ Sλ(W) = max

(R1,R2)∈R(W)

  • λR1 + R2}

= max

p1(x1)p2(x2)

  • (λ − 1)I(X1; Y |X2) + I(X1, X2; Y )
  • ❆s ❜❡❢♦r❡✱ R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢

Sλ(W ⊗ W) = 2Sλ(W) ∀ W, λ ≥ 1. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ (λ − 1)I(X11, X12; Y1, Y2|X21, X22) + I(X11, X12, X21, X22; Y1, Y2) ≤ (λ − 1)I(X11; Y1|X21) + I(X11, X21; Y1) + (λ − 1)I(X12; Y2|X22) + I(X12, X22; Y2) ❖♥❡ ❝❛♥ ❡st❛❜❧✐s❤ t❤✐s ✐♥ s❛♠❡ ✇❛② ❛s ♣♦✐♥t✲t♦✲♣♦✐♥t s❡tt✐♥❣✳

✼

slide-25
SLIDE 25

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

❉❡✜♥❡✱ ❢♦r λ ≥ 1✱ Sλ(W) = max

(R1,R2)∈R(W)

  • λR1 + R2}

= max

p1(x1)p2(x2)

  • (λ − 1)I(X1; Y |X2) + I(X1, X2; Y )
  • ❆s ❜❡❢♦r❡✱ R(W) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢

Sλ(W ⊗ W) = 2Sλ(W) ∀ W, λ ≥ 1. ❚❤❡ ❛❜♦✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♦❧❧♦✇s ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❤♦❧❞s✿ (λ − 1)I(X11, X12; Y1, Y2|X21, X22) + I(X11, X12, X21, X22; Y1, Y2) ≤ (λ − 1)I(X11; Y1|X21) + I(X11, X21; Y1) + (λ − 1)I(X12; Y2|X22) + I(X12, X22; Y2) ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ Sλ(W) ✐s r❡❧❛t✐✈❡❧② ❡❛s② s✐♥❝❡

  • (λ − 1)I(X1; Y |X2) + I(X1, X2; Y )
  • ✐s ❝♦♥❝❛✈❡ ✐♥ p1(x1), p2(x2)✳

✼

slide-26
SLIDE 26

✷✳ ❇r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✭❞♦✇♥❧✐♥❦✮ ✭❈♦✈❡r ✬✼✷✮

(M0, M1, M2) ❊♥❝♦❞❡r Xn Wa(y1|x) Wb(y2|x) Y n

1

Y n

2

❉❡❝♦❞❡r ✶ ❉❡❝♦❞❡r ✷ ˆ M0, ˆ M1 ˜ M0, ˜ M2

r❢✇✐r❡❧❡ss✲✇♦r❧❞

▼❛rt♦♥

❙✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ❛♥❞ r❛♥❞♦♠ ❤❛s❤✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ tr✐♣❧❡s t❤❛t s❛t✐s❢② ❢♦r s♦♠❡ ✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ ✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② ✭♦♣t✐♠❛❧✮ r❡❣✐♦♥❄ ✭❖♣❡♥✮ ✭s✐♥❝❡ ▼❛rt♦♥ ✬✼✾✮

✽

slide-27
SLIDE 27

✷✳ ❇r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✭❞♦✇♥❧✐♥❦✮ ✭❈♦✈❡r ✬✼✷✮

(M0, M1, M2) ❊♥❝♦❞❡r Xn Wa(y1|x) Wb(y2|x) Y n

1

Y n

2

❉❡❝♦❞❡r ✶ ❉❡❝♦❞❡r ✷ ˆ M0, ˆ M1 ˜ M0, ˜ M2

r❢✇✐r❡❧❡ss✲✇♦r❧❞

▼❛rt♦♥

❙✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ❛♥❞ r❛♥❞♦♠ ❤❛s❤✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ tr✐♣❧❡s (R0, R1, R2) t❤❛t s❛t✐s❢②

R0 ≤ min{I(Q; Y1), I(Q; Y2)} R0 + R1 ≤ I(U, Q; Y1) R0 + R2 ≤ I(V, Q; Y2) R0 + R1 + R2 ≤ min{I(Q; Y1), I(Q; Y2)} + I(U; Y1|Q) + I(V ; Y2|Q) − I(U; V |Q)

❢♦r s♦♠❡ p(q, u, v, x)✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ R(Wa, Wb)✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② ✭♦♣t✐♠❛❧✮ r❡❣✐♦♥❄ ✭❖♣❡♥✮ ✭s✐♥❝❡ ▼❛rt♦♥ ✬✼✾✮

✽

slide-28
SLIDE 28

✷✳ ❇r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✭❞♦✇♥❧✐♥❦✮ ✭❈♦✈❡r ✬✼✷✮

(M0, M1, M2) ❊♥❝♦❞❡r Xn Wa(y1|x) Wb(y2|x) Y n

1

Y n

2

❉❡❝♦❞❡r ✶ ❉❡❝♦❞❡r ✷ ˆ M0, ˆ M1 ˜ M0, ˜ M2

r❢✇✐r❡❧❡ss✲✇♦r❧❞

▼❛rt♦♥

❙✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ❛♥❞ r❛♥❞♦♠ ❤❛s❤✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ tr✐♣❧❡s (R0, R1, R2) t❤❛t s❛t✐s❢②

R0 ≤ min{I(Q; Y1), I(Q; Y2)} R0 + R1 ≤ I(U, Q; Y1) R0 + R2 ≤ I(V, Q; Y2) R0 + R1 + R2 ≤ min{I(Q; Y1), I(Q; Y2)} + I(U; Y1|Q) + I(V ; Y2|Q) − I(U; V |Q)

❢♦r s♦♠❡ p(q, u, v, x)✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ R(Wa, Wb)✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② ✭♦♣t✐♠❛❧✮ r❡❣✐♦♥❄ ✭❖♣❡♥✮ ✭s✐♥❝❡ ▼❛rt♦♥ ✬✼✾✮

✽

slide-29
SLIDE 29

✷✳ ❇r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✭❞♦✇♥❧✐♥❦✮ ✭❈♦✈❡r ✬✼✷✮

(M0, M1, M2) ❊♥❝♦❞❡r Xn Wa(y1|x) Wb(y2|x) Y n

1

Y n

2

❉❡❝♦❞❡r ✶ ❉❡❝♦❞❡r ✷ ˆ M0, ˆ M1 ˜ M0, ˜ M2

r❢✇✐r❡❧❡ss✲✇♦r❧❞

▼❛rt♦♥

❙✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ❛♥❞ r❛♥❞♦♠ ❤❛s❤✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ tr✐♣❧❡s (R0, R1, R2) t❤❛t s❛t✐s❢②

R0 ≤ min{I(Q; Y1), I(Q; Y2)} R0 + R1 ≤ I(U, Q; Y1) R0 + R2 ≤ I(V, Q; Y2) R0 + R1 + R2 ≤ min{I(Q; Y1), I(Q; Y2)} + I(U; Y1|Q) + I(V ; Y2|Q) − I(U; V |Q)

❢♦r s♦♠❡ p(q, u, v, x)✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ R(Wa, Wb)✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② ✭♦♣t✐♠❛❧✮ r❡❣✐♦♥❄ ✭❖♣❡♥✮ ✭s✐♥❝❡ ▼❛rt♦♥ ✬✼✾✮

✽

slide-30
SLIDE 30

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t② ✭R0 = 0✮

❉❡✜♥❡✱ ❢♦r λ ≥ 1✱ Sλ(W) = max

(R1,R2)∈R(Wa,Wb){λR1 + R2}

= max

p(u,v,w,x)

  • (λ − 1)I(U, Q; Y1) + min{I(Q; Y1), I(Q; Y2)} + I(U; Y1|Q)

+ I(V ; Y2|Q) − I(U; V |Q)

  • = min

α∈[0,1]

max

p(u,v,w,x)

  • (λ − α)I(Q; Y1) + αI(Q; Y2) + λI(U; Y1|Q)

+ I(V ; Y2|Q) − I(U; V |Q)

  • ❆s ❜❡❢♦r❡✱ R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢

Sλ(Wa ⊗ Wa, Wb ⊗ Wb) = 2Sλ(Wa, Wb) ∀ Wa, Wb, λ ≥ 1. ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ ✐s ❛ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✳

✾

slide-31
SLIDE 31

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t② ✭R0 = 0✮

❉❡✜♥❡✱ ❢♦r λ ≥ 1✱ Sλ(W) = max

(R1,R2)∈R(Wa,Wb){λR1 + R2}

= max

p(u,v,w,x)

  • (λ − 1)I(U, Q; Y1) + min{I(Q; Y1), I(Q; Y2)} + I(U; Y1|Q)

+ I(V ; Y2|Q) − I(U; V |Q)

  • =

min

α∈[0,1]

max

p(u,v,w,x)

  • (λ − α)I(Q; Y1) + αI(Q; Y2) + λI(U; Y1|Q)

+I(V ; Y2|Q) − I(U; V |Q)

  • ❆s ❜❡❢♦r❡✱ R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢

Sλ(Wa ⊗ Wa, Wb ⊗ Wb) = 2Sλ(Wa, Wb) ∀ Wa, Wb, λ ≥ 1. ◆♦t❡✿ ❈♦♠♣✉t✐♥❣ Sλ(Wa, Wb) ✐s ❛ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✳

✾

slide-32
SLIDE 32

❙✉❝❝❡ss❡s

■♥ s♣✐t❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♦♥✲❝♦♥✈❡①

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ♦♥ R1 = 0 ✭♦r R2 = 0✮

⋆ ❉❡❣r❛❞❡❞ ♠❡ss❛❣❡ s❡ts✿ ❑♦r♥❡r ❛♥❞ ▼❛rt♦♥ ✭✬✼✼✮

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ❢♦r s♦♠❡ ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s

⋆ ●❛❧❧❛❣❡r ✬✼✹✱ ❑♦r♥❡r ❛♥❞ ▼❛rt♦♥ ✭✬✼✺✱ ✬✼✼✱ ✬✼✾✮✱ ●❡❧❢❛♥❞ ❛♥❞ P✐♥s❦❡r ✭✬✼✽✮✱ P♦❧t②r❡✈ ✭✬✼✽✮✱ ❊❧ ●❛♠❛❧ ✭✬✼✾✱ ✬✽✵✮ ⋆ ❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❤❛♠❛✐ ✬✵✻✱ ◆❛✐r ✬✶✵✱ ●❡♥❣ ❛♥❞ ●♦❤❛r✐ ❛♥❞ ◆❛✐r ❛♥❞ ❨✉ ✬✶✹✱ ●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹

  • ◆♦✈❡❧ ✐❞❡❛s ❛♥❞ t❡❝❤♥✐q✉❡s ✇❡r❡ ♥❡❡❞❡❞ t♦ ❡st❛❜❧✐s❤ t❤❡s❡ ❝❛♣❛❝✐t② r❡❣✐♦♥s

❈♦✈❡r ✬✼✷✿ ❞❡✈❡❧♦♣♠❡♥t ♦❢ s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ str❛t❡❣②

  • ❛❧❧❛❣❡r ✬✼✹✿ ❝♦♥✈❡rs❡ t♦ t❤❡ ❞❡❣r❛❞❡❞ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✭ s✉❜✲❛❞❞✐t✐✈✐t② ✮

❲❡✐♥❣❛rt❡♥✲❙t❡✐♥❜❡r❣✲❙❤❛♠❛✐ ✬✵✻✿ ❖♣t✐♠❛❧✐t② ♦❢ ✭♦♥ ✮ ❢♦r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❀ ❞❡✈❡❧♦♣✐♥❣ ❛ ❢❛♠✐❧② ♦❢ t✐❣❤t ❝♦♥✈❡① r❡❧❛①❛t✐♦♥s t♦ ❝♦♠♣✉t❡ t❤❡ ♦♣t✐♠❛❧ ✈❛❧✉❡ ♦❢ ❛ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠

  • ❡♥❣✲◆❛✐r ✬✶✹✿ ❖♣t✐♠❛❧✐t② ♦❢

❢♦r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ ❚❡❝❤♥✐q✉❡ ❢♦r ❡st❛❜❧✐s❤✐♥❣ ❡①tr❡♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥s ✉s✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧s

✶✵

slide-33
SLIDE 33

❙✉❝❝❡ss❡s

■♥ s♣✐t❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♦♥✲❝♦♥✈❡①

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ♦♥ R1 = 0 ✭♦r R2 = 0✮

⋆ ❉❡❣r❛❞❡❞ ♠❡ss❛❣❡ s❡ts✿ ❑♦r♥❡r ❛♥❞ ▼❛rt♦♥ ✭✬✼✼✮

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ❢♦r s♦♠❡ ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s

⋆ ●❛❧❧❛❣❡r ✬✼✹✱ ❑♦r♥❡r ❛♥❞ ▼❛rt♦♥ ✭✬✼✺✱ ✬✼✼✱ ✬✼✾✮✱ ●❡❧❢❛♥❞ ❛♥❞ P✐♥s❦❡r ✭✬✼✽✮✱ P♦❧t②r❡✈ ✭✬✼✽✮✱ ❊❧ ●❛♠❛❧ ✭✬✼✾✱ ✬✽✵✮ ⋆ ❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❤❛♠❛✐ ✬✵✻✱ ◆❛✐r ✬✶✵✱ ●❡♥❣ ❛♥❞ ●♦❤❛r✐ ❛♥❞ ◆❛✐r ❛♥❞ ❨✉ ✬✶✹✱ ●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹

  • ◆♦✈❡❧ ✐❞❡❛s ❛♥❞ t❡❝❤♥✐q✉❡s ✇❡r❡ ♥❡❡❞❡❞ t♦ ❡st❛❜❧✐s❤ t❤❡s❡ ❝❛♣❛❝✐t② r❡❣✐♦♥s

⋆ ❈♦✈❡r ✬✼✷✿ ❞❡✈❡❧♦♣♠❡♥t ♦❢ s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ str❛t❡❣②

  • ❛❧❧❛❣❡r ✬✼✹✿ ❝♦♥✈❡rs❡ t♦ t❤❡ ❞❡❣r❛❞❡❞ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✭ s✉❜✲❛❞❞✐t✐✈✐t② ✮

❲❡✐♥❣❛rt❡♥✲❙t❡✐♥❜❡r❣✲❙❤❛♠❛✐ ✬✵✻✿ ❖♣t✐♠❛❧✐t② ♦❢ ✭♦♥ ✮ ❢♦r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❀ ❞❡✈❡❧♦♣✐♥❣ ❛ ❢❛♠✐❧② ♦❢ t✐❣❤t ❝♦♥✈❡① r❡❧❛①❛t✐♦♥s t♦ ❝♦♠♣✉t❡ t❤❡ ♦♣t✐♠❛❧ ✈❛❧✉❡ ♦❢ ❛ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠

  • ❡♥❣✲◆❛✐r ✬✶✹✿ ❖♣t✐♠❛❧✐t② ♦❢

❢♦r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ ❚❡❝❤♥✐q✉❡ ❢♦r ❡st❛❜❧✐s❤✐♥❣ ❡①tr❡♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥s ✉s✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧s

✶✵

slide-34
SLIDE 34

❙✉❝❝❡ss❡s

■♥ s♣✐t❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♦♥✲❝♦♥✈❡①

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ♦♥ R1 = 0 ✭♦r R2 = 0✮

⋆ ❉❡❣r❛❞❡❞ ♠❡ss❛❣❡ s❡ts✿ ❑♦r♥❡r ❛♥❞ ▼❛rt♦♥ ✭✬✼✼✮

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ❢♦r s♦♠❡ ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s

⋆ ●❛❧❧❛❣❡r ✬✼✹✱ ❑♦r♥❡r ❛♥❞ ▼❛rt♦♥ ✭✬✼✺✱ ✬✼✼✱ ✬✼✾✮✱ ●❡❧❢❛♥❞ ❛♥❞ P✐♥s❦❡r ✭✬✼✽✮✱ P♦❧t②r❡✈ ✭✬✼✽✮✱ ❊❧ ●❛♠❛❧ ✭✬✼✾✱ ✬✽✵✮ ⋆ ❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❤❛♠❛✐ ✬✵✻✱ ◆❛✐r ✬✶✵✱ ●❡♥❣ ❛♥❞ ●♦❤❛r✐ ❛♥❞ ◆❛✐r ❛♥❞ ❨✉ ✬✶✹✱ ●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹

  • ◆♦✈❡❧ ✐❞❡❛s ❛♥❞ t❡❝❤♥✐q✉❡s ✇❡r❡ ♥❡❡❞❡❞ t♦ ❡st❛❜❧✐s❤ t❤❡s❡ ❝❛♣❛❝✐t② r❡❣✐♦♥s

⋆ ❈♦✈❡r ✬✼✷✿ ❞❡✈❡❧♦♣♠❡♥t ♦❢ s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ str❛t❡❣② ⋆ ●❛❧❧❛❣❡r ✬✼✹✿ ❝♦♥✈❡rs❡ t♦ t❤❡ ❞❡❣r❛❞❡❞ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✭ s✉❜✲❛❞❞✐t✐✈✐t② ✮ ❲❡✐♥❣❛rt❡♥✲❙t❡✐♥❜❡r❣✲❙❤❛♠❛✐ ✬✵✻✿ ❖♣t✐♠❛❧✐t② ♦❢ ✭♦♥ ✮ ❢♦r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❀ ❞❡✈❡❧♦♣✐♥❣ ❛ ❢❛♠✐❧② ♦❢ t✐❣❤t ❝♦♥✈❡① r❡❧❛①❛t✐♦♥s t♦ ❝♦♠♣✉t❡ t❤❡ ♦♣t✐♠❛❧ ✈❛❧✉❡ ♦❢ ❛ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠

  • ❡♥❣✲◆❛✐r ✬✶✹✿ ❖♣t✐♠❛❧✐t② ♦❢

❢♦r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ ❚❡❝❤♥✐q✉❡ ❢♦r ❡st❛❜❧✐s❤✐♥❣ ❡①tr❡♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥s ✉s✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧s

✶✵

slide-35
SLIDE 35

❙✉❝❝❡ss❡s

■♥ s♣✐t❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♦♥✲❝♦♥✈❡①

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ♦♥ R1 = 0 ✭♦r R2 = 0✮

⋆ ❉❡❣r❛❞❡❞ ♠❡ss❛❣❡ s❡ts✿ ❑♦r♥❡r ❛♥❞ ▼❛rt♦♥ ✭✬✼✼✮

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ❢♦r s♦♠❡ ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s

⋆ ●❛❧❧❛❣❡r ✬✼✹✱ ❑♦r♥❡r ❛♥❞ ▼❛rt♦♥ ✭✬✼✺✱ ✬✼✼✱ ✬✼✾✮✱ ●❡❧❢❛♥❞ ❛♥❞ P✐♥s❦❡r ✭✬✼✽✮✱ P♦❧t②r❡✈ ✭✬✼✽✮✱ ❊❧ ●❛♠❛❧ ✭✬✼✾✱ ✬✽✵✮ ⋆ ❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❤❛♠❛✐ ✬✵✻✱ ◆❛✐r ✬✶✵✱ ●❡♥❣ ❛♥❞ ●♦❤❛r✐ ❛♥❞ ◆❛✐r ❛♥❞ ❨✉ ✬✶✹✱ ●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹

  • ◆♦✈❡❧ ✐❞❡❛s ❛♥❞ t❡❝❤♥✐q✉❡s ✇❡r❡ ♥❡❡❞❡❞ t♦ ❡st❛❜❧✐s❤ t❤❡s❡ ❝❛♣❛❝✐t② r❡❣✐♦♥s

⋆ ❈♦✈❡r ✬✼✷✿ ❞❡✈❡❧♦♣♠❡♥t ♦❢ s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ str❛t❡❣② ⋆ ●❛❧❧❛❣❡r ✬✼✹✿ ❝♦♥✈❡rs❡ t♦ t❤❡ ❞❡❣r❛❞❡❞ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✭ s✉❜✲❛❞❞✐t✐✈✐t② ✮ ⋆ ❲❡✐♥❣❛rt❡♥✲❙t❡✐♥❜❡r❣✲❙❤❛♠❛✐ ✬✵✻✿ ❖♣t✐♠❛❧✐t② ♦❢ R(Wa, Wb) ✭♦♥ R0 = 0✮ ❢♦r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❀ ❞❡✈❡❧♦♣✐♥❣ ❛ ❢❛♠✐❧② ♦❢ t✐❣❤t ❝♦♥✈❡① r❡❧❛①❛t✐♦♥s t♦ ❝♦♠♣✉t❡ t❤❡ ♦♣t✐♠❛❧ ✈❛❧✉❡ ♦❢ ❛ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠

  • ❡♥❣✲◆❛✐r ✬✶✹✿ ❖♣t✐♠❛❧✐t② ♦❢

❢♦r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ ❚❡❝❤♥✐q✉❡ ❢♦r ❡st❛❜❧✐s❤✐♥❣ ❡①tr❡♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥s ✉s✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧s

✶✵

slide-36
SLIDE 36

❙✉❝❝❡ss❡s

■♥ s♣✐t❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♦♥✲❝♦♥✈❡①

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ♦♥ R1 = 0 ✭♦r R2 = 0✮

⋆ ❉❡❣r❛❞❡❞ ♠❡ss❛❣❡ s❡ts✿ ❑♦r♥❡r ❛♥❞ ▼❛rt♦♥ ✭✬✼✼✮

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ❢♦r s♦♠❡ ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s

⋆ ●❛❧❧❛❣❡r ✬✼✹✱ ❑♦r♥❡r ❛♥❞ ▼❛rt♦♥ ✭✬✼✺✱ ✬✼✼✱ ✬✼✾✮✱ ●❡❧❢❛♥❞ ❛♥❞ P✐♥s❦❡r ✭✬✼✽✮✱ P♦❧t②r❡✈ ✭✬✼✽✮✱ ❊❧ ●❛♠❛❧ ✭✬✼✾✱ ✬✽✵✮ ⋆ ❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❤❛♠❛✐ ✬✵✻✱ ◆❛✐r ✬✶✵✱ ●❡♥❣ ❛♥❞ ●♦❤❛r✐ ❛♥❞ ◆❛✐r ❛♥❞ ❨✉ ✬✶✹✱ ●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹

  • ◆♦✈❡❧ ✐❞❡❛s ❛♥❞ t❡❝❤♥✐q✉❡s ✇❡r❡ ♥❡❡❞❡❞ t♦ ❡st❛❜❧✐s❤ t❤❡s❡ ❝❛♣❛❝✐t② r❡❣✐♦♥s

⋆ ❈♦✈❡r ✬✼✷✿ ❞❡✈❡❧♦♣♠❡♥t ♦❢ s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ str❛t❡❣② ⋆ ●❛❧❧❛❣❡r ✬✼✹✿ ❝♦♥✈❡rs❡ t♦ t❤❡ ❞❡❣r❛❞❡❞ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✭ s✉❜✲❛❞❞✐t✐✈✐t② ✮ ⋆ ❲❡✐♥❣❛rt❡♥✲❙t❡✐♥❜❡r❣✲❙❤❛♠❛✐ ✬✵✻✿ ❖♣t✐♠❛❧✐t② ♦❢ R(Wa, Wb) ✭♦♥ R0 = 0✮ ❢♦r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❀ ❞❡✈❡❧♦♣✐♥❣ ❛ ❢❛♠✐❧② ♦❢ t✐❣❤t ❝♦♥✈❡① r❡❧❛①❛t✐♦♥s t♦ ❝♦♠♣✉t❡ t❤❡ ♦♣t✐♠❛❧ ✈❛❧✉❡ ♦❢ ❛ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ⋆ ●❡♥❣✲◆❛✐r ✬✶✹✿ ❖♣t✐♠❛❧✐t② ♦❢ R(Wa, Wb) ❢♦r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ ❚❡❝❤♥✐q✉❡ ❢♦r ❡st❛❜❧✐s❤✐♥❣ ❡①tr❡♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥s ✉s✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♦❢ ❢✉♥❝t✐♦♥❛❧s

✶✵

slide-37
SLIDE 37

✸✳ ■♥t❡r❢❡r❡♥❝❡ ❈❤❛♥♥❡❧ ✭❆❤❧s✇❡❞❡ ✬✼✹✮

❈r❡❞✐t✿ ✇✇✇✳♣❡rs♦♥❛❧✳♣s✉✳❡❞✉✴❜①❣✷✶✺✴r❡s❡❛r❝❤✳❤t♠❧

M1 M2 ❊♥❝♦❞❡r ✶ ❊♥❝♦❞❡r ✷ Xn

1

Xn

2

Wb(y2|x1, x2) Wa(y1|x1, x2) Y n

1

Y n

2

❉❡❝♦❞❡r ✶ ❉❡❝♦❞❡r ✷ ˆ M1 ˆ M2

✶✶

slide-38
SLIDE 38

✸✳ ■♥t❡r❢❡r❡♥❝❡ ❈❤❛♥♥❡❧ ✭❆❤❧s✇❡❞❡ ✬✼✹✮

❍❛♥ ❑♦❜❛②❛s❤✐

❙✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣✱ ♠❡ss❛❣❡ s♣❧✐tt✐♥❣✱ ❝♦❞❡❞ t✐♠❡✲s❤❛r✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ ♣❛✐rs (R1, R2) t❤❛t s❛t✐s❢②

R1 < I(X1; Y1|U2, Q), R2 < I(X2; Y2|U1, Q), R1 + R2 < I(X1, U2; Y1|Q) + I(X2; Y2|U1, U2, Q), R1 + R2 < I(X2, U1; Y2|Q) + I(X1; Y1|U1, U2, Q), R1 + R2 < I(X1, U2; Y1|U1, Q) + I(X2, U1; Y2|U2, Q), 2R1 + R2 < I(X1, U2; Y1|Q) + I(X1; Y1|U1, U2, Q) + I(X2, U1; Y2|U2, Q), R1 + 2R2 < I(X2, U1; Y2|Q) + I(X2; Y2|U1, U2, Q) + I(X1, U2; Y1|U1, Q)

❢♦r s♦♠❡ ♣♠❢ p(q)p(u1, x1|q)p(u2, x2|q)✱ ✇❤❡r❡ |U1| ≤ |X1| + 4✱ |U2| ≤ |X2| + 4✱ ❛♥❞ |Q| ≤ 7✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ R(Wa, Wb)✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥❄

✶✶

slide-39
SLIDE 39

✸✳ ■♥t❡r❢❡r❡♥❝❡ ❈❤❛♥♥❡❧ ✭❆❤❧s✇❡❞❡ ✬✼✹✮

❍❛♥ ❑♦❜❛②❛s❤✐

❙✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣✱ ♠❡ss❛❣❡ s♣❧✐tt✐♥❣✱ ❝♦❞❡❞ t✐♠❡✲s❤❛r✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ ♣❛✐rs (R1, R2) t❤❛t s❛t✐s❢②

R1 < I(X1; Y1|U2, Q), R2 < I(X2; Y2|U1, Q), R1 + R2 < I(X1, U2; Y1|Q) + I(X2; Y2|U1, U2, Q), R1 + R2 < I(X2, U1; Y2|Q) + I(X1; Y1|U1, U2, Q), R1 + R2 < I(X1, U2; Y1|U1, Q) + I(X2, U1; Y2|U2, Q), 2R1 + R2 < I(X1, U2; Y1|Q) + I(X1; Y1|U1, U2, Q) + I(X2, U1; Y2|U2, Q), R1 + 2R2 < I(X2, U1; Y2|Q) + I(X2; Y2|U1, U2, Q) + I(X1, U2; Y1|U1, Q)

❢♦r s♦♠❡ ♣♠❢ p(q)p(u1, x1|q)p(u2, x2|q)✱ ✇❤❡r❡ |U1| ≤ |X1| + 4✱ |U2| ≤ |X2| + 4✱ ❛♥❞ |Q| ≤ 7✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ R(Wa, Wb)✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥❄

✶✶

slide-40
SLIDE 40

✸✳ ■♥t❡r❢❡r❡♥❝❡ ❈❤❛♥♥❡❧ ✭❆❤❧s✇❡❞❡ ✬✼✹✮

❍❛♥ ❑♦❜❛②❛s❤✐

❙✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣✱ ♠❡ss❛❣❡ s♣❧✐tt✐♥❣✱ ❝♦❞❡❞ t✐♠❡✲s❤❛r✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❛❝❤✐❡✈❡ r❛t❡ ♣❛✐rs (R1, R2) t❤❛t s❛t✐s❢②

R1 < I(X1; Y1|U2, Q), R2 < I(X2; Y2|U1, Q), R1 + R2 < I(X1, U2; Y1|Q) + I(X2; Y2|U1, U2, Q), R1 + R2 < I(X2, U1; Y2|Q) + I(X1; Y1|U1, U2, Q), R1 + R2 < I(X1, U2; Y1|U1, Q) + I(X2, U1; Y2|U2, Q), 2R1 + R2 < I(X1, U2; Y1|Q) + I(X1; Y1|U1, U2, Q) + I(X2, U1; Y2|U2, Q), R1 + 2R2 < I(X2, U1; Y2|Q) + I(X2; Y2|U1, U2, Q) + I(X1, U2; Y1|U1, Q)

❢♦r s♦♠❡ ♣♠❢ p(q)p(u1, x1|q)p(u2, x2|q)✱ ✇❤❡r❡ |U1| ≤ |X1| + 4✱ |U2| ≤ |X2| + 4✱ ❛♥❞ |Q| ≤ 7✳ ❈❛❧❧ t❤✐s r❡❣✐♦♥ R(Wa, Wb)✳ ◗✉❡st✐♦♥✿ ■s t❤✐s t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥❄ ❍❛❞ ❜❡❡♥ ♦♣❡♥ ✭s✐♥❝❡ ❍❛♥ ❛♥❞ ❑♦❜❛②❛s❤✐ ✬✽✶✮

✶✶

slide-41
SLIDE 41

❙✉❝❝❡ss❡s

■♥ s♣✐t❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♦♥✲❝♦♥✈❡①

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ❢♦r s♦♠❡ ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s

⋆ ❈❛r❧❡✐❛❧ ✬✼✺✱ ❙❛t♦ ✬✽✶✱ ❊❧ ●❛♠❛❧ ❛♥❞ ❈♦st❛ ✭✬✽✶ ❛♥❞ ✬✽✻✮

  • R(Wa, Wb) ✐s ❝❧♦s❡ t♦ ♦♣t✐♠❛❧ ❢♦r ●❛✉ss✐❛♥ ■♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧

⋆ ❊t❦✐♥ ❛♥❞ ❚s❡ ❛♥❞ ❲❛♥❣ ✭✬✵✾✮

  • ◆♦✈❡❧ ✐❞❡❛s ❛♥❞ ♠❛t❤❡♠❛t✐❝❛❧ r❡s✉❧ts ❝❛♠❡ ♦✉t ❢r♦♠ t❤❡ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢

♦♣t✐♠❛❧✐t② ⋆ ❈♦♥❝❛✈✐t② ♦❢ ❡♥tr♦♣② ♣♦✇❡r ✭❈♦st❛ ✬✽✺✮ ⋆ ●❡♥✐❡ ❜❛s❡❞ ❛♣♣r♦❛❝❤ t♦ ♣r♦✈❡ s✉❜✲❛❞❞✐t✐✈✐t② ✭❊❧ ●❛♠❛❧ ❛♥❞ ❈♦st❛ ✬✽✶✱ ❑r❛♠❡r ✬✵✷✮ ✐s ♥♦t ♦♣t✐♠❛❧ ✐♥ ❣❡♥❡r❛❧ ✭◆❛✐r✱ ❳✐❛✱ ❨❛③❞❛♥♣❛♥❛❤ ✬✶✺✮ ❇r♦❛❞❝❛st ❛♥❞ ✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧s ❛r❡ ❢❛r t♦♦ ✐♠♣♦rt❛♥t ❚♦ ❧❡t ♥♦♥✲❝♦♥✈❡①✐t② ❞✐ss✉❛❞❡ ✉s ❚♦ ♥♦t ✐♥✈❡st✐❣❛t❡ s✐♠♣❧❡ ❝❧❛ss❡s t❤❛t r❡q✉✐r❡ ♥❡✇ ✐❞❡❛s

✶✷

slide-42
SLIDE 42

❙✉❝❝❡ss❡s

■♥ s♣✐t❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♦♥✲❝♦♥✈❡①

  • R(Wa, Wb) ✐s ♦♣t✐♠❛❧ ❢♦r s♦♠❡ ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s

⋆ ❈❛r❧❡✐❛❧ ✬✼✺✱ ❙❛t♦ ✬✽✶✱ ❊❧ ●❛♠❛❧ ❛♥❞ ❈♦st❛ ✭✬✽✶ ❛♥❞ ✬✽✻✮

  • R(Wa, Wb) ✐s ❝❧♦s❡ t♦ ♦♣t✐♠❛❧ ❢♦r ●❛✉ss✐❛♥ ■♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧

⋆ ❊t❦✐♥ ❛♥❞ ❚s❡ ❛♥❞ ❲❛♥❣ ✭✬✵✾✮

  • ◆♦✈❡❧ ✐❞❡❛s ❛♥❞ ♠❛t❤❡♠❛t✐❝❛❧ r❡s✉❧ts ❝❛♠❡ ♦✉t ❢r♦♠ t❤❡ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢

♦♣t✐♠❛❧✐t② ⋆ ❈♦♥❝❛✈✐t② ♦❢ ❡♥tr♦♣② ♣♦✇❡r ✭❈♦st❛ ✬✽✺✮ ⋆ ●❡♥✐❡ ❜❛s❡❞ ❛♣♣r♦❛❝❤ t♦ ♣r♦✈❡ s✉❜✲❛❞❞✐t✐✈✐t② ✭❊❧ ●❛♠❛❧ ❛♥❞ ❈♦st❛ ✬✽✶✱ ❑r❛♠❡r ✬✵✷✮

  • R(Wa, Wb) ✐s ♥♦t ♦♣t✐♠❛❧ ✐♥ ❣❡♥❡r❛❧ ✭◆❛✐r✱ ❳✐❛✱ ❨❛③❞❛♥♣❛♥❛❤ ✬✶✺✮

❇r♦❛❞❝❛st ❛♥❞ ✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧s ❛r❡ ❢❛r t♦♦ ✐♠♣♦rt❛♥t

  • ❚♦ ❧❡t ♥♦♥✲❝♦♥✈❡①✐t② ❞✐ss✉❛❞❡ ✉s
  • ❚♦ ♥♦t ✐♥✈❡st✐❣❛t❡ s✐♠♣❧❡ ❝❧❛ss❡s t❤❛t r❡q✉✐r❡ ♥❡✇ ✐❞❡❛s

✶✷

slide-43
SLIDE 43

❆ ❝❧❛ss ♦❢ ♦♣❡♥ ♣r♦❜❧❡♠s

❆ s✉❜✲❝♦❧❧❡❝t✐♦♥ ♦❢ t❤❡ ✶✺ ♦♣❡♥ ♣r♦❜❧❡♠s ❧✐st❡❞ ✐♥ ❈❤❛♣s✳ ✺✲✾✳

✺✳✶ ❲❤❛t ✐s t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ♦❢ ❧❡ss ♥♦✐s② ❜r♦❛❞❝❛st✲❝❤❛♥♥❡❧s ✇✐t❤ ❢♦✉r ♦r ♠♦r❡ r❡❝❡✐✈❡rs❄ ✭t✇♦✲r❡❝❡✐✈❡r✿ ❑♦r♥❡r✲▼❛rt♦♥ ✬✼✻✱ t❤r❡❡✲r❡❝❡✐✈❡r✿ ◆❛✐r✲❲❛♥❣ ✬✶✵✮ ✺✳✷ ❲❤❛t ✐s t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ♦❢ ♠♦r❡ ❝❛♣❛❜❧❡ ❜r♦❛❞❝❛st✲❝❤❛♥♥❡❧s ✇✐t❤ t❤r❡❡ ♦r ♠♦r❡ r❡❝❡✐✈❡rs❄ ✭t✇♦✲r❡❝❡✐✈❡r✿ ❊❧ ●❛♠❛❧ ✬✼✾✮ ✻✳✶ ❲❤❛t ✐s t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ♦❢ t❤❡ ●❛✉ss✐❛♥ ■♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧ ✇✐t❤ ✇❡❛❦ ✐♥t❡r❢❡r❡♥❝❡❄ ✭str♦♥❣✲✐♥t❡r❢❡r❡♥❝❡✿ ❙❛t♦ ✬✼✾❀ ♠✐①❡❞✲✐♥t❡r❢❡r❡♥❝❡ ❝♦r♥❡r✲♣♦✐♥ts✿ ❙❛t♦✬ ✽✶✱ ❈♦st❛✬✽✺❀ ✇❡❛❦✲✐♥t❡r❢❡r❡♥❝❡ ❝♦r♥❡r✲♣♦✐♥ts✿ r❛t❡✲s✉♠ ✭♣❛rt✐❛❧✮✿ t❤r❡❡✲❣r♦✉♣s ✬✵✾ ✮ ✻✳✹ ■s t❤❡ ❍❛♥✲❑♦❜❛②❛s❤✐ ✐♥♥❡r ❜♦✉♥❞ t✐❣❤t ✐♥ ❣❡♥❡r❛❧ ❢♦r ✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧s❄ ✽✳✷ ■s s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ♦♣t✐♠❛❧ ❢♦r t❤❡ ❣❡♥❡r❛❧ ✸✲r❡❝❡✐✈❡r ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✇✐t❤ ♦♥❡ ♠❡ss❛❣❡ t♦ ❛❧❧ t❤r❡❡ r❡❝❡✐✈❡rs ❛♥❞ ❛♥♦t❤❡r ♠❡ss❛❣❡ t♦ t✇♦ r❡❝❡✐✈❡rs❄ ✽✳✸ ❲❤❛t ✐s t❤❡ s✉♠✲❝❛♣❛❝✐t② ♦❢ t❤❡ ❜✐♥❛r② s❦❡✇✲s②♠♠❡tr✐❝ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❄ ✽✳✹ ■s t❤❡ ▼❛rt♦♥ ✐♥♥❡r ❜♦✉♥❞ t✐❣❤t ✐♥ ❣❡♥❡r❛❧ ❢♦r ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s❄ ✾✳✷ ❈❛♥ t❤❡ ❝♦♥✈❡rs❡ ❢♦r t❤❡ ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ❜❡ ♣r♦✈❡❞ ❞✐r❡❝t❧② ❜② ♦♣t✐♠✐③✐♥❣ t❤❡ ◆❛✐r✲❊❧ ●❛♠❛❧ ♦✉t❡r ❜♦✉♥❞❄ ✾✳✸ ❲❤❛t ✐s t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ♦❢ t❤❡ ✷✲r❡❝❡✐✈❡r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✇✐t❤ ❝♦♠♠♦♥ ♠❡ss❛❣❡❄

✶✸

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SLIDE 44

❆ ❝❧❛ss ♦❢ ♦♣❡♥ ♣r♦❜❧❡♠s

▼② r❡❢♦r♠✉❧❛t✐♦♥s ♦❢ ❛ ❢❡✇ ♦❢ t❤❡♠✳

✺✳✶ ■s s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ♦♣t✐♠❛❧ ❢♦r ❧❡ss✲♥♦✐s② ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s ✇✐t❤ ❢♦✉r ♦r ♠♦r❡ r❡❝❡✐✈❡rs❄ ✺✳✷ ■s s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ♦♣t✐♠❛❧ ❢♦r ♠♦r❡✲❝❛♣❛❜❧❡ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s ✇✐t❤ t❤r❡❡ ♦r ♠♦r❡ r❡❝❡✐✈❡rs❄ ✻✳✶ ■s t❤❡ ❍❛♥✕❑♦❜❛②❛s❤✐ s❝❤❡♠❡ ✇✐t❤ ●❛✉ss✐❛♥ s✐❣♥❛❧✐♥❣ t✐❣❤t ❢♦r t❤❡ ●❛✉ss✐❛♥ ■♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧ ✇✐t❤ ✇❡❛❦ ✐♥t❡r❢❡r❡♥❝❡❄ ✻✳✹ ■s t❤❡ ❍❛♥✲❑♦❜❛②❛s❤✐ ✐♥♥❡r ❜♦✉♥❞ t✐❣❤t ✐♥ ❣❡♥❡r❛❧ ❢♦r ✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧s❄ ✽✳✷ ■s s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ♦♣t✐♠❛❧ ❢♦r t❤❡ ❣❡♥❡r❛❧ ✸✲r❡❝❡✐✈❡r ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✇✐t❤ ♦♥❡ ♠❡ss❛❣❡ t♦ ❛❧❧ t❤r❡❡ r❡❝❡✐✈❡rs ❛♥❞ ❛♥♦t❤❡r ♠❡ss❛❣❡ t♦ t✇♦ r❡❝❡✐✈❡rs❄ ✽✳✸ ❉♦❡s t❤❡ ▼❛rt♦♥ ✐♥♥❡r ❜♦✉♥❞ ❛❝❤✐❡✈❡ t❤❡ s✉♠✲❝❛♣❛❝✐t② ♦❢ t❤❡ ❜✐♥❛r② s❦❡✇✲s②♠♠❡tr✐❝ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❄ ✽✳✹ ■s t❤❡ ▼❛rt♦♥ ✐♥♥❡r ❜♦✉♥❞ t✐❣❤t ✐♥ ❣❡♥❡r❛❧ ❢♦r ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s❄ ✾✳✷ ❈❛♥ t❤❡ ❝♦♥✈❡rs❡ ❢♦r t❤❡ ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ❜❡ ♣r♦✈❡❞ ❞✐r❡❝t❧② ❜② ♦♣t✐♠✐③✐♥❣ t❤❡ ◆❛✐r✲❊❧ ●❛♠❛❧ ♦✉t❡r ❜♦✉♥❞❄ ✾✳✸ ❉♦❡s t❤❡ ▼❛rt♦♥ ✐♥♥❡r ❜♦✉♥❞ ❛❝❤✐❡✈❡ t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ♦❢ t❤❡ ✷✲r❡❝❡✐✈❡r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✇✐t❤ ❝♦♠♠♦♥ ♠❡ss❛❣❡❄

✶✸

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SLIDE 45

❚❤❡ ❝♦♠♠♦♥ t❤❡♠❡ t♦ t❤❡s❡ ✭r❡❢♦r♠✉❧❛t❡❞✮ q✉❡st✐♦♥s

❈♦♠♠♦♥ t❤❡♠❡ ■s ❛ ❝❛♥❞✐❞❛t❡ r❛t❡ r❡❣✐♦♥ ♦♣t✐♠❛❧❄ ■❞❡❛ ❢♦r t❡st✐♥❣ ♦♣t✐♠❛❧✐t②✿

  • Sλ(W ⊗ W) ?

= 2Sλ(W)

  • ❉❡t❡r♠✐♥❡ s✉❜✲❛❞❞✐t✐✈✐t② ♦❢ ❛♥ ❛ss♦❝✐❛t❡❞ ♥♦♥✲❝♦♥✈❡① ❢✉♥❝t✐♦♥❛❧

✶✹

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SLIDE 46

❙t❛t✉s ♦❢ t❤❡ ♦♣❡♥ ♣r♦❜❧❡♠s

✺✳✶ ■s s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ♦♣t✐♠❛❧ ❢♦r ❧❡ss✲♥♦✐s② ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s ✇✐t❤ ❢♦✉r ♦r ♠♦r❡ r❡❝❡✐✈❡rs❄✭❖P❊◆✮ ✺✳✷ ■s s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ♦♣t✐♠❛❧ ❢♦r ♠♦r❡✲❝❛♣❛❜❧❡ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s ✇✐t❤ t❤r❡❡ ♦r ♠♦r❡ r❡❝❡✐✈❡rs❄ ✭◆❖✿ ◆❛✐r✲❳✐❛ ✬✶✷✮ ✻✳✶ ■s t❤❡ ❍❛♥✕❑♦❜❛②❛s❤✐ s❝❤❡♠❡ ✇✐t❤ ●❛✉ss✐❛♥ s✐❣♥❛❧✐♥❣ t✐❣❤t ❢♦r t❤❡ ●❛✉ss✐❛♥ ■♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧ ✇✐t❤ ✇❡❛❦ ✐♥t❡r❢❡r❡♥❝❡❄✭❖P❊◆✮ ✭❨❊❙✿ ❝♦r♥❡r ♣♦✐♥ts ✉s✐♥❣ ✐❞❡❛s ✐♥ ♠❡❛s✉r❡ tr❛♥s♣♦rt❛t✐♦♥ ❜② P♦❧②❛♥s❦✐②✲❲✉ ✬✶✺✮ ✻✳✹ ■s t❤❡ ❍❛♥✲❑♦❜❛②❛s❤✐ ✐♥♥❡r ❜♦✉♥❞ t✐❣❤t ✐♥ ❣❡♥❡r❛❧ ❢♦r ✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧s❄ ✭◆❖✿ ◆❛✐r✲❳✐❛✲❨❛③❞❛♥♣❛♥❛❤ ✬✶✺✮ ✽✳✷ ■s s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ♦♣t✐♠❛❧ ❢♦r t❤❡ ❣❡♥❡r❛❧ ✸✲r❡❝❡✐✈❡r ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✇✐t❤ ♦♥❡ ♠❡ss❛❣❡ t♦ ❛❧❧ t❤r❡❡ r❡❝❡✐✈❡rs ❛♥❞ ❛♥♦t❤❡r ♠❡ss❛❣❡ t♦ t✇♦ r❡❝❡✐✈❡rs❄ ✭◆❖✿ ◆❛✐r✲❨❛③❞❛♥♣❛♥❛❤ ✬✶✼✮ ✽✳✸ ❉♦❡s t❤❡ ▼❛rt♦♥ ✐♥♥❡r ❜♦✉♥❞ ❛❝❤✐❡✈❡ t❤❡ s✉♠✲❝❛♣❛❝✐t② ♦❢ t❤❡ ❜✐♥❛r② s❦❡✇✲s②♠♠❡tr✐❝ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❄✭❖P❊◆✮ ✽✳✹ ■s t❤❡ ▼❛rt♦♥ ✐♥♥❡r ❜♦✉♥❞ t✐❣❤t ✐♥ ❣❡♥❡r❛❧ ❢♦r ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s❄✭❖P❊◆✮ ✾✳✷ ❈❛♥ t❤❡ ❝♦♥✈❡rs❡ ❢♦r t❤❡ ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ❜❡ ♣r♦✈❡❞ ❞✐r❡❝t❧② ❜② ♦♣t✐♠✐③✐♥❣ t❤❡ ◆❛✐r✲❊❧ ●❛♠❛❧ ♦✉t❡r ❜♦✉♥❞❄✭❨❊❙✿ ●❡♥❣✲◆❛✐r ✬✶✹✮ ✾✳✸ ❉♦❡s t❤❡ ▼❛rt♦♥ ✐♥♥❡r ❜♦✉♥❞ ❛❝❤✐❡✈❡ t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ♦❢ t❤❡ ✷✲r❡❝❡✐✈❡r ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧ ✇✐t❤ ❝♦♠♠♦♥ ♠❡ss❛❣❡❄✭❨❊❙✿ ●❡♥❣✲◆❛✐r ✬✶✹✮

✶✺

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SLIDE 47

❖✉t❧✐♥❡

  • ❇r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ ❊st❛❜❧✐s❤✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ▼❛rt♦♥✬s r❡❣✐♦♥ ❢♦r ▼■▼❖

❜r♦❛❞❝❛st ❝❤❛♥♥❡❧

  • ■♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧✿ ❙✉❜✲♦♣t✐♠❛❧✐t② ♦❢ t❤❡ ❍❛♥✕❑♦❜❛②❛s❤✐ r❡❣✐♦♥
  • ❋❛♠✐❧② ♦❢ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s

⋆ ❘❡❧❛t✐♦♥ t♦ ♣r♦❜❧❡♠s ♦❢ ✐♥t❡r❡st ✐♥ ♦t❤❡r ✜❡❧❞s ⋆ ❯♥✐❢②✐♥❣ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ s♦♠❡ ❝♦♥❥❡❝t✉r❡s

✶✻

slide-48
SLIDE 48

▼■▼❖ ✭❱❡❝t♦r✮ ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧

(M0, M1, M2) ❊♥❝♦❞❡r Xn Wa(y1|x) Wb(y2|x) Y n

1

Y n

2

❉❡❝♦❞❡r ✶ ❉❡❝♦❞❡r ✷ ˆ M0, ˆ M1 ˜ M0, ˜ M2 ▼■▼❖ ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ❛❞❞✐t✐✈❡ ♥♦✐s❡✳ ❱❡r② ✐♠♣♦rt❛♥t ❝❤❛♥♥❡❧ ❝❧❛ss ✐♥ ✇✐r❡❧❡ss ❝♦♠♠✉♥✐❝❛t✐♦♥ ▼♦❞❡❧s✿ ♠✉❧t✐✲❛♥t❡♥♥❛ tr❛♥s♠✐tt❡r✴r❡❝❡✐✈❡rs ✭❞♦✇♥❧✐♥❦✮

✶✼

slide-49
SLIDE 49

▼■▼❖ ✭❱❡❝t♦r✮ ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧

(M0, M1, M2) ❊♥❝♦❞❡r Xn Wa(y1|x) Wb(y2|x) Y n

1

Y n

2

❉❡❝♦❞❡r ✶ ❉❡❝♦❞❡r ✷ ˆ M0, ˆ M1 ˜ M0, ˜ M2 ▼■▼❖ ●❛✉ss✐❛♥ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ Y1 = AX + Z Y2 = BX + Z ✇❤❡r❡ Z ∼ N(0, I) ❞❡♥♦t❡s t❤❡ ❛❞❞✐t✐✈❡ ♥♦✐s❡✳ ❱❡r② ✐♠♣♦rt❛♥t ❝❤❛♥♥❡❧ ❝❧❛ss ✐♥ ✇✐r❡❧❡ss ❝♦♠♠✉♥✐❝❛t✐♦♥ ▼♦❞❡❧s✿ ♠✉❧t✐✲❛♥t❡♥♥❛ tr❛♥s♠✐tt❡r✴r❡❝❡✐✈❡rs ✭❞♦✇♥❧✐♥❦✮

✶✼

slide-50
SLIDE 50

❍✐st♦r②

❖♣t✐♠❛❧✐t② ♦❢ ▼❛rt♦♥✬s ❜♦✉♥❞✱ R(Wa, Wb)✱ ✇❛s ❡st❛❜❧✐s❤❡❞✿

  • ❙❝❛❧❛r ❝❛s❡ ✭❇❡r❣♠❛♥s ✬✼✸✮ ✭❊♥tr♦♣② P♦✇❡r ■♥❡q✉❛❧✐t②✮
  • ❘❡✈❡rs❡❧② ❞❡❣r❛❞❡❞ s❡tt✐♥❣ ✭P♦❧t②r❡✈ ✬✼✽✱ ❊❧ ●❛♠❛❧ ✬✽✶✮
  • ❖♣t✐♠❛❧✐t② ♦♥ R0 = 0 ✭❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❤❛♠❛✐ ✬✵✻✮

⋆ ❇✉✐❧❞s ♦♥ ✐❞❡❛s ✐♥ P♦❧t②r❡✈ ⋆ ❚♦✉r ❞❡ ❢♦r❝❡ ✐♥ ♦♣t✐♠✐③❛t✐♦♥

  • ✐st ✿ ❙❤♦✇✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❢♦r ❛ ♥♦♥✲❝♦♥✈❡①

♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❘❡♠❛r❦✿ ■❞❡❛s ❞♦ ♥♦t ❡①t❡♥❞ t♦ s❤♦✇ ♦♣t✐♠❛❧✐t② ✇❤❡♥ t❤❡r❡ ✐s ❝♦♠♠♦♥ ♠❡ss❛❣❡✱ ✐✳❡✳ ❖♣t✐♠❛❧✐t② ✐♥ ❣❡♥❡r❛❧ ✭●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹✮

  • ✐st ✿ ❉❡✈❡❧♦♣ ❛ t❡❝❤♥✐q✉❡ ❢♦r ♣r♦✈✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ r❛♥❞♦♠

✈❛r✐❛❜❧❡s ✭❢r♦♠ s✉❜✲❛❞❞✐t✐✈✐t②✮ ❊①♣❧❛✐♥ ♦✉r t❡❝❤♥✐q✉❡ ♦♥ ✭❢♦r s✐♠♣❧✐❝✐t②✮

✶✽

slide-51
SLIDE 51

❍✐st♦r②

❖♣t✐♠❛❧✐t② ♦❢ ▼❛rt♦♥✬s ❜♦✉♥❞✱ R(Wa, Wb)✱ ✇❛s ❡st❛❜❧✐s❤❡❞✿

  • ❙❝❛❧❛r ❝❛s❡ ✭❇❡r❣♠❛♥s ✬✼✸✮ ✭❊♥tr♦♣② P♦✇❡r ■♥❡q✉❛❧✐t②✮
  • ❘❡✈❡rs❡❧② ❞❡❣r❛❞❡❞ s❡tt✐♥❣ ✭P♦❧t②r❡✈ ✬✼✽✱ ❊❧ ●❛♠❛❧ ✬✽✶✮
  • ❖♣t✐♠❛❧✐t② ♦♥ R0 = 0 ✭❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❤❛♠❛✐ ✬✵✻✮

⋆ ❇✉✐❧❞s ♦♥ ✐❞❡❛s ✐♥ P♦❧t②r❡✈ ⋆ ❚♦✉r ❞❡ ❢♦r❝❡ ✐♥ ♦♣t✐♠✐③❛t✐♦♥ ⋆ ●✐st ✿ ❙❤♦✇✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❢♦r ❛ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ⋆ ❘❡♠❛r❦✿ ■❞❡❛s ❞♦ ♥♦t ❡①t❡♥❞ t♦ s❤♦✇ ♦♣t✐♠❛❧✐t② ✇❤❡♥ t❤❡r❡ ✐s ❝♦♠♠♦♥ ♠❡ss❛❣❡✱ ✐✳❡✳ R0 = 0 ❖♣t✐♠❛❧✐t② ✐♥ ❣❡♥❡r❛❧ ✭●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹✮

  • ✐st ✿ ❉❡✈❡❧♦♣ ❛ t❡❝❤♥✐q✉❡ ❢♦r ♣r♦✈✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ r❛♥❞♦♠

✈❛r✐❛❜❧❡s ✭❢r♦♠ s✉❜✲❛❞❞✐t✐✈✐t②✮ ❊①♣❧❛✐♥ ♦✉r t❡❝❤♥✐q✉❡ ♦♥ ✭❢♦r s✐♠♣❧✐❝✐t②✮

✶✽

slide-52
SLIDE 52

❍✐st♦r②

❖♣t✐♠❛❧✐t② ♦❢ ▼❛rt♦♥✬s ❜♦✉♥❞✱ R(Wa, Wb)✱ ✇❛s ❡st❛❜❧✐s❤❡❞✿

  • ❙❝❛❧❛r ❝❛s❡ ✭❇❡r❣♠❛♥s ✬✼✸✮ ✭❊♥tr♦♣② P♦✇❡r ■♥❡q✉❛❧✐t②✮
  • ❘❡✈❡rs❡❧② ❞❡❣r❛❞❡❞ s❡tt✐♥❣ ✭P♦❧t②r❡✈ ✬✼✽✱ ❊❧ ●❛♠❛❧ ✬✽✶✮
  • ❖♣t✐♠❛❧✐t② ♦♥ R0 = 0 ✭❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❤❛♠❛✐ ✬✵✻✮

⋆ ❇✉✐❧❞s ♦♥ ✐❞❡❛s ✐♥ P♦❧t②r❡✈ ⋆ ❚♦✉r ❞❡ ❢♦r❝❡ ✐♥ ♦♣t✐♠✐③❛t✐♦♥ ⋆ ●✐st ✿ ❙❤♦✇✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❢♦r ❛ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ⋆ ❘❡♠❛r❦✿ ■❞❡❛s ❞♦ ♥♦t ❡①t❡♥❞ t♦ s❤♦✇ ♦♣t✐♠❛❧✐t② ✇❤❡♥ t❤❡r❡ ✐s ❝♦♠♠♦♥ ♠❡ss❛❣❡✱ ✐✳❡✳ R0 = 0

  • ❖♣t✐♠❛❧✐t② ✐♥ ❣❡♥❡r❛❧ ✭●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹✮

⋆ ●✐st ✿ ❉❡✈❡❧♦♣ ❛ t❡❝❤♥✐q✉❡ ❢♦r ♣r♦✈✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✭❢r♦♠ s✉❜✲❛❞❞✐t✐✈✐t②✮ ❊①♣❧❛✐♥ ♦✉r t❡❝❤♥✐q✉❡ ♦♥ ✭❢♦r s✐♠♣❧✐❝✐t②✮

✶✽

slide-53
SLIDE 53

❍✐st♦r②

❖♣t✐♠❛❧✐t② ♦❢ ▼❛rt♦♥✬s ❜♦✉♥❞✱ R(Wa, Wb)✱ ✇❛s ❡st❛❜❧✐s❤❡❞✿

  • ❙❝❛❧❛r ❝❛s❡ ✭❇❡r❣♠❛♥s ✬✼✸✮ ✭❊♥tr♦♣② P♦✇❡r ■♥❡q✉❛❧✐t②✮
  • ❘❡✈❡rs❡❧② ❞❡❣r❛❞❡❞ s❡tt✐♥❣ ✭P♦❧t②r❡✈ ✬✼✽✱ ❊❧ ●❛♠❛❧ ✬✽✶✮
  • ❖♣t✐♠❛❧✐t② ♦♥ R0 = 0 ✭❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❤❛♠❛✐ ✬✵✻✮

⋆ ❇✉✐❧❞s ♦♥ ✐❞❡❛s ✐♥ P♦❧t②r❡✈ ⋆ ❚♦✉r ❞❡ ❢♦r❝❡ ✐♥ ♦♣t✐♠✐③❛t✐♦♥ ⋆ ●✐st ✿ ❙❤♦✇✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❢♦r ❛ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ⋆ ❘❡♠❛r❦✿ ■❞❡❛s ❞♦ ♥♦t ❡①t❡♥❞ t♦ s❤♦✇ ♦♣t✐♠❛❧✐t② ✇❤❡♥ t❤❡r❡ ✐s ❝♦♠♠♦♥ ♠❡ss❛❣❡✱ ✐✳❡✳ R0 = 0

  • ❖♣t✐♠❛❧✐t② ✐♥ ❣❡♥❡r❛❧ ✭●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹✮

⋆ ●✐st ✿ ❉❡✈❡❧♦♣ ❛ t❡❝❤♥✐q✉❡ ❢♦r ♣r♦✈✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✭❢r♦♠ s✉❜✲❛❞❞✐t✐✈✐t②✮ ❊①♣❧❛✐♥ ♦✉r t❡❝❤♥✐q✉❡ ♦♥ R0 = 0 ✭❢♦r s✐♠♣❧✐❝✐t②✮

✶✽

slide-54
SLIDE 54

❖✉t❡r ❜♦✉♥❞ ✭❑♦r♥❡r✲▼❛rt♦♥ ✬✼✾✮

❚❤❡ s❡t ♦❢ r❛t❡ ♣❛✐rs (R1, R2) s❛t✐s❢②✐♥❣ R2 ≤ I(U; Y2) R1 + R2 ≤ I(U; Y2) + I(X; Y1|U) ❢♦r s♦♠❡ p(u, x)✱ ✇❤❡r❡ E(X2) ≤ P ❢♦r♠s ❛♥ ♦✉t❡r ❜♦✉♥❞ t♦ t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥✳ ❉❡♥♦t❡ t❤✐s r❡❣✐♦♥ ❛s O(Wa, Wb)✳ ❋♦r ✱ ❧❡t ✭◆❛✐r ✬✶✸✮

✉♣♣❡r ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡✿

◆♦t ❡❛s② t♦ ❝♦♠♣✉t❡ ✭✐♥ ❣❡♥❡r❛❧✮

✶✾

slide-55
SLIDE 55

❖✉t❡r ❜♦✉♥❞ ✭❑♦r♥❡r✲▼❛rt♦♥ ✬✼✾✮

❚❤❡ s❡t ♦❢ r❛t❡ ♣❛✐rs (R1, R2) s❛t✐s❢②✐♥❣ R2 ≤ I(U; Y2) R1 + R2 ≤ I(U; Y2) + I(X; Y1|U) ❢♦r s♦♠❡ p(u, x)✱ ✇❤❡r❡ E(X2) ≤ P ❢♦r♠s ❛♥ ♦✉t❡r ❜♦✉♥❞ t♦ t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥✳ ❉❡♥♦t❡ t❤✐s r❡❣✐♦♥ ❛s O(Wa, Wb)✳ ❋♦r λ > 1✱ ❧❡t Sλ(Wa, Wb) := max

(R1,R2)∈O R1 + λR2

= max

p(u,x) λI(U; Y2) + I(X; Y1|U)

✭◆❛✐r ✬✶✸✮

✉♣♣❡r ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡✿

◆♦t ❡❛s② t♦ ❝♦♠♣✉t❡ ✭✐♥ ❣❡♥❡r❛❧✮

✶✾

slide-56
SLIDE 56

❖✉t❡r ❜♦✉♥❞ ✭❑♦r♥❡r✲▼❛rt♦♥ ✬✼✾✮

❚❤❡ s❡t ♦❢ r❛t❡ ♣❛✐rs (R1, R2) s❛t✐s❢②✐♥❣ R2 ≤ I(U; Y2) R1 + R2 ≤ I(U; Y2) + I(X; Y1|U) ❢♦r s♦♠❡ p(u, x)✱ ✇❤❡r❡ E(X2) ≤ P ❢♦r♠s ❛♥ ♦✉t❡r ❜♦✉♥❞ t♦ t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥✳ ❉❡♥♦t❡ t❤✐s r❡❣✐♦♥ ❛s O(Wa, Wb)✳ ❋♦r λ > 1✱ ❧❡t Sλ(Wa, Wb) := max

(R1,R2)∈O R1 + λR2

= max

p(u,x) λI(U; Y2) + I(X; Y1|U)

= max

p(x)

  • λI(X; Z) + CµX[I(X; Y ) − λI(X; Z)]
  • ✭◆❛✐r ✬✶✸✮

✉♣♣❡r ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡✿

◆♦t ❡❛s② t♦ ❝♦♠♣✉t❡ ✭✐♥ ❣❡♥❡r❛❧✮

✶✾

slide-57
SLIDE 57

❖✉t❡r ❜♦✉♥❞ ✭❑♦r♥❡r✲▼❛rt♦♥ ✬✼✾✮

❚❤❡ s❡t ♦❢ r❛t❡ ♣❛✐rs (R1, R2) s❛t✐s❢②✐♥❣ R2 ≤ I(U; Y2) R1 + R2 ≤ I(U; Y2) + I(X; Y1|U) ❢♦r s♦♠❡ p(u, x)✱ ✇❤❡r❡ E(X2) ≤ P ❢♦r♠s ❛♥ ♦✉t❡r ❜♦✉♥❞ t♦ t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥✳ ❉❡♥♦t❡ t❤✐s r❡❣✐♦♥ ❛s O(Wa, Wb)✳ ❋♦r λ > 1✱ ❧❡t Sλ(Wa, Wb) := max

(R1,R2)∈O R1 + λR2

= max

p(u,x) λI(U; Y2) + I(X; Y1|U)

= max

p(x)

  • λI(X; Z) + CµX[I(X; Y ) − λI(X; Z)]
  • ✭◆❛✐r ✬✶✸✮

x

f(x)

x

✉♣♣❡r ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡✿ Cx[f]

◆♦t ❡❛s② t♦ ❝♦♠♣✉t❡ ✭✐♥ ❣❡♥❡r❛❧✮

✶✾

slide-58
SLIDE 58

❖♥❡ ❝❛♥ s❤♦✇ t❤❛t ✐❢ ●❛✉ss✐❛♥s ♠❛①✐♠✐③❡ CµX[h(Y1) − λh(Y2)] t❤❡♥ ▼❛rt♦♥✬s ✐♥♥❡r ❜♦✉♥❞ ✐s ♦♣t✐♠❛❧ ✭♦♥ R0 = 0✮ ❍❡r❡ h(X) ✐s t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡♥tr♦♣②✿ h(X) := −

  • f(x) log f(x)dx,

✇❤❡r❡ f(x) ✐s t❤❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦❢ X✳ ❆ s✐♠✐❧❛r ✭♠♦r❡✲✐♥✈♦❧✈❡❞✮ ♣r♦❜❧❡♠ s❤♦✇s ✉♣ ✇❤❡♥ R0 = 0 ❆♥ ✐❞❡♥t✐❝❛❧ t❡❝❤♥✐q✉❡ ✭t♦ t❤❡ ♦♥❡ ■ ❛♠ ❣♦✐♥❣ t♦ ❞❡♠♦♥str❛t❡✮ ❡st❛❜❧✐s❤❡s t❤❛t ❛❧s♦

✷✵

slide-59
SLIDE 59
  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t② ✭●❡♥❣✲◆❛✐r ✬✶✹✮

▼❛①✐♠✐③❡✱ ❢♦r λ > 1✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧ CµX[h(AX + Z) − λh(BX + Z)] ♦✈❡r X : ❊(XXT ) K✱ ✇❤❡r❡ A, B ❛r❡ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐❝❡s ❛♥❞ Z ∼ N(0, I)✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s h(AX∗ + Z) − λh(BX∗ + Z), ✇❤❡r❡ X∗ ∼ N(0, K′) ❢♦r s♦♠❡ K′ K✳ ▲❡♠♠❛✿ ✐s s✉❜✲❛❞❞✐t✐✈❡✳ Pr♦♦❢✿ ❋♦r ❛♥②

✷✶

slide-60
SLIDE 60
  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t② ✭●❡♥❣✲◆❛✐r ✬✶✹✮

▼❛①✐♠✐③❡✱ ❢♦r λ > 1✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧ CµX[h(AX + Z) − λh(BX + Z)] ♦✈❡r X : ❊(XXT ) K✱ ✇❤❡r❡ A, B ❛r❡ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐❝❡s ❛♥❞ Z ∼ N(0, I)✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s h(AX∗ + Z) − λh(BX∗ + Z), ✇❤❡r❡ X∗ ∼ N(0, K′) ❢♦r s♦♠❡ K′ K✳ ▲❡♠♠❛✿ CµX[h(AX + Z) − λh(BX + Z)] ✐s s✉❜✲❛❞❞✐t✐✈❡✳ Pr♦♦❢✿ ❋♦r ❛♥② µX1,X2 h(AX1 + Z1, AX2 + Z2|U) − λh(BX1 + Z1, BX2 + Z2|U) = h(AX1 + Z1|U, AX2 + Z2) − λh(BX1 + Z1|U, AX2 + Z2) + h(AX2 + Z2|U, BX1 + Z1) − λh(BX2 + Z2|U, BX1 + Z1) − (λ − 1)I(AX2 + Z2; BX1 + Z1|U)

✷✶

slide-61
SLIDE 61
  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t② ✭●❡♥❣✲◆❛✐r ✬✶✹✮

▼❛①✐♠✐③❡✱ ❢♦r λ > 1✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧ CµX[h(AX + Z) − λh(BX + Z)] ♦✈❡r X : ❊(XXT ) K✱ ✇❤❡r❡ A, B ❛r❡ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐❝❡s ❛♥❞ Z ∼ N(0, I)✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s h(AX∗ + Z) − λh(BX∗ + Z), ✇❤❡r❡ X∗ ∼ N(0, K′) ❢♦r s♦♠❡ K′ K✳ ▲❡♠♠❛✿ CµX[h(AX + Z) − λh(BX + Z)] ✐s s✉❜✲❛❞❞✐t✐✈❡✳ Pr♦♦❢✿ ❋♦r ❛♥② µX1,X2 CµX1,X2[h(AX1 + Z1, AX2 + Z2) − λh(BX1 + Z1, BX2 + Z2)] ≤ CµX1[h(AX1 + Z1) − λh(BX1 + Z1)] + CµX2[h(AX2 + Z2) − λh(BX2 + Z2)]

✷✶

slide-62
SLIDE 62
  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t② ✭●❡♥❣✲◆❛✐r ✬✶✹✮

▼❛①✐♠✐③❡✱ ❢♦r λ > 1✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧ CµX[h(AX + Z) − λh(BX + Z)] ♦✈❡r X : ❊(XXT ) K✱ ✇❤❡r❡ A, B ❛r❡ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐❝❡s ❛♥❞ Z ∼ N(0, I)✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s h(AX∗ + Z) − λh(BX∗ + Z), ✇❤❡r❡ X∗ ∼ N(0, K′) ❢♦r s♦♠❡ K′ K✳ ▲❡♠♠❛✿ CµX[h(AX + Z) − λh(BX + Z)] ✐s s✉❜✲❛❞❞✐t✐✈❡✳ Pr♦♦❢✿ ❋♦r ❛♥② µX1,X2 h(AX1 + Z1, AX2 + Z2|U) − λh(BX1 + Z1, BX2 + Z2|U) = h(AX1 + Z1|U, AX2 + Z2) − λh(BX1 + Z1|U, AX2 + Z2) + h(AX2 + Z2|U, BX1 + Z1) − λh(BX2 + Z2|U, BX1 + Z1) −(λ − 1)I(AX2 + Z2; BX1 + Z1|U)

✷✶

slide-63
SLIDE 63
  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t②✿ ❝t❞✳✳

▲❡t (U†, X†) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max

µX CµX[h(AX + Z) − λh(BX + Z)] = h(AX† + Z|U†) − λh(BX† + Z|U†).

▲❡t (Xa, Ua) ❛♥❞ (Xb, Ub) ❜❡ ✐✳✐✳❞✳ ❛❝❝♦r❞✐♥❣ t♦ (U†, X†)✳ ◆♦t❡✿ ❚❤✉s✱ ❝♦♥❞✐t✐♦♥❡❞ ♦♥ ✿ ✭❢r♦♠ ❝♦♥str✉❝t✐♦♥✮ ✭❢r♦♠ ♣r♦♦❢ ♦❢ s✉❜✲❛❞❞✐t✐✈✐t②✮ ■♠♣❧✐❡s t❤❛t ❝♦♥❞✐t✐♦♥❡❞ ♦♥ ✿ ❛r❡ ●❛✉ss✐❛♥ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ Pr♦♦❢✿ ❯s✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥s ✭❋♦✉r✐❡r tr❛♥s❢♦r♠s✮ ❚❤✐s t❡❝❤♥✐q✉❡ ❤❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♦t❤❡rs ✐♥ ✈❛r✐♦✉s ♦t❤❡r ✐♥st❛♥❝❡s✳ ◆♦t❡✿ ❚❤❡r❡ ❛r❡ s♦♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ ✇♦r❦ ♦❢ ▲✐❡❜ ❛♥❞ ❇❛rt❤❡ ✭✾✵s✮ ❚❤❡② ❛❧s♦ ✉s❡ r♦t❛t✐♦♥s ✭❜✉t ♥♦t ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮

✷✷

slide-64
SLIDE 64
  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t②✿ ❝t❞✳✳

▲❡t (U†, X†) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max

µX CµX[h(AX + Z) − λh(BX + Z)] = h(AX† + Z|U†) − λh(BX† + Z|U†).

▲❡t (Xa, Ua) ❛♥❞ (Xb, Ub) ❜❡ ✐✳✐✳❞✳ ❛❝❝♦r❞✐♥❣ t♦ (U†, X†)✳ ❙❡tt✐♥❣ U = (Ua, Ub)✱ X+ = Xa+Xb

√ 2

❛♥❞ X− = Xa−Xb

√ 2

t❤❡ ♣r♦♦❢ ♦❢ s✉❜✲❛❞❞✐t✐✈✐t② ②✐❡❧❞s 2V = CµX1,X2[h(AX1 + Z1, AX2 + Z2) − λh(BX1 + Z1, BX2 + Z2)]

  • (µX+,X−)

≤ CµX1[h(AX1 + Z1) − λh(BX1 + Z1)]

  • µX+

+ CµX2[h(AX2 + Z2) − λh(BX2 + Z2)]

  • µX−

−(λ − 1)I(AX− + Z2; BX+ + Z1|Ua, Ub) ≤ V + V ❚❤❡r❡❢♦r❡✿ ✇❡ ❣❡t t❤❛t ❝♦♥❞✐t✐♦♥❡❞ ♦♥ ✿ ✳ ◆♦t❡✿ ❚❤✉s✱ ❝♦♥❞✐t✐♦♥❡❞ ♦♥ ✿ ✭❢r♦♠ ❝♦♥str✉❝t✐♦♥✮ ✭❢r♦♠ ♣r♦♦❢ ♦❢ s✉❜✲❛❞❞✐t✐✈✐t②✮ ■♠♣❧✐❡s t❤❛t ❝♦♥❞✐t✐♦♥❡❞ ♦♥ ✿ ❛r❡ ●❛✉ss✐❛♥ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ Pr♦♦❢✿ ❯s✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥s ✭❋♦✉r✐❡r tr❛♥s❢♦r♠s✮ ❚❤✐s t❡❝❤♥✐q✉❡ ❤❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♦t❤❡rs ✐♥ ✈❛r✐♦✉s ♦t❤❡r ✐♥st❛♥❝❡s✳ ◆♦t❡✿ ❚❤❡r❡ ❛r❡ s♦♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ ✇♦r❦ ♦❢ ▲✐❡❜ ❛♥❞ ❇❛rt❤❡ ✭✾✵s✮ ❚❤❡② ❛❧s♦ ✉s❡ r♦t❛t✐♦♥s ✭❜✉t ♥♦t ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮

✷✷

slide-65
SLIDE 65
  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t②✿ ❝t❞✳✳

▲❡t (U†, X†) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max

µX CµX[h(AX + Z) − λh(BX + Z)] = h(AX† + Z|U†) − λh(BX† + Z|U†).

▲❡t (Xa, Ua) ❛♥❞ (Xb, Ub) ❜❡ ✐✳✐✳❞✳ ❛❝❝♦r❞✐♥❣ t♦ (U†, X†)✳ ❙❡tt✐♥❣ U = (Ua, Ub)✱ X+ = Xa+Xb

√ 2

❛♥❞ X− = Xa−Xb

√ 2

t❤❡ ♣r♦♦❢ ♦❢ s✉❜✲❛❞❞✐t✐✈✐t② ②✐❡❧❞s 2V = CµX1,X2[h(AX1 + Z1, AX2 + Z2) − λh(BX1 + Z1, BX2 + Z2)]

  • (µX+,X−)

≤ CµX1[h(AX1 + Z1) − λh(BX1 + Z1)]

  • µX+

+ CµX2[h(AX2 + Z2) − λh(BX2 + Z2)]

  • µX−

−(λ − 1)I(AX− + Z2; BX+ + Z1|Ua, Ub) ≤ V + V ❚❤❡r❡❢♦r❡✿ ✇❡ ❣❡t t❤❛t ❝♦♥❞✐t✐♦♥❡❞ ♦♥ (Ua, Ub)✿ X+ ⊥ X−✳ ◆♦t❡✿ ❚❤✉s✱ ❝♦♥❞✐t✐♦♥❡❞ ♦♥ ✿ ✭❢r♦♠ ❝♦♥str✉❝t✐♦♥✮ ✭❢r♦♠ ♣r♦♦❢ ♦❢ s✉❜✲❛❞❞✐t✐✈✐t②✮ ■♠♣❧✐❡s t❤❛t ❝♦♥❞✐t✐♦♥❡❞ ♦♥ ✿ ❛r❡ ●❛✉ss✐❛♥ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ Pr♦♦❢✿ ❯s✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥s ✭❋♦✉r✐❡r tr❛♥s❢♦r♠s✮ ❚❤✐s t❡❝❤♥✐q✉❡ ❤❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♦t❤❡rs ✐♥ ✈❛r✐♦✉s ♦t❤❡r ✐♥st❛♥❝❡s✳ ◆♦t❡✿ ❚❤❡r❡ ❛r❡ s♦♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ ✇♦r❦ ♦❢ ▲✐❡❜ ❛♥❞ ❇❛rt❤❡ ✭✾✵s✮ ❚❤❡② ❛❧s♦ ✉s❡ r♦t❛t✐♦♥s ✭❜✉t ♥♦t ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮

✷✷

slide-66
SLIDE 66
  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t②✿ ❝t❞✳✳

▲❡t (U†, X†) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max

µX CµX[h(AX + Z) − λh(BX + Z)] = h(AX† + Z|U†) − λh(BX† + Z|U†).

▲❡t (Xa, Ua) ❛♥❞ (Xb, Ub) ❜❡ ✐✳✐✳❞✳ ❛❝❝♦r❞✐♥❣ t♦ (U†, X†)✳ ◆♦t❡✿ ❚❤✉s✱ ❝♦♥❞✐t✐♦♥❡❞ ♦♥ (Ua, Ub)✿

  • Xa ⊥ Xb ✭❢r♦♠ ❝♦♥str✉❝t✐♦♥✮
  • (Xa + Xb) ⊥ (Xa − Xb) ✭❢r♦♠ ♣r♦♦❢ ♦❢ s✉❜✲❛❞❞✐t✐✈✐t②✮

■♠♣❧✐❡s t❤❛t ❝♦♥❞✐t✐♦♥❡❞ ♦♥ ✿ ❛r❡ ●❛✉ss✐❛♥ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ Pr♦♦❢✿ ❯s✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥s ✭❋♦✉r✐❡r tr❛♥s❢♦r♠s✮ ❚❤✐s t❡❝❤♥✐q✉❡ ❤❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♦t❤❡rs ✐♥ ✈❛r✐♦✉s ♦t❤❡r ✐♥st❛♥❝❡s✳ ◆♦t❡✿ ❚❤❡r❡ ❛r❡ s♦♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ ✇♦r❦ ♦❢ ▲✐❡❜ ❛♥❞ ❇❛rt❤❡ ✭✾✵s✮ ❚❤❡② ❛❧s♦ ✉s❡ r♦t❛t✐♦♥s ✭❜✉t ♥♦t ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮

✷✷

slide-67
SLIDE 67
  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t②✿ ❝t❞✳✳

▲❡t (U†, X†) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max

µX CµX[h(AX + Z) − λh(BX + Z)] = h(AX† + Z|U†) − λh(BX† + Z|U†).

▲❡t (Xa, Ua) ❛♥❞ (Xb, Ub) ❜❡ ✐✳✐✳❞✳ ❛❝❝♦r❞✐♥❣ t♦ (U†, X†)✳ ◆♦t❡✿ ❚❤✉s✱ ❝♦♥❞✐t✐♦♥❡❞ ♦♥ (Ua, Ub)✿

  • Xa ⊥ Xb ✭❢r♦♠ ❝♦♥str✉❝t✐♦♥✮
  • (Xa + Xb) ⊥ (Xa − Xb) ✭❢r♦♠ ♣r♦♦❢ ♦❢ s✉❜✲❛❞❞✐t✐✈✐t②✮
  • ■♠♣❧✐❡s t❤❛t ❝♦♥❞✐t✐♦♥❡❞ ♦♥ (Ua, Ub)✿ Xa, Xb ❛r❡ ●❛✉ss✐❛♥

⋆ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ ⋆ Pr♦♦❢✿ ❯s✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥s ✭❋♦✉r✐❡r tr❛♥s❢♦r♠s✮ ❚❤✐s t❡❝❤♥✐q✉❡ ❤❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♦t❤❡rs ✐♥ ✈❛r✐♦✉s ♦t❤❡r ✐♥st❛♥❝❡s✳ ◆♦t❡✿ ❚❤❡r❡ ❛r❡ s♦♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ ✇♦r❦ ♦❢ ▲✐❡❜ ❛♥❞ ❇❛rt❤❡ ✭✾✵s✮ ❚❤❡② ❛❧s♦ ✉s❡ r♦t❛t✐♦♥s ✭❜✉t ♥♦t ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮

✷✷

slide-68
SLIDE 68
  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t②✿ ❝t❞✳✳

▲❡t (U†, X†) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max

µX CµX[h(AX + Z) − λh(BX + Z)] = h(AX† + Z|U†) − λh(BX† + Z|U†).

▲❡t (Xa, Ua) ❛♥❞ (Xb, Ub) ❜❡ ✐✳✐✳❞✳ ❛❝❝♦r❞✐♥❣ t♦ (U†, X†)✳ ◆♦t❡✿ ❚❤✉s✱ ❝♦♥❞✐t✐♦♥❡❞ ♦♥ (Ua, Ub)✿

  • Xa ⊥ Xb ✭❢r♦♠ ❝♦♥str✉❝t✐♦♥✮
  • (Xa + Xb) ⊥ (Xa − Xb) ✭❢r♦♠ ♣r♦♦❢ ♦❢ s✉❜✲❛❞❞✐t✐✈✐t②✮
  • ■♠♣❧✐❡s t❤❛t ❝♦♥❞✐t✐♦♥❡❞ ♦♥ (Ua, Ub)✿ Xa, Xb ❛r❡ ●❛✉ss✐❛♥

⋆ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ ⋆ Pr♦♦❢✿ ❯s✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥s ✭❋♦✉r✐❡r tr❛♥s❢♦r♠s✮ ❚❤✐s t❡❝❤♥✐q✉❡ ❤❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♦t❤❡rs ✐♥ ✈❛r✐♦✉s ♦t❤❡r ✐♥st❛♥❝❡s✳ ◆♦t❡✿ ❚❤❡r❡ ❛r❡ s♦♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ ✇♦r❦ ♦❢ ▲✐❡❜ ❛♥❞ ❇❛rt❤❡ ✭✾✵s✮ ❚❤❡② ❛❧s♦ ✉s❡ r♦t❛t✐♦♥s ✭❜✉t ♥♦t ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮

✷✷

slide-69
SLIDE 69

❆♥ ♦♣❡♥ q✉❡st✐♦♥

❲❡ ❤❛✈❡ s❡❡♥ ✭②❡st❡r❞❛② ❛♥❞ t♦❞❛②✮ ❤♦✇ s✉❜✲❛❞❞✐t✐✈✐t② ✐♠♣❧✐❡s ●❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t② ❖♣❡♥ q✉❡st✐♦♥ ❋♦r α, a ∈ (0, 1)✱ ❞♦ ●❛✉ss✐❛♥s ♠❛①✐♠✐③❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ αh(X2 + aX1 + Z) + (1 − α)h(X1 + Z) − h(aX1 + Z) ♦✈❡r X1 ⊥ X2✱ s✉❜❥❡❝t t♦ E(X2

1) ≤ P1✱ E(X2 2) ≤ P2✳ ❍❡r❡ Z ∼ N(0, 1) ✐s ✐♥❞❡♣❡♥❞❡♥t

♦❢ X1, X2✳ ❆✣r♠❛t✐✈❡ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥❛❧ s✉❜✲❛❞❞✐t✐✈❡❄ ❲❤② s❤♦✉❧❞ s♦♠❡♦♥❡ ❝❛r❡❄ ■❢ tr✉❡✱ s♦❧✈❡s t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ❢♦r t❤❡ ●❛✉ss✐❛♥ ❩✲✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧ ❘❡❧❛t❡❞ t♦ r❡✈❡rs❡ ❊P■s✱ ❤②♣❡r♣❧❛♥❡ ❝♦♥❥❡❝t✉r❡✱ ❡t❝✳

✷✸

slide-70
SLIDE 70

❆♥ ♦♣❡♥ q✉❡st✐♦♥

❲❡ ❤❛✈❡ s❡❡♥ ✭②❡st❡r❞❛② ❛♥❞ t♦❞❛②✮ ❤♦✇ s✉❜✲❛❞❞✐t✐✈✐t② ✐♠♣❧✐❡s ●❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t② ❖♣❡♥ q✉❡st✐♦♥ ❋♦r α, a ∈ (0, 1)✱ ❞♦ ●❛✉ss✐❛♥s ♠❛①✐♠✐③❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ αh(X2 + aX1 + Z) + (1 − α)h(X1 + Z) − h(aX1 + Z) ♦✈❡r X1 ⊥ X2✱ s✉❜❥❡❝t t♦ E(X2

1) ≤ P1✱ E(X2 2) ≤ P2✳ ❍❡r❡ Z ∼ N(0, 1) ✐s ✐♥❞❡♣❡♥❞❡♥t

♦❢ X1, X2✳ ❆✣r♠❛t✐✈❡ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥❛❧ s✉❜✲❛❞❞✐t✐✈❡❄ CµX1

  • αh(X2 + aX1 + Z) + (1 − α)h(X1 + Z) − h(aX1 + Z)
  • ❲❤② s❤♦✉❧❞ s♦♠❡♦♥❡ ❝❛r❡❄

■❢ tr✉❡✱ s♦❧✈❡s t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ❢♦r t❤❡ ●❛✉ss✐❛♥ ❩✲✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧ ❘❡❧❛t❡❞ t♦ r❡✈❡rs❡ ❊P■s✱ ❤②♣❡r♣❧❛♥❡ ❝♦♥❥❡❝t✉r❡✱ ❡t❝✳

✷✸

slide-71
SLIDE 71

❆♥ ♦♣❡♥ q✉❡st✐♦♥

❲❡ ❤❛✈❡ s❡❡♥ ✭②❡st❡r❞❛② ❛♥❞ t♦❞❛②✮ ❤♦✇ s✉❜✲❛❞❞✐t✐✈✐t② ✐♠♣❧✐❡s ●❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t② ❖♣❡♥ q✉❡st✐♦♥ ❋♦r α, a ∈ (0, 1)✱ ❞♦ ●❛✉ss✐❛♥s ♠❛①✐♠✐③❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ αh(X2 + aX1 + Z) + (1 − α)h(X1 + Z) − h(aX1 + Z) ♦✈❡r X1 ⊥ X2✱ s✉❜❥❡❝t t♦ E(X2

1) ≤ P1✱ E(X2 2) ≤ P2✳ ❍❡r❡ Z ∼ N(0, 1) ✐s ✐♥❞❡♣❡♥❞❡♥t

♦❢ X1, X2✳ ❆✣r♠❛t✐✈❡ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥❛❧ s✉❜✲❛❞❞✐t✐✈❡❄ CµX1

  • αh(X2 + aX1 + Z) + (1 − α)h(X1 + Z) − h(aX1 + Z)
  • ❲❤② s❤♦✉❧❞ s♦♠❡♦♥❡ ❝❛r❡❄
  • ■❢ tr✉❡✱ s♦❧✈❡s t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ❢♦r t❤❡ ●❛✉ss✐❛♥ ❩✲✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧
  • ❘❡❧❛t❡❞ t♦ r❡✈❡rs❡ ❊P■s✱ ❤②♣❡r♣❧❛♥❡ ❝♦♥❥❡❝t✉r❡✱ ❡t❝✳

✷✸

slide-72
SLIDE 72

❖✉t❧✐♥❡

  • ❇r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ ❊st❛❜❧✐s❤✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ▼❛rt♦♥✬s ❢♦r ▼■▼❖ ❜r♦❛❞❝❛st

❝❤❛♥♥❡❧

  • ■♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧✿ ❙✉❜✲♦♣t✐♠❛❧✐t② ♦❢ t❤❡ ❍❛♥✕❑♦❜❛②❛s❤✐ r❡❣✐♦♥
  • ❋❛♠✐❧② ♦❢ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s

⋆ ❘❡❧❛t✐♦♥ t♦ ♣r♦❜❧❡♠s ♦❢ ✐♥t❡r❡st ✐♥ ♦t❤❡r ✜❡❧❞s ⋆ ❯♥✐❢②✐♥❣ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ s♦♠❡ ❝♦♥❥❡❝t✉r❡s

✷✹

slide-73
SLIDE 73

■♥t❡r❢❡r❡♥❝❡ ❈❤❛♥♥❡❧ ✭❆❤❧s✇❡❞❡ ✬✼✹✮

M1 M2 ❊♥❝♦❞❡r ✶ ❊♥❝♦❞❡r ✷ Xn

1

Xn

2

Wb(y2|x1, x2) Wa(y1|x1, x2) Y n

1

Y n

2

❉❡❝♦❞❡r ✶ ❉❡❝♦❞❡r ✷ ˆ M1 ˆ M2

✷✺

slide-74
SLIDE 74

❍❛♥✲❑♦❜❛②❛s❤✐ ❛❝❤✐❡✈❛❜❧❡ r❡❣✐♦♥ ✭✶✾✽✶✮ á ❧❛ ❈❤♦♥❣ ❡t✳ ❛❧✳

❆ r❛t❡✲♣❛✐r (R1, R2) ✐s ❛❝❤✐❡✈❛❜❧❡ ❢♦r t❤❡ ✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧ ✐❢ R1 < I(X1; Y1|U2, Q), R2 < I(X2; Y2|U1, Q), R1 + R2 < I(X1, U2; Y1|Q) + I(X2; Y2|U1, U2, Q), R1 + R2 < I(X2, U1; Y2|Q) + I(X1; Y1|U1, U2, Q), R1 + R2 < I(X1, U2; Y1|U1, Q) + I(X2, U1; Y2|U2, Q), 2R1 + R2 < I(X1, U2; Y1|Q) + I(X1; Y1|U1, U2, Q) + I(X2, U1; Y2|U2, Q), R1 + 2R2 < I(X2, U1; Y2|Q) + I(X2; Y2|U1, U2, Q) + I(X1, U2; Y1|U1, Q) ❢♦r s♦♠❡ ♣♠❢ p(q)p(u1, x1|q)p(u2, x2|q)✱ ✇❤❡r❡ |U1| ≤ |X1| + 4✱ |U2| ≤ |X2| + 4✱ ❛♥❞ |Q| ≤ 7✳ ❉❡♥♦t❡ t❤❡ ✭❝❧♦s✉r❡ ♦❢✮ r❡❣✐♦♥ ❛s R(Wa, Wb)✳ ◆✉♠❡r✐❝❛❧❧② ✐♥❢❡❛s✐❜❧❡ t♦ ❝♦♠♣✉t❡ R(WaWb) ❡✈❡♥ ❢♦r ❣❡♥❡r✐❝ ❜✐♥❛r②✲✐♥♣✉t ❜✐♥❛r②✲♦✉t♣✉t ✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧s ❋✐rst st❡♣✿ ❋✐♥❞ ❛ ❝❤❛♥♥❡❧ ❝❧❛ss ✇❤❡r❡ ❍❑ r❡❣✐♦♥ s✐♠♣❧✐✜❡s ❆◆❉ ②❡t ♥♦t t♦♦ tr✐✈✐❛❧

✷✻

slide-75
SLIDE 75

❍❛♥✲❑♦❜❛②❛s❤✐ ❛❝❤✐❡✈❛❜❧❡ r❡❣✐♦♥ ✭✶✾✽✶✮ á ❧❛ ❈❤♦♥❣ ❡t✳ ❛❧✳

❆ r❛t❡✲♣❛✐r (R1, R2) ✐s ❛❝❤✐❡✈❛❜❧❡ ❢♦r t❤❡ ✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧ ✐❢ R1 < I(X1; Y1|U2, Q), R2 < I(X2; Y2|U1, Q), R1 + R2 < I(X1, U2; Y1|Q) + I(X2; Y2|U1, U2, Q), R1 + R2 < I(X2, U1; Y2|Q) + I(X1; Y1|U1, U2, Q), R1 + R2 < I(X1, U2; Y1|U1, Q) + I(X2, U1; Y2|U2, Q), 2R1 + R2 < I(X1, U2; Y1|Q) + I(X1; Y1|U1, U2, Q) + I(X2, U1; Y2|U2, Q), R1 + 2R2 < I(X2, U1; Y2|Q) + I(X2; Y2|U1, U2, Q) + I(X1, U2; Y1|U1, Q) ❢♦r s♦♠❡ ♣♠❢ p(q)p(u1, x1|q)p(u2, x2|q)✱ ✇❤❡r❡ |U1| ≤ |X1| + 4✱ |U2| ≤ |X2| + 4✱ ❛♥❞ |Q| ≤ 7✳ ❉❡♥♦t❡ t❤❡ ✭❝❧♦s✉r❡ ♦❢✮ r❡❣✐♦♥ ❛s R(Wa, Wb)✳ ◆✉♠❡r✐❝❛❧❧② ✐♥❢❡❛s✐❜❧❡ t♦ ❝♦♠♣✉t❡ R(WaWb) ❡✈❡♥ ❢♦r ❣❡♥❡r✐❝ ❜✐♥❛r②✲✐♥♣✉t ❜✐♥❛r②✲♦✉t♣✉t ✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧s ❋✐rst st❡♣✿

  • ❋✐♥❞ ❛ ❝❤❛♥♥❡❧ ❝❧❛ss ✇❤❡r❡ ❍❑ r❡❣✐♦♥ s✐♠♣❧✐✜❡s ❆◆❉ ②❡t ♥♦t t♦♦ tr✐✈✐❛❧

✷✻

slide-76
SLIDE 76

❈❧❡❛♥ ❩ ■♥t❡r❢❡r❡♥❝❡ ❈❤❛♥♥❡❧ ✭❈❩■❈✮ ▼♦❞❡❧

M1 M2 ❊♥❝♦❞❡r ✶ ❊♥❝♦❞❡r ✷ Xn

1

Xn

2

Wa(y1|x1, x2) Y n

1

Y n

2 = Xn 2

❉❡❝♦❞❡r ✶ ❉❡❝♦❞❡r ✷ ˆ M1 ˆ M2

❈❧❡❛♥ ❩✲✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧

▲❡♠♠❛✿ ❆ r❛t❡✲♣❛✐r (R1, R2) ❜❡❧♦♥❣s t♦ ❍❛♥✲❑♦❜❛②❛s❤✐ r❡❣✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ R1 < I(X1; Y1|U2, Q), R2 < H(X2|Q), R1 + R2 < I(X1, U2; Y1|Q) + H(X2|U2, Q), ❢♦r s♦♠❡ ♣♠❢ p(q)p(x1|q)p(u2, x2|q)✱ ✇❤❡r❡ |U2| ≤ |X2| ❛♥❞ |Q| ≤ 2✳ ❉❡♥♦t❡ r❡❣✐♦♥✿ R(Wa)

✷✼

slide-77
SLIDE 77

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

❊q✉✐✈❛❧❡♥t t♦ t❡st ✐❢ Sλ(Wa ⊗ Wa) = 2Sλ(Wa), ∀ Wa, λ ≥ 0, ✇❤❡r❡ Sλ(Wa) := max

(R1,R2)∈R(Wa) λR1 + R2.

❋♦r ✱ ✐s ❣✐✈❡♥ ❜② ▲❡♠♠❛ ✭s✉❜✲❛❞❞✐t✐✈✐t②✮ ✭◆❛✐r✲❳✐❛✲❨❛③❞❛♥♣❛♥❛❤ ✬✶✺✮✿ ■♠♣❧✐❡s ♦♣t✐♠❛❧✐t② ♦❢ ✱ ✳

✷✽

slide-78
SLIDE 78

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

❊q✉✐✈❛❧❡♥t t♦ t❡st ✐❢ Sλ(Wa ⊗ Wa) = 2Sλ(Wa), ∀ Wa, λ ≥ 0, ✇❤❡r❡ Sλ(Wa) := max

(R1,R2)∈R(Wa) λR1 + R2.

❋♦r λ ∈ [0, 1]✱ Sλ(Wa) ✐s ❣✐✈❡♥ ❜② max

p1(x1)p2(u2,x2)

  • (1 − λ)H(X2) + λI(X1, U2; Y1) + λH(X2|U2)
  • =

max

p1(x1)p2(x2)

  • H(X2) + λI(X1; Y1)
  • ▲❡♠♠❛ ✭s✉❜✲❛❞❞✐t✐✈✐t②✮ ✭◆❛✐r✲❳✐❛✲❨❛③❞❛♥♣❛♥❛❤ ✬✶✺✮✿

■♠♣❧✐❡s ♦♣t✐♠❛❧✐t② ♦❢ ✱ ✳

✷✽

slide-79
SLIDE 79

❚❡st✐♥❣ ♦♣t✐♠❛❧✐t②

❊q✉✐✈❛❧❡♥t t♦ t❡st ✐❢ Sλ(Wa ⊗ Wa) = 2Sλ(Wa), ∀ Wa, λ ≥ 0, ✇❤❡r❡ Sλ(Wa) := max

(R1,R2)∈R(Wa) λR1 + R2.

❋♦r λ ∈ [0, 1]✱ Sλ(Wa) ✐s ❣✐✈❡♥ ❜② max

p1(x1)p2(u2,x2)

  • (1 − λ)H(X2) + λI(X1, U2; Y1) + λH(X2|U2)
  • =

max

p1(x1)p2(x2)

  • H(X2) + λI(X1; Y1)
  • ▲❡♠♠❛ ✭s✉❜✲❛❞❞✐t✐✈✐t②✮ ✭◆❛✐r✲❳✐❛✲❨❛③❞❛♥♣❛♥❛❤ ✬✶✺✮✿

H(X21, X22) + λI(X11, X12; Y11, Y12) ≤

  • H(X21) + λI(X11; Y11)
  • +
  • H(X22) + λI(X12; Y12)
  • − (1 − λ)I(X21; X22).

■♠♣❧✐❡s ♦♣t✐♠❛❧✐t② ♦❢ Sλ(Wa)✱ λ ∈ [0, 1]✳

✷✽

slide-80
SLIDE 80

❲❤❛t ❛❜♦✉t λ > 1❄

❋♦r λ ≥ 1✱ Sλ(Wa) ✐s ❣✐✈❡♥ ❜②

max

p1(x1)p2(u2,x2)

  • I(X1, U2; Y1) + H(X2|U2) + (λ − 1)I(X1; Y1|U2)
  • =

max

p1(x1)p2(x2)

  • I(X1, X2; Y1) + CX2[(λ − 1)I(X1; Y1) + H(X2) − I(X2; Y1|X1)]
  • ◗✉❡st✐♦♥✿ ❈❛♥ ✇❡ ♥✉♠❡r✐❝❛❧❧② t❡st ✐❢

❄ ✐s ❛ ❜✐♥❛r② r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✭✐✳❡✳ ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡ ♦✈❡r s✐♥❣❧❡ ✈❛r✐❛❜❧❡✮ ✿ ❤❛s ❛t ♠♦st ✷ ✐♥✢❡①✐♦♥ ♣♦✐♥ts

✷✾

slide-81
SLIDE 81

❲❤❛t ❛❜♦✉t λ > 1❄

❋♦r λ ≥ 1✱ Sλ(Wa) ✐s ❣✐✈❡♥ ❜②

max

p1(x1)p2(u2,x2)

  • I(X1, U2; Y1) + H(X2|U2) + (λ − 1)I(X1; Y1|U2)
  • =

max

p1(x1)p2(x2)

  • I(X1, X2; Y1) + CX2[(λ − 1)I(X1; Y1) + H(X2) − I(X2; Y1|X1)]
  • ◗✉❡st✐♦♥✿ ❈❛♥ ✇❡ ♥✉♠❡r✐❝❛❧❧② t❡st ✐❢ Sλ(Wa ⊗ Wa) = 2Sλ(Wa) ❄

X2 ✐s ❛ ❜✐♥❛r② r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✭✐✳❡✳ ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡ ♦✈❡r s✐♥❣❧❡ ✈❛r✐❛❜❧❡✮ (λ − 1)I(X1; Y1) + H(X2) − I(X2; Y1|X1)✿ ❤❛s ❛t ♠♦st ✷ ✐♥✢❡①✐♦♥ ♣♦✐♥ts

✷✾

slide-82
SLIDE 82

❲❤❛t ❛❜♦✉t λ > 1❄

P(X2) H(X2) + (λ − 1)H(Y1) − λH(Y1|X1) C

P2(X2)[H(X2) + (λ − 1)H(Y1) − λH(Y1|X1)]

❚❤❡ s❤❛♣❡ ♦❢ ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡ ❢♦r ❛ ❣❡♥❡r✐❝ ❜✐♥❛r② ❈❩■❈

✷✾

slide-83
SLIDE 83

❙✉❜✲♦♣t✐♠❛❧✐t② ♦❢ t❤❡ ❍❛♥✲❑♦❜❛②❛s❤✐ r❡❣✐♦♥

λ W(Y1 = 0|X1, X2) A ❍❑

λ

(W)

1 2A ❚■◆ λ

(W ⊗2) ✷ 1 0.5 1

  • ✶✳✶✵✼✺✶✻

✶✳✶✵✽✶✹✶ ✾ 0.12 0.89 0.21 0.62

  • ✶✳✵✼✹✹✽✹

✶✳✵✼✺✺✹✹ ✶✷ 0.01 0.58 0.20 0.74

  • ✶✳✷✽✾✽✸✵

✶✳✷✾✸✼✻✵ ✶✹ 0.78 0.07 0.46 0.05

  • ✶✳✹✷✻✺✷✻

✶✳✹✸✷✹✶✾ ✶✺ 0.91 0.22 0.66 0.15

  • ✶✳✸✷✸✼✻✻

✶✳✸✸✾✵✻✺ ✶✻ 0.91 0.13 0.62 0.06

  • ✶✳✺✶✺✹✷✶

✶✳✺✸✹✼✷✹ ✶✽ 0.38 0.87 0.12 0.79

  • ✶✳✹✹✾✾✺✾

✶✳✹✻✽✺✼✼

❈♦✉♥t❡r❡①❛♠♣❧❡s t♦ t❤❡ ♦♣t✐♠❛❧✐t② ♦❢ ❍❛♥✲❑♦❜❛②❛s❤✐ r❡❣✐♦♥✳ ◆♦t❡✿ ❋♦r t❤❡ ✜rst ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❤❡ ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡ ❛♥❛❧②t✐❝❛❧❧②✳

✸✵

slide-84
SLIDE 84

P❛rt✐❝✉❧❛r ❈❤❛♥♥❡❧

X2 = 0 X2 = 1 X1 1 Y1 1 X1 1 Y1 1

1 2 1 2

  • ❲❡ ❝♦♠♣✉t❡ max

HK λR1 + R2 ❢♦r λ = 2

max

p1(x1)p2(x2)

  • H(Y1) +

C

p2(x2)

  • H(X2) + 2H(Y1) − H(Y1|X1)
  • ▲❡t

❛♥❞ r❡s♣❡❝t✐✈❡❧② ❞❡♥♦t❡ ❛♥❞ ✇❤❡r❡ ✐s t❤❡ ❜✐♥❛r② ❡♥tr♦♣② ❢✉♥❝t✐♦♥

✸✶

slide-85
SLIDE 85

P❛rt✐❝✉❧❛r ❈❤❛♥♥❡❧

X2 = 0 X2 = 1 X1 1 Y1 1 X1 1 Y1 1

1 2 1 2

  • ❲❡ ❝♦♠♣✉t❡ max

HK λR1 + R2 ❢♦r λ = 2

max

p1(x1)p2(x2)

  • H(Y1) +

C

p2(x2)

  • H(X2) + 2H(Y1) − H(Y1|X1)
  • ▲❡t p ❛♥❞ q r❡s♣❡❝t✐✈❡❧② ❞❡♥♦t❡ Pr(X1 = 0) ❛♥❞ Pr(X2 = 0)

f(p, q) = (1 − 2¯ p)hb(q) + hb(q + p 2 ¯ q) − 2phb(q + 1 2 ) ✇❤❡r❡ hb(.) ✐s t❤❡ ❜✐♥❛r② ❡♥tr♦♣② ❢✉♥❝t✐♦♥

✸✶

f(p, q)

slide-86
SLIDE 86

P❛rt✐❝✉❧❛r ❝❤❛♥♥❡❧ ❝♦♥t✐♥✉❡❞

f(p, q) ✐s ❝♦♥❝❛✈❡ ✐♥ q ❢♦r p ≥ 1

2 ❛♥❞ ❢♦r 0 ≤ p < 1 2

C

q[f(p, q)] =

   f(p, q) q > 1 − 2p f(p, 1 − 2p) − f(p, 0) 1 − 2p q + f(p, 0) q ∈ [0, 1 − 2p]

✸✷

slide-87
SLIDE 87

P❛rt✐❝✉❧❛r ❝❤❛♥♥❡❧ ❝♦♥t✐♥✉❡❞

f(p, q) ✐s ❝♦♥❝❛✈❡ ✐♥ q ❢♦r p ≥ 1

2 ❛♥❞ ❢♦r 0 ≤ p < 1 2

C

q[f(p, q)] =

   f(p, q) q > 1 − 2p f(p, 1 − 2p) − f(p, 0) 1 − 2p q + f(p, 0) q ∈ [0, 1 − 2p]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ✵ ✵✳✵✺ ✵✳✶ ✵✳✶✺ ✵✳✷ ✵✳✷✺ ✵✳✸ q f(0.2, q) C

q[f(0.2, q)]

✸✷

slide-88
SLIDE 88

P❛rt✐❝✉❧❛r ❝❤❛♥♥❡❧ ❝♦♥t✐♥✉❡❞

f(p, q) ✐s ❝♦♥❝❛✈❡ ✐♥ q ❢♦r p ≥ 1

2 ❛♥❞ ❢♦r 0 ≤ p < 1 2

C

q[f(p, q)] =

   f(p, q) q > 1 − 2p f(p, 1 − 2p) − f(p, 0) 1 − 2p q + f(p, 0) q ∈ [0, 1 − 2p] ❈♦r♦❧❧❛r② ▼❛①✐♠✉♠ ♦❢ 2R1 + R2 ❢♦r t❤❡ ❍❛♥✕❑♦❜❛②❛s❤✐ r❡❣✐♦♥ ✐s ❡q✉❛❧ t♦ t❤❡ ♠❛①✐♠✉♠ ♦❢ T(p, q) ❢♦r (p, q) ∈ [0, 1] × [0, 1]✱ ✇❤❡r❡ T(p, q) =    hb(q + p

2 ¯

q) + f(p, q) q ≥ min{0, 1 − 2p} hb(q + p

2 ¯

q) + f(p, 1 − 2p) − f(p, 0) 1 − 2p q + f(p, 0)

  • .w.,

✇❤❡r❡ f(p, q) = (1 − 2¯ p)hb(q) + hb(q + p

2 ¯

q) − 2phb( q+1

2 ) ✸✷

slide-89
SLIDE 89

P❧♦t ♦❢ T(p, q)

◆✉♠❡r✐❝❛❧ s❡❛r❝❤ ✐♥❞✐❝❛t❡s✿ maxp,q T(p, q) = 1.107516.. ❛t p = 0.5078.. ❛♥❞ q = 0.4365..

✸✸

slide-90
SLIDE 90

P❛rt✐❝✉❧❛r ❝❤❛♥♥❡❧ ❝♦♥t✐♥✉❡❞

  • ■♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝ ✐s ❛ ♠❡t❤♦❞ t♦ ♦❜t❛✐♥ ❢♦r♠❛❧ ❜♦✉♥❞s ❢♦r ❢✉♥❝t✐♦♥s ❝♦♥s✐st✐♥❣

♦❢ ❜❛s✐❝ ❛r✐t❤♠❡t✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ ❝♦♠♠♦♥❧② ✉s❡❞ ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ❧♦❣❛r✐t❤♠s ❛♥❞ tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✳

  • T(p, q) ♦♥❧② ✐♥❝❧✉❞❡s ❜❛s✐❝ ❛r✐t❤♠❡t✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ ❧♦❣❛r✐t❤♠✳
  • ❲❡ ✉s❡❞ ❏✉❧✐❛ ❜❛s❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤✐s ❢♦r♠❛❧ ♠❡t❤♦❞ t♦ ♦❜t❛✐♥

max T(p, q) ∈ [1.10751, 1.10769]

  • ❚❤❡ ✷✲❧❡tt❡r ❚■◆ ❛❝❤✐❡✈❡s 2R1 + R2 = 1.108141 ❛t t❤❡ ❞✐str✐❜✉t✐♦♥

P((X11, X12) = (0, 0)) = p P((X11, X12) = (1, 1)) = 1 − p P((X21, X22) = (0, 0)) = 0.36q P((X21, X22) = (1, 1)) = 1 − 1.64q P((X21, X22) = (0, 1)) = 0.64q P((X21, X22) = (1, 0)) = 0.64q ✇❤❡r❡ p = 0.507829413✱ q = 0.436538150

  • ❘❡♣❡t✐t✐♦♥ ❝♦❞✐♥❣ ❛❝r♦ss t✐♠❡ s❡❡♠s t♦ ♦✉t♣❡r❢♦r♠ ♠❡♠♦r②❧❡ss ❝♦❞✐♥❣

✸✹

slide-91
SLIDE 91

❲❤❛t ❛❜♦✉t ▼❛rt♦♥✬s r❡❣✐♦♥ ❢♦r t❤❡ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❄

■s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥❛❧ s✉❜✲❛❞❞✐t✐✈❡ ♦r ✐s t❤❡r❡ ❛♥ ❡①❛♠♣❧❡ ✇❤❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡❄ ▲❡t Wa(y|x) ❛♥❞ Wb(z|x) ❜❡ ❣✐✈❡♥ ❝❤❛♥♥❡❧s✱ α ∈ [0, 1]✱ ❛♥❞ λ ≥ 1✳ CµX

  • (λ − α)H(Y ) − αH(Z) + max

p(u,v|x) {λI(U; Y ) + I(V ; Z) − I(U; V )}

  • ■❢ s✉❜✲❛❞❞✐t✐✈❡✱ t❤❡♥ ▼❛rt♦♥✬s r❡❣✐♦♥ ✐s ♦♣t✐♠❛❧ ❢♦r ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧

■❢ ❡①❛♠♣❧❡ ✇❤❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡✱ t❤❡♥ ♦♥❡ s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ❞❡❞✉❝❡ ❛ ❝❤❛♥♥❡❧ ✇❤❡r❡ ▼❛rt♦♥✬s r❡❣✐♦♥ ✐s ♥♦t ♦♣t✐♠❛❧ ❘❡♠❛r❦s✿ ❈♦♥❥❡❝t✉r❡❞ t♦ ❜❡ s✉❜✲❛❞❞✐t✐✈❡ ✭❆♥❛♥t❤❛r❛♠✲●♦❤❛r✐✲◆❛✐r ✬✶✸✮ ❚♦ ❡✈❛❧✉❛t❡ t❤❡ ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡ ❙✉✣❝❡s t♦ ❝♦♥s✐❞❡r ✿ ✳ ❲❡ ❞✐❞ ♥♦t ❣❡t ❛♥② ❝♦♥tr❛❞✐❝t✐♦♥ t♦ s✉❜✲❛❞❞✐✈✐t② ❢♦r ❜✐♥❛r② ✐♥♣✉t ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s ❈❛♥ ♣r♦✈❡ s✉❜✲❛❞❞✐t✐✈✐t② ✇❤❡♥ ♦r ✳

✸✺

slide-92
SLIDE 92

❲❤❛t ❛❜♦✉t ▼❛rt♦♥✬s r❡❣✐♦♥ ❢♦r t❤❡ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❄

■s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥❛❧ s✉❜✲❛❞❞✐t✐✈❡ ♦r ✐s t❤❡r❡ ❛♥ ❡①❛♠♣❧❡ ✇❤❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡❄ ▲❡t Wa(y|x) ❛♥❞ Wb(z|x) ❜❡ ❣✐✈❡♥ ❝❤❛♥♥❡❧s✱ α ∈ [0, 1]✱ ❛♥❞ λ ≥ 1✳ CµX

  • (λ − α)H(Y ) − αH(Z) + max

p(u,v|x) {λI(U; Y ) + I(V ; Z) − I(U; V )}

  • ■❢ s✉❜✲❛❞❞✐t✐✈❡✱ t❤❡♥ ▼❛rt♦♥✬s r❡❣✐♦♥ ✐s ♦♣t✐♠❛❧ ❢♦r ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧
  • ■❢ ∃ ❡①❛♠♣❧❡ ✇❤❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡✱ t❤❡♥ ♦♥❡ s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ❞❡❞✉❝❡ ❛

❝❤❛♥♥❡❧ ✇❤❡r❡ ▼❛rt♦♥✬s r❡❣✐♦♥ ✐s ♥♦t ♦♣t✐♠❛❧ ❘❡♠❛r❦s✿ ❈♦♥❥❡❝t✉r❡❞ t♦ ❜❡ s✉❜✲❛❞❞✐t✐✈❡ ✭❆♥❛♥t❤❛r❛♠✲●♦❤❛r✐✲◆❛✐r ✬✶✸✮ ❚♦ ❡✈❛❧✉❛t❡ t❤❡ ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡ ❙✉✣❝❡s t♦ ❝♦♥s✐❞❡r ✿ ✳ ❲❡ ❞✐❞ ♥♦t ❣❡t ❛♥② ❝♦♥tr❛❞✐❝t✐♦♥ t♦ s✉❜✲❛❞❞✐✈✐t② ❢♦r ❜✐♥❛r② ✐♥♣✉t ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s ❈❛♥ ♣r♦✈❡ s✉❜✲❛❞❞✐t✐✈✐t② ✇❤❡♥ ♦r ✳

✸✺

slide-93
SLIDE 93

❲❤❛t ❛❜♦✉t ▼❛rt♦♥✬s r❡❣✐♦♥ ❢♦r t❤❡ ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧❄

■s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥❛❧ s✉❜✲❛❞❞✐t✐✈❡ ♦r ✐s t❤❡r❡ ❛♥ ❡①❛♠♣❧❡ ✇❤❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡❄ ▲❡t Wa(y|x) ❛♥❞ Wb(z|x) ❜❡ ❣✐✈❡♥ ❝❤❛♥♥❡❧s✱ α ∈ [0, 1]✱ ❛♥❞ λ ≥ 1✳ CµX

  • (λ − α)H(Y ) − αH(Z) + max

p(u,v|x) {λI(U; Y ) + I(V ; Z) − I(U; V )}

  • ■❢ s✉❜✲❛❞❞✐t✐✈❡✱ t❤❡♥ ▼❛rt♦♥✬s r❡❣✐♦♥ ✐s ♦♣t✐♠❛❧ ❢♦r ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧
  • ■❢ ∃ ❡①❛♠♣❧❡ ✇❤❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡✱ t❤❡♥ ♦♥❡ s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ❞❡❞✉❝❡ ❛

❝❤❛♥♥❡❧ ✇❤❡r❡ ▼❛rt♦♥✬s r❡❣✐♦♥ ✐s ♥♦t ♦♣t✐♠❛❧ ❘❡♠❛r❦s✿

  • ❈♦♥❥❡❝t✉r❡❞ t♦ ❜❡ s✉❜✲❛❞❞✐t✐✈❡ ✭❆♥❛♥t❤❛r❛♠✲●♦❤❛r✐✲◆❛✐r ✬✶✸✮
  • ❚♦ ❡✈❛❧✉❛t❡ t❤❡ ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡

⋆ ❙✉✣❝❡s t♦ ❝♦♥s✐❞❡r (U, V )✿ |U| + |V | ≤ |X| + 1✳ ⋆ ❲❡ ❞✐❞ ♥♦t ❣❡t ❛♥② ❝♦♥tr❛❞✐❝t✐♦♥ t♦ s✉❜✲❛❞❞✐✈✐t② ❢♦r ❜✐♥❛r② ✐♥♣✉t ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s

  • ❈❛♥ ♣r♦✈❡ s✉❜✲❛❞❞✐t✐✈✐t② ✇❤❡♥ α = 0 ♦r α = 1✳

✸✺

slide-94
SLIDE 94

❘❡♠❛r❦s

  • ■❞❡❛✿ ❚♦ ❞❡♠♦♥str❛t❡ s✉♣❡r✲❛❞❞✐t✐✈✐t②
  • ❉✐✣❝✉❧t②✿ ■❞❡♥t✐❢② ❛ s✉✣❝✐❡♥t❧② s✐♠♣❧❡ ❝❧❛ss ✇❤❡r❡

⋆ ❊✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ r❡❣✐♦♥ ✐s ♣♦ss✐❜❧❡ ✿ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ⋆ ❙✉♣❡r✲❛❞❞✐t✐✈✐t② ❤♦❧❞s ❚❤✐s ✐❞❡❛ ✇❛s ❛❧s♦ ✉s❡❞ t♦ r❡s♦❧✈❡ ✽✳✷ ■s s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ♦♣t✐♠❛❧ ❢♦r t❤❡ ❣❡♥❡r❛❧ ✸✲r❡❝❡✐✈❡r ❉▼✲❇❈ ✇✐t❤ ♦♥❡ ♠❡ss❛❣❡ t♦ ❛❧❧ t❤r❡❡ r❡❝❡✐✈❡rs ❛♥❞ ❛♥♦t❤❡r ♠❡ss❛❣❡ t♦ t✇♦ r❡❝❡✐✈❡rs❄ ◆❖ ✭◆❛✐r✱❨❛③❞❛♥♣❛♥❛❤ ✬✶✼✮

✸✻

slide-95
SLIDE 95

❘❡♠❛r❦s

  • ■❞❡❛✿ ❚♦ ❞❡♠♦♥str❛t❡ s✉♣❡r✲❛❞❞✐t✐✈✐t②
  • ❉✐✣❝✉❧t②✿ ■❞❡♥t✐❢② ❛ s✉✣❝✐❡♥t❧② s✐♠♣❧❡ ❝❧❛ss ✇❤❡r❡

⋆ ❊✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ r❡❣✐♦♥ ✐s ♣♦ss✐❜❧❡ ✿ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ⋆ ❙✉♣❡r✲❛❞❞✐t✐✈✐t② ❤♦❧❞s ❚❤✐s ✐❞❡❛ ✇❛s ❛❧s♦ ✉s❡❞ t♦ r❡s♦❧✈❡ ✽✳✷ ■s s✉♣❡r♣♦s✐t✐♦♥ ❝♦❞✐♥❣ ♦♣t✐♠❛❧ ❢♦r t❤❡ ❣❡♥❡r❛❧ ✸✲r❡❝❡✐✈❡r ❉▼✲❇❈ ✇✐t❤ ♦♥❡ ♠❡ss❛❣❡ t♦ ❛❧❧ t❤r❡❡ r❡❝❡✐✈❡rs ❛♥❞ ❛♥♦t❤❡r ♠❡ss❛❣❡ t♦ t✇♦ r❡❝❡✐✈❡rs❄ ◆❖ ✭◆❛✐r✱❨❛③❞❛♥♣❛♥❛❤ ✬✶✼✮

✸✻

slide-96
SLIDE 96

❖✉t❧✐♥❡

  • ❇r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ ❊st❛❜❧✐s❤✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ▼❛rt♦♥✬s ❢♦r ▼■▼❖ ❜r♦❛❞❝❛st

❝❤❛♥♥❡❧

  • ■♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧✿ ❙✉❜✲♦♣t✐♠❛❧✐t② ♦❢ t❤❡ ❍❛♥✕❑♦❜❛②❛s❤✐ r❡❣✐♦♥
  • ❋❛♠✐❧② ♦❢ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s

⋆ ❘❡❧❛t✐♦♥ t♦ ♣r♦❜❧❡♠s ♦❢ ✐♥t❡r❡st ✐♥ ♦t❤❡r ✜❡❧❞s ⋆ ❯♥✐❢②✐♥❣ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ s♦♠❡ ❝♦♥❥❡❝t✉r❡s

✸✼

slide-97
SLIDE 97

❆ s♣❡❝✐✜❝ ❢❛♠✐❧② ♦❢ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s

❙❤♦✇s ✉♣✿ ❚❡st✐♥❣ t❤❡ ♦♣t✐♠❛❧✐t② ♦❢ ❝♦❞✐♥❣ s❝❤❡♠❡s ❚❡st✐♥❣ ♦♣t✐♠❛❧✐t② ✭✉s✉❛❧❧②✮ r❡❞✉❝❡s t♦ t❡st✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♦❢ CνX

S⊆[n]

αSH(XS)

  • , αS ∈ R.

❯s✐♥❣ ❋❡♥❝❤❡❧ ❞✉❛❧✐t② t❤✐s ✐s s❛♠❡ ❛s

✸✽

slide-98
SLIDE 98

❆ s♣❡❝✐✜❝ ❢❛♠✐❧② ♦❢ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s

❙❤♦✇s ✉♣✿ ❚❡st✐♥❣ t❤❡ ♦♣t✐♠❛❧✐t② ♦❢ ❝♦❞✐♥❣ s❝❤❡♠❡s ❚❡st✐♥❣ ♦♣t✐♠❛❧✐t② ✭✉s✉❛❧❧②✮ r❡❞✉❝❡s t♦ t❡st✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♦❢ CνX

S⊆[n]

αSH(XS)

  • , αS ∈ R.

❯s✐♥❣ ❋❡♥❝❤❡❧ ❞✉❛❧✐t② t❤✐s ✐s s❛♠❡ ❛s G1(γ1) := max

µX

  • S⊆[n]

αSH(XS) − E(γ1(X)) G2(γ2) := max

µX

  • S⊆[n]

αSH(XS) − E(γ2(X)) G12(γ1, γ2) := max

µX1,X2

  • S⊆[n]

αSH(X1S, X2S) − E(γ1(X1)) − E(γ2(X2)) ■s G12(γ1, γ2) = G1(γ1) + G2(γ2) ∀ γ1, γ2 ❄ ✐✳❡✳ ■s t❤❡ ♠❛①✐♠✐③❡r ♦❢ G12 ❛ ♣r♦❞✉❝t ❞✐str✐❜✉t✐♦♥❄

✸✽

slide-99
SLIDE 99

❆ s♣❡❝✐✜❝ ❢❛♠✐❧② ♦❢ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s

❙❤♦✇s ✉♣✿ ❚❡st✐♥❣ t❤❡ ♦♣t✐♠❛❧✐t② ♦❢ ❝♦❞✐♥❣ s❝❤❡♠❡s ❚❡st✐♥❣ ♦♣t✐♠❛❧✐t② ✭✉s✉❛❧❧②✮ r❡❞✉❝❡s t♦ t❡st✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♦❢ CνX

S⊆[n]

αSH(XS)

  • , αS ∈ R.

❯s✐♥❣ ❋❡♥❝❤❡❧ ❞✉❛❧✐t② t❤✐s ✐s s❛♠❡ ❛s G1(γ1) := max

µX

  • S⊆[n]

αSH(XS) − E(γ1(X)) G2(γ2) := max

µX

  • S⊆[n]

αSH(XS) − E(γ2(X)) G12(γ1, γ2) := max

µX1,X2

  • S⊆[n]

αSH(X1S, X2S) − E(γ1(X1)) − E(γ2(X2)) ❆r❡ t❤❡r❡ ♦t❤❡r ✜❡❧❞s ✇❤❡r❡ t❤❡ s❛♠❡ ❢❛♠✐❧② s❤♦✇s ✉♣❄

✸✽

slide-100
SLIDE 100

❍②♣❡r❝♦♥tr❛❝t✐✈✐t②

❙t✉❞✐❡❞ ✐♥ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ❝s t❤❡♦r②✱ ❡t❝✳ ❉❡✜♥✐t✐♦♥ (X, Y ) ∼ µXY ✐s (p, q)✲❤②♣❡r❝♦♥tr❛❝t✐✈❡ ❢♦r 1 ≤ q ≤ p ✐❢ Tgp ≤ gq ∀g(Y ) ✇❤❡r❡ T ✐s t❤❡ ▼❛r❦♦✈ ♦♣❡r❛t♦r ❝❤❛r❛❝t❡r✐③❡❞ ❜② µY |X ❍❡r❡ Zp = E(|Z|p)

1 p ✳

❚❤✐s ✭s❡r❡♥❞✐♣✐t♦✉s✮ r❡❞✐s❝♦✈❡r② ♦❢ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ t❤❡s❡ ❡q✉✐✈❛❧❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ✐s s♣✉rr✐♥❣ ❛ ❧♦t ♦❢ ✇♦r❦

✸✾

slide-101
SLIDE 101

❍②♣❡r❝♦♥tr❛❝t✐✈✐t②

❙t✉❞✐❡❞ ✐♥ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ❝s t❤❡♦r②✱ ❡t❝✳ ❉❡✜♥✐t✐♦♥ (X, Y ) ∼ µXY ✐s (p, q)✲❤②♣❡r❝♦♥tr❛❝t✐✈❡ ❢♦r 1 ≤ q ≤ p ✐❢ Tgp ≤ gq ∀g(Y ) ✇❤❡r❡ T ✐s t❤❡ ▼❛r❦♦✈ ♦♣❡r❛t♦r ❝❤❛r❛❝t❡r✐③❡❞ ❜② µY |X ❍❡r❡ Zp = E(|Z|p)

1 p ✳

❚❤❡r❡ ✐s ❛ ❧♦t ♦❢ ✐♥t❡r❡st ✐♥ ❡✈❛❧✉t✐♥❣ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs ❢♦r ❞✐str✐❜✉t✐♦♥s✳ ❚❤❡♦r❡♠ ✭◆❛✐r ✬✶✹✮ (X, Y ) ∼ µXY ✐s (p, q)✲❤②♣❡r❝♦♥tr❛❝t✐✈❡ ❢♦r 1 ≤ q ≤ p ✐❢ ❛♥❞ ♦♥❧② ✐❢ CνX,Y

  • H(X, Y ) − (1 − 1

p)H(X) − 1 q H(Y )

  • µX,Y

= H(X, Y ) − (1 − 1 p)H(X) − 1 q H(Y ) ❚❤✐s ✭s❡r❡♥❞✐♣✐t♦✉s✮ r❡❞✐s❝♦✈❡r② ♦❢ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ t❤❡s❡ ❡q✉✐✈❛❧❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ✐s s♣✉rr✐♥❣ ❛ ❧♦t ♦❢ ✇♦r❦

✸✾

slide-102
SLIDE 102

❍②♣❡r❝♦♥tr❛❝t✐✈✐t②

❙t✉❞✐❡❞ ✐♥ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ❝s t❤❡♦r②✱ ❡t❝✳ ❉❡✜♥✐t✐♦♥ (X, Y ) ∼ µXY ✐s (p, q)✲❤②♣❡r❝♦♥tr❛❝t✐✈❡ ❢♦r 1 ≤ q ≤ p ✐❢ Tgp ≤ gq ∀g(Y ) ✇❤❡r❡ T ✐s t❤❡ ▼❛r❦♦✈ ♦♣❡r❛t♦r ❝❤❛r❛❝t❡r✐③❡❞ ❜② µY |X ❍❡r❡ Zp = E(|Z|p)

1 p ✳

❍②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs s❛t✐s✜❡s ❛ ♣r♦♣❡rt② ❝❛❧❧❡❞ t❡♥s♦r✐③❛t✐♦♥ ✿ ■❢ (X1, Y1) ⊥ (X2, Y2) ❛r❡ ❜♦t❤ (p, q)✲❤②♣❡r❝♦♥tr❛❝t✐✈❡✱ t❤❡♥ ((X1, X2), (Y1, Y2)) ✐s ❛❧s♦ (p, q)✲❤②♣❡r❝♦♥tr❛❝t✐✈❡

  • ❡ts ❛r♦✉♥❞ t❤❡ ❝✉rs❡ ♦❢ ❞✐♠❡♥s✐♦♥❛❧✐t②✳

❚❤✐s ✭s❡r❡♥❞✐♣✐t♦✉s✮ r❡❞✐s❝♦✈❡r② ♦❢ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ t❤❡s❡ ❡q✉✐✈❛❧❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ✐s s♣✉rr✐♥❣ ❛ ❧♦t ♦❢ ✇♦r❦

✸✾

slide-103
SLIDE 103

❍②♣❡r❝♦♥tr❛❝t✐✈✐t②

❙t✉❞✐❡❞ ✐♥ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ❝s t❤❡♦r②✱ ❡t❝✳ ❉❡✜♥✐t✐♦♥ (X, Y ) ∼ µXY ✐s (p, q)✲❤②♣❡r❝♦♥tr❛❝t✐✈❡ ❢♦r 1 ≤ q ≤ p ✐❢ Tgp ≤ gq ∀g(Y ) ✇❤❡r❡ T ✐s t❤❡ ▼❛r❦♦✈ ♦♣❡r❛t♦r ❝❤❛r❛❝t❡r✐③❡❞ ❜② µY |X ❍❡r❡ Zp = E(|Z|p)

1 p ✳

❘❛t❤❡r ✐♠♠❡❞✐❛t❡ t❤❛t s✉❜✲❛❞❞✐t✐✈✐t②✱ ✐✳❡✳

CµX1Y1X2Y2 [H(X1Y1X2Y2) − λ1H(X1X2) − λ2H(Y1Y2)] ≤ CµX1Y1 [H(X1Y1) − λ1H(X1) − λ2H(Y1)] + CµX2Y2 [H(X2Y2) − λ1H(X2) − λ2H(Y2)]

✐s ❡q✉✐✈❛❧❡♥t t♦ t❡♥s♦r✐③❛t✐♦♥ ♦❢ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs ❚❤✐s ✭s❡r❡♥❞✐♣✐t♦✉s✮ r❡❞✐s❝♦✈❡r② ♦❢ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ t❤❡s❡ ❡q✉✐✈❛❧❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ✐s s♣✉rr✐♥❣ ❛ ❧♦t ♦❢ ✇♦r❦

✸✾

slide-104
SLIDE 104

❍②♣❡r❝♦♥tr❛❝t✐✈✐t②

❙t✉❞✐❡❞ ✐♥ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ❝s t❤❡♦r②✱ ❡t❝✳ ❉❡✜♥✐t✐♦♥ (X, Y ) ∼ µXY ✐s (p, q)✲❤②♣❡r❝♦♥tr❛❝t✐✈❡ ❢♦r 1 ≤ q ≤ p ✐❢ Tgp ≤ gq ∀g(Y ) ✇❤❡r❡ T ✐s t❤❡ ▼❛r❦♦✈ ♦♣❡r❛t♦r ❝❤❛r❛❝t❡r✐③❡❞ ❜② µY |X ❍❡r❡ Zp = E(|Z|p)

1 p ✳

❚❤✐s ✭s❡r❡♥❞✐♣✐t♦✉s✮ r❡❞✐s❝♦✈❡r② ♦❢ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ♠❡❛s✉r❡s ❛♥❞ t❤❡s❡ ❡q✉✐✈❛❧❡♥t ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ✐s s♣✉rr✐♥❣ ❛ ❧♦t ♦❢ ✇♦r❦

✸✾

slide-105
SLIDE 105

❈♦♥s❡q✉❡♥❝❡s

❈♦♠♣✉t❛t✐♦♥ ♦❢ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs ✐s ❝♦♥s✐❞❡r❡❞ ❤❛r❞

  • X ✐s ✉♥✐❢♦r♠ ❛♥❞ µY |X ✐s ❜✐♥❛r② s②♠♠❡tr✐❝ ❝❤❛♥♥❡❧

⋆ ✭❇♦♥❛♠✐✲❇❡❝❦♥❡r ✐♥❡q✉❛❧✐t② ✬✼✵s✱ ❇♦rr❡❧❧ ✬✽✷✮

  • (X, Y ) ❏♦✐♥t❧② ●❛✉ss✐❛♥ ✭●r♦ss ✬✼✺✮

❊✈❛❧✉❛t✐♦♥ ♦❢ ❛❝❤✐❡✈❛❜❧❡ r❡❣✐♦♥s ✐s ♦❢ s✐♠✐❧❛r ❞✐✣❝✉❧t② ❛s ❞❡t❡r♠✐♥✐♥❣ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ✭s❛♠❡ ❢❛♠✐❧② ❛♥❞ s✐♠✐❧❛r t❡r♠s✮ ❋♦r t❡st✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ s❝❤❡♠❡s ✇❡ ❤❛❞ t♦ ❞❡✈❡❧♦♣ t♦♦❧s ❢♦r ❡✈❛❧✉❛t✐♥❣ ❛❝❤✐❡✈❛❜❧❡ r❡❣✐♦♥s ❢♦r ❝❡rt❛✐♥ ❝❤❛♥♥❡❧s ❈❛♥ ✇❡ ✉s❡ ♦✉r t❡❝❤♥✐q✉❡s t♦ ❡✈❛❧✉❛t❡ ♥❡✇ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs❄ ❨❡s✱ ✇❡ ❝❛♥✳ ❊✳❣✳✿ ✐s ✉♥✐❢♦r♠ ❛♥❞ ✐s ❜✐♥❛r② ❡r❛s✉r❡ ❝❤❛♥♥❡❧ ✭◆❛✐r✲❲❛♥❣ ✬✶✻✱✬✶✼✮ ❖t❤❡r t❡❝❤♥✐q✉❡s ✇❡ ✉s❡❞ t♦ s♦❧✈❡ t❤❡s❡ ♥♦♥✲❝♦♥✈❡① ♣r♦❜❧❡♠s✿ ■❞❡♥t✐❢② ❛ ❧♦✇❡r ❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞ t❤❛t ❝♦♥t❛✐♥s ❛❧❧ t❤❡ st❛t✐♦♥❛r② ♣♦✐♥ts ❆♥❛❧②③❡ t❤❡ ❢✉♥❝t✐♦♥ ❞✐r❡❝t❧② ♦♥ t❤✐s ♠❛♥✐❢♦❧❞ ♦r ❯s❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♣♦✐♥ts ♦♥ t❤✐s ♠❛♥✐❢♦❧❞ t♦ ❞❡❞✉❝❡ s✉❜✲❛❞❞✐t✐✈✐t②

✹✵

slide-106
SLIDE 106

❈♦♥s❡q✉❡♥❝❡s

❈♦♠♣✉t❛t✐♦♥ ♦❢ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs ✐s ❝♦♥s✐❞❡r❡❞ ❤❛r❞

  • X ✐s ✉♥✐❢♦r♠ ❛♥❞ µY |X ✐s ❜✐♥❛r② s②♠♠❡tr✐❝ ❝❤❛♥♥❡❧

⋆ ✭❇♦♥❛♠✐✲❇❡❝❦♥❡r ✐♥❡q✉❛❧✐t② ✬✼✵s✱ ❇♦rr❡❧❧ ✬✽✷✮

  • (X, Y ) ❏♦✐♥t❧② ●❛✉ss✐❛♥ ✭●r♦ss ✬✼✺✮

❊✈❛❧✉❛t✐♦♥ ♦❢ ❛❝❤✐❡✈❛❜❧❡ r❡❣✐♦♥s ✐s ♦❢ s✐♠✐❧❛r ❞✐✣❝✉❧t② ❛s ❞❡t❡r♠✐♥✐♥❣ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ✭s❛♠❡ ❢❛♠✐❧② ❛♥❞ s✐♠✐❧❛r t❡r♠s✮ ❋♦r t❡st✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ s❝❤❡♠❡s ✇❡ ❤❛❞ t♦ ❞❡✈❡❧♦♣ t♦♦❧s ❢♦r ❡✈❛❧✉❛t✐♥❣ ❛❝❤✐❡✈❛❜❧❡ r❡❣✐♦♥s ❢♦r ❝❡rt❛✐♥ ❝❤❛♥♥❡❧s ❈❛♥ ✇❡ ✉s❡ ♦✉r t❡❝❤♥✐q✉❡s t♦ ❡✈❛❧✉❛t❡ ♥❡✇ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs❄ ❨❡s✱ ✇❡ ❝❛♥✳ ❊✳❣✳✿ X ✐s ✉♥✐❢♦r♠ ❛♥❞ µY |X ✐s ❜✐♥❛r② ❡r❛s✉r❡ ❝❤❛♥♥❡❧ ✭◆❛✐r✲❲❛♥❣ ✬✶✻✱✬✶✼✮ ❖t❤❡r t❡❝❤♥✐q✉❡s ✇❡ ✉s❡❞ t♦ s♦❧✈❡ t❤❡s❡ ♥♦♥✲❝♦♥✈❡① ♣r♦❜❧❡♠s✿ ■❞❡♥t✐❢② ❛ ❧♦✇❡r ❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞ t❤❛t ❝♦♥t❛✐♥s ❛❧❧ t❤❡ st❛t✐♦♥❛r② ♣♦✐♥ts ❆♥❛❧②③❡ t❤❡ ❢✉♥❝t✐♦♥ ❞✐r❡❝t❧② ♦♥ t❤✐s ♠❛♥✐❢♦❧❞ ♦r ❯s❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♣♦✐♥ts ♦♥ t❤✐s ♠❛♥✐❢♦❧❞ t♦ ❞❡❞✉❝❡ s✉❜✲❛❞❞✐t✐✈✐t②

✹✵

slide-107
SLIDE 107

❈♦♥s❡q✉❡♥❝❡s

❈♦♠♣✉t❛t✐♦♥ ♦❢ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs ✐s ❝♦♥s✐❞❡r❡❞ ❤❛r❞

  • X ✐s ✉♥✐❢♦r♠ ❛♥❞ µY |X ✐s ❜✐♥❛r② s②♠♠❡tr✐❝ ❝❤❛♥♥❡❧

⋆ ✭❇♦♥❛♠✐✲❇❡❝❦♥❡r ✐♥❡q✉❛❧✐t② ✬✼✵s✱ ❇♦rr❡❧❧ ✬✽✷✮

  • (X, Y ) ❏♦✐♥t❧② ●❛✉ss✐❛♥ ✭●r♦ss ✬✼✺✮

❊✈❛❧✉❛t✐♦♥ ♦❢ ❛❝❤✐❡✈❛❜❧❡ r❡❣✐♦♥s ✐s ♦❢ s✐♠✐❧❛r ❞✐✣❝✉❧t② ❛s ❞❡t❡r♠✐♥✐♥❣ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ✭s❛♠❡ ❢❛♠✐❧② ❛♥❞ s✐♠✐❧❛r t❡r♠s✮ ❋♦r t❡st✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ s❝❤❡♠❡s ✇❡ ❤❛❞ t♦ ❞❡✈❡❧♦♣ t♦♦❧s ❢♦r ❡✈❛❧✉❛t✐♥❣ ❛❝❤✐❡✈❛❜❧❡ r❡❣✐♦♥s ❢♦r ❝❡rt❛✐♥ ❝❤❛♥♥❡❧s ❈❛♥ ✇❡ ✉s❡ ♦✉r t❡❝❤♥✐q✉❡s t♦ ❡✈❛❧✉❛t❡ ♥❡✇ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs❄ ❨❡s✱ ✇❡ ❝❛♥✳ ❊✳❣✳✿ X ✐s ✉♥✐❢♦r♠ ❛♥❞ µY |X ✐s ❜✐♥❛r② ❡r❛s✉r❡ ❝❤❛♥♥❡❧ ✭◆❛✐r✲❲❛♥❣ ✬✶✻✱✬✶✼✮ ❖t❤❡r t❡❝❤♥✐q✉❡s ✇❡ ✉s❡❞ t♦ s♦❧✈❡ t❤❡s❡ ♥♦♥✲❝♦♥✈❡① ♣r♦❜❧❡♠s✿

  • ■❞❡♥t✐❢② ❛ ❧♦✇❡r ❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞ t❤❛t ❝♦♥t❛✐♥s ❛❧❧ t❤❡ st❛t✐♦♥❛r② ♣♦✐♥ts
  • ❆♥❛❧②③❡ t❤❡ ❢✉♥❝t✐♦♥ ❞✐r❡❝t❧② ♦♥ t❤✐s ♠❛♥✐❢♦❧❞ ♦r
  • ❯s❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♣♦✐♥ts ♦♥ t❤✐s ♠❛♥✐❢♦❧❞ t♦ ❞❡❞✉❝❡ s✉❜✲❛❞❞✐t✐✈✐t②

✹✵

slide-108
SLIDE 108

❘❡❝❛♣

❚❡st t❤❡ ♦♣t✐♠❛❧✐t② ♦❢ ❝♦❞✐♥❣ s❝❤❡♠❡s ✐♥ ♥❡t✇♦r❦ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

  • ❘❡s♦❧✈❡❞ s♦♠❡ ♦♣❡♥ q✉❡st✐♦♥s
  • ▼❛♥② r❡♠❛✐♥ ♦♣❡♥

❈♦♠♣✉t❡❞ t❤❡ ♦♣t✐♠✐③❡rs ♦❢ s❡✈❡r❛❧ ♥♦♥✲❝♦♥✈❡① ❢✉♥❝t✐♦♥❛❧s ❉❡✈❡❧♦♣❡❞ s♦♠❡ ♥❡✇ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s

  • ❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t②

❖♣t✐♠❛❧ ❛✉①✐❧✐❛r✐❡s ❝♦rr❡s♣♦♥❞ t♦ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡s ▼✐♥✲♠❛① t❤❡♦r❡♠ ▼♦r❡ ✐❞❡❛s ❛♥❞ t♦♦❧s s❡❡♠ ♥❡❝❡ss❛r② ❚❤❡s❡ ✭s♣❡❝✐✜❝ ❢❛♠✐❧②✮ ♥♦♥✲❝♦♥✈❡① ❢✉♥❝t✐♦♥❛❧s ❛❧s♦ ❛♣♣❡❛r ✐♥ ♦t❤❡r ✜❡❧❞s ❚❤❡ t♦♦❧s ✭❛❧r❡❛❞②✮ ❞❡✈❡❧♦♣❡❞ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❣❡t s♦♠❡ ♥❡✇ r❡s✉❧ts

✹✶

slide-109
SLIDE 109

❘❡❝❛♣

❚❡st t❤❡ ♦♣t✐♠❛❧✐t② ♦❢ ❝♦❞✐♥❣ s❝❤❡♠❡s ✐♥ ♥❡t✇♦r❦ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

  • ❘❡s♦❧✈❡❞ s♦♠❡ ♦♣❡♥ q✉❡st✐♦♥s
  • ▼❛♥② r❡♠❛✐♥ ♦♣❡♥

❈♦♠♣✉t❡❞ t❤❡ ♦♣t✐♠✐③❡rs ♦❢ s❡✈❡r❛❧ ♥♦♥✲❝♦♥✈❡① ❢✉♥❝t✐♦♥❛❧s

  • ❉❡✈❡❧♦♣❡❞ s♦♠❡ ♥❡✇ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s

⋆ ●❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t② ⋆ ❖♣t✐♠❛❧ ❛✉①✐❧✐❛r✐❡s ❝♦rr❡s♣♦♥❞ t♦ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡s ⋆ ▼✐♥✲♠❛① t❤❡♦r❡♠

  • ▼♦r❡ ✐❞❡❛s ❛♥❞ t♦♦❧s s❡❡♠ ♥❡❝❡ss❛r②

❚❤❡s❡ ✭s♣❡❝✐✜❝ ❢❛♠✐❧②✮ ♥♦♥✲❝♦♥✈❡① ❢✉♥❝t✐♦♥❛❧s ❛❧s♦ ❛♣♣❡❛r ✐♥ ♦t❤❡r ✜❡❧❞s ❚❤❡ t♦♦❧s ✭❛❧r❡❛❞②✮ ❞❡✈❡❧♦♣❡❞ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❣❡t s♦♠❡ ♥❡✇ r❡s✉❧ts

✹✶

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SLIDE 110

❘❡❝❛♣

❚❡st t❤❡ ♦♣t✐♠❛❧✐t② ♦❢ ❝♦❞✐♥❣ s❝❤❡♠❡s ✐♥ ♥❡t✇♦r❦ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

  • ❘❡s♦❧✈❡❞ s♦♠❡ ♦♣❡♥ q✉❡st✐♦♥s
  • ▼❛♥② r❡♠❛✐♥ ♦♣❡♥

❈♦♠♣✉t❡❞ t❤❡ ♦♣t✐♠✐③❡rs ♦❢ s❡✈❡r❛❧ ♥♦♥✲❝♦♥✈❡① ❢✉♥❝t✐♦♥❛❧s

  • ❉❡✈❡❧♦♣❡❞ s♦♠❡ ♥❡✇ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s

⋆ ●❛✉ss✐❛♥ ♦♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t② ⋆ ❖♣t✐♠❛❧ ❛✉①✐❧✐❛r✐❡s ❝♦rr❡s♣♦♥❞ t♦ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❝♦♥❝❛✈❡ ❡♥✈❡❧♦♣❡s ⋆ ▼✐♥✲♠❛① t❤❡♦r❡♠

  • ▼♦r❡ ✐❞❡❛s ❛♥❞ t♦♦❧s s❡❡♠ ♥❡❝❡ss❛r②

❚❤❡s❡ ✭s♣❡❝✐✜❝ ❢❛♠✐❧②✮ ♥♦♥✲❝♦♥✈❡① ❢✉♥❝t✐♦♥❛❧s ❛❧s♦ ❛♣♣❡❛r ✐♥ ♦t❤❡r ✜❡❧❞s

  • ❚❤❡ t♦♦❧s ✭❛❧r❡❛❞②✮ ❞❡✈❡❧♦♣❡❞ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❣❡t s♦♠❡ ♥❡✇ r❡s✉❧ts

✹✶

slide-111
SLIDE 111

❖✉t❧✐♥❡

  • ❇r♦❛❞❝❛st ❝❤❛♥♥❡❧✿ ❊st❛❜❧✐s❤✐♥❣ ♦♣t✐♠❛❧✐t② ♦❢ ▼❛rt♦♥✬s ❢♦r ▼■▼❖ ❜r♦❛❞❝❛st

❝❤❛♥♥❡❧

  • ■♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧✿ ❙✉❜✲♦♣t✐♠❛❧✐t② ♦❢ t❤❡ ❍❛♥✕❑♦❜❛②❛s❤✐ r❡❣✐♦♥
  • ❋❛♠✐❧② ♦❢ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s

⋆ ❘❡❧❛t✐♦♥ t♦ ♣r♦❜❧❡♠s ♦❢ ✐♥t❡r❡st ✐♥ ♦t❤❡r ✜❡❧❞s ⋆ ❯♥✐❢②✐♥❣ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ s♦♠❡ ❝♦♥❥❡❝t✉r❡s

✹✷

slide-112
SLIDE 112

❆♥ ❖❜s❡r✈❛t✐♦♥

❘❡♠✐♥❞❡r✿ ❋❛♠✐❧② ♦❢ ❢✉♥❝t✐♦♥❛❧s t❤❛t s❤♦✇❡❞ ✉♣ ✐♥ ♥❡t✇♦r❦ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②

  • S⊆[n]

αSH(XS), αS ∈ R. ❯s✉❛❧❧②✱ ♦♥❡ ✐s ✐♥t❡r❡st❡❞ ✐♥ t❡st✐♥❣ t❤❡ s✉❜✲❛❞❞✐t✐✈✐t② ♦❢ CµX[αSH(XS)]. ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ t❡st✐♥❣ ❛ ❣❧♦❜❛❧ t❡♥s♦r✐③❛t✐♦♥ ♣r♦♣❡rt②✳ ❉❡✜♥✐t✐♦♥ ❆ ❢✉♥❝t✐♦♥❛❧

S⊆[n] αSH(XS) ✐s s❛✐❞ t♦ s❛t✐s❢② ❣❧♦❜❛❧ t❡♥s♦r✐③❛t✐♦♥ ✐❢ ❛ ♣r♦❞✉❝t

❞✐str✐❜✉t✐♦♥ ♠❛①✐♠✐③❡s Gµ

12(γ1, γ2) ❢♦r ❛❧❧ γ1, γ2✱ ✇❤❡r❡

Gµ

12(γ1, γ2) :=

  • S⊆[n]

αSH(X1S, X2S) − E(γ1(X1)) − E(γ2(X2))

✹✸

slide-113
SLIDE 113

❆♥ ❖❜s❡r✈❛t✐♦♥

❉❡✜♥✐t✐♦♥ ❆ ❢✉♥❝t✐♦♥❛❧

S⊆[n] αSH(XS) ✐s s❛✐❞ t♦ s❛t✐s❢② ❧♦❝❛❧ t❡♥s♦r✐③❛t✐♦♥ ✐❢ t❤❡ ♣r♦❞✉❝t

♦❢ ❧♦❝❛❧ ♠❛①✐♠✐③❡rs ♦❢ Gµ1(γ1) ❛♥❞ Gµ2(γ2) ✐s ❛ ❧♦❝❛❧ ♠❛①✐♠✐③❡r ♦❢ Gµ

12(γ1, γ2)

❢♦r ❛❧❧ γ1, γ2✱ ✇❤❡r❡ Gµ

1(γ1) :=

  • S⊆[n]

αSH(X1S) − E(γ1(X1)) Gµ

2(γ2) :=

  • S⊆[n]

αSH(X2S) − E(γ2(X2)) Gµ

12(γ1, γ2) :=

  • S⊆[n]

αSH(X1S, X2S) − E(γ1(X1)) − E(γ2(X2)) ❖❜s❡r✈❛t✐♦♥ ✭❈♦♥❥❡❝t✉r❡✮ ❋♦r ❢✉♥❝t✐♦♥❛❧s ✐♥ t❤✐s ❢❛♠✐❧② ❣❧♦❜❛❧ t❡♥s♦r✐③❛t✐♦♥ ❤♦❧❞s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❧♦❝❛❧ t❡♥s♦r✐③❛t✐♦♥ ❤♦❧❞s ◆♦t❡✿ ❙✐♠✐❧❛r✐t② t♦ t❡st✐♥❣ ❝♦♥❝❛✈✐t② ✉s✐♥❣ ❛ ❧♦❝❛❧ ✭s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✮ ❝♦♥❞✐t✐♦♥

✹✸

slide-114
SLIDE 114

❆♥ ❖❜s❡r✈❛t✐♦♥

❉❡✜♥✐t✐♦♥ ❆ ❢✉♥❝t✐♦♥❛❧

S⊆[n] αSH(XS) ✐s s❛✐❞ t♦ s❛t✐s❢② ❧♦❝❛❧ t❡♥s♦r✐③❛t✐♦♥ ✐❢ t❤❡ ♣r♦❞✉❝t

♦❢ ❧♦❝❛❧ ♠❛①✐♠✐③❡rs ♦❢ Gµ1(γ1) ❛♥❞ Gµ2(γ2) ✐s ❛ ❧♦❝❛❧ ♠❛①✐♠✐③❡r ♦❢ Gµ

12(γ1, γ2)

❢♦r ❛❧❧ γ1, γ2✱ ✇❤❡r❡ Gµ

1(γ1) :=

  • S⊆[n]

αSH(X1S) − E(γ1(X1)) Gµ

2(γ2) :=

  • S⊆[n]

αSH(X2S) − E(γ2(X2)) Gµ

12(γ1, γ2) :=

  • S⊆[n]

αSH(X1S, X2S) − E(γ1(X1)) − E(γ2(X2)) ❖❜s❡r✈❛t✐♦♥ ✭❈♦♥❥❡❝t✉r❡✮ ❋♦r ❢✉♥❝t✐♦♥❛❧s ✐♥ t❤✐s ❢❛♠✐❧② ❣❧♦❜❛❧ t❡♥s♦r✐③❛t✐♦♥ ❤♦❧❞s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❧♦❝❛❧ t❡♥s♦r✐③❛t✐♦♥ ❤♦❧❞s ◆♦t❡✿ ❙✐♠✐❧❛r✐t② t♦ t❡st✐♥❣ ❝♦♥❝❛✈✐t② ✉s✐♥❣ ❛ ❧♦❝❛❧ ✭s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✮ ❝♦♥❞✐t✐♦♥

✹✸

slide-115
SLIDE 115

◆♦t❡s

❋♦r s♦♠❡ ♦❢ t❤❡ r❡♠❛✐♥✐♥❣ ♦♣❡♥ ♣r♦❜❧❡♠s ✭♠❡♥t✐♦♥❡❞ ❡❛r❧✐❡r✮✱ ✇❡ ❝❛♥ ❡st❛❜❧✐s❤ ❧♦❝❛❧✲t❡♥s♦r✐③❛t✐♦♥

  • ▼❛rt♦♥✬s ✐♥♥❡r ❜♦✉♥❞ ❢♦r ❜✐♥❛r② ✐♥♣✉t ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s
  • ●❛✉ss✐❛♥ ❩✲✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧

❚❤❡r❡❢♦r❡✱ ✐❢ t❤❡ ❈♦♥❥❡❝t✉r❡ ✐s tr✉❡✱ t❤❡♥ ✇❡ ✇♦✉❧❞ ❡st❛❜❧✐s❤ t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ❢♦r t❤❡s❡ s❡tt✐♥❣s ◗✉❡st✐♦♥✿ ❍♦✇ ♠❛② t❤❡s❡ t✇♦ ♣❤❡♥♦♠❡♥❛ ❜❡ ❝♦♥♥❡❝t❡❞❄ ❆ ♣♦ss✐❜❧❡ ❛♥s✇❡r ✐s ✭❛❣❛✐♥✮ s✉❣❣❡st❡❞ ❜② ♦✉r ❝♦♠♣✉t❛t✐♦♥s ✐♥ ✈❛r✐♦✉s ❡①❛♠♣❧❡s

✹✹

slide-116
SLIDE 116

◆♦t❡s

❋♦r s♦♠❡ ♦❢ t❤❡ r❡♠❛✐♥✐♥❣ ♦♣❡♥ ♣r♦❜❧❡♠s ✭♠❡♥t✐♦♥❡❞ ❡❛r❧✐❡r✮✱ ✇❡ ❝❛♥ ❡st❛❜❧✐s❤ ❧♦❝❛❧✲t❡♥s♦r✐③❛t✐♦♥

  • ▼❛rt♦♥✬s ✐♥♥❡r ❜♦✉♥❞ ❢♦r ❜✐♥❛r② ✐♥♣✉t ❜r♦❛❞❝❛st ❝❤❛♥♥❡❧s
  • ●❛✉ss✐❛♥ ❩✲✐♥t❡r❢❡r❡♥❝❡ ❝❤❛♥♥❡❧

❚❤❡r❡❢♦r❡✱ ✐❢ t❤❡ ❈♦♥❥❡❝t✉r❡ ✐s tr✉❡✱ t❤❡♥ ✇❡ ✇♦✉❧❞ ❡st❛❜❧✐s❤ t❤❡ ❝❛♣❛❝✐t② r❡❣✐♦♥ ❢♦r t❤❡s❡ s❡tt✐♥❣s ◗✉❡st✐♦♥✿ ❍♦✇ ♠❛② t❤❡s❡ t✇♦ ♣❤❡♥♦♠❡♥❛ ❜❡ ❝♦♥♥❡❝t❡❞❄ ❆ ♣♦ss✐❜❧❡ ❛♥s✇❡r ✐s ✭❛❣❛✐♥✮ s✉❣❣❡st❡❞ ❜② ♦✉r ❝♦♠♣✉t❛t✐♦♥s ✐♥ ✈❛r✐♦✉s ❡①❛♠♣❧❡s

✹✹

slide-117
SLIDE 117

❈♦♥❥❡❝t✉r❡ ✷

❈♦♥❥❡❝t✉r❡ ✷ ❈♦♥s✐❞❡r fα(γ) = max

µX

  • S⊆[n]

αSH(XS) − E(γ(X)), αS ∈ R. ❙✉♣♣♦s❡ α(0)

S

❛♥❞ α(1)

S

❤❛✈❡ ✐♥t❡r✐♦r ❣❧♦❜❛❧ ♠❛①✐♠✐③❡rs✳ ▲❡t α(t)

S = (1 − t)α(0) S + tα(1) S ✱ t ∈ [0, 1]✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ♣❛t❤ ✐♥ t❤❡

s✐♠♣❧❡① s✉❝❤ t❤❛t µ(t) ✐s ❛ ❣❧♦❜❛❧ ♠❛①✐♠✐③❡r ♦❢ fα(t)(γ) ❢♦r ❛❧❧ t ∈ [0, 1]✳ ❈♦♥s❡q✉❡♥❝❡s✿ ■♥❢♦r♠❛t✐♦♥ t❤❡♦r②✿ ❈♦♥❥❡❝t✉r❡ ✷ ✭♣❧✉s ♠✐❧❞ r❡❣✉❧❛r✐t② ❝♦♥❞✐t✐♦♥s✮ ✐♠♣❧✐❡s t❤❡ ❈♦♥❥❡❝t✉r❡ t❤❛t ❧♦❝❛❧ t❡♥s♦r✐③❛t✐♦♥ ✐♠♣❧✐❡s ❣❧♦❜❛❧ t❡♥s♦r✐③❛t✐♦♥ ❆❧❣♦r✐t❤♠s✿ ❙✉♣♣♦s❡ ♦♥❡ ✇❛♥ts t♦ ❛♣♣r♦①✐♠❛t❡ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs ❙t❛rt ✇✐t❤ ❆♣♣r♦①✐♠❛t❡ t❤❡ ♠❛①✐♠✐③✐♥❣ ❞✐str✐❜✉t✐♦♥ ❛t t❤✐s ❜♦✉♥❞❛r② ✈❛❧✉❡ ♦❢ ♥♦r♠✳ ❉❡❝r❡❛s❡ ❛♥❞ tr❛❝❦ t❤❡ ❣❧♦❜❛❧ ♠❛①✐♠✐③❡rs ❜② ❧♦❝❛❧ s❡❛r❝❤✳

✹✺

slide-118
SLIDE 118

❈♦♥❥❡❝t✉r❡ ✷

❈♦♥❥❡❝t✉r❡ ✷ ❈♦♥s✐❞❡r fα(γ) = max

µX

  • S⊆[n]

αSH(XS) − E(γ(X)), αS ∈ R. ❙✉♣♣♦s❡ α(0)

S

❛♥❞ α(1)

S

❤❛✈❡ ✐♥t❡r✐♦r ❣❧♦❜❛❧ ♠❛①✐♠✐③❡rs✳ ▲❡t α(t)

S = (1 − t)α(0) S + tα(1) S ✱ t ∈ [0, 1]✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ♣❛t❤ ✐♥ t❤❡

s✐♠♣❧❡① s✉❝❤ t❤❛t µ(t) ✐s ❛ ❣❧♦❜❛❧ ♠❛①✐♠✐③❡r ♦❢ fα(t)(γ) ❢♦r ❛❧❧ t ∈ [0, 1]✳ ❈♦♥s❡q✉❡♥❝❡s✿

  • ■♥❢♦r♠❛t✐♦♥ t❤❡♦r②✿ ❈♦♥❥❡❝t✉r❡ ✷ ✭♣❧✉s ♠✐❧❞ r❡❣✉❧❛r✐t② ❝♦♥❞✐t✐♦♥s✮ ✐♠♣❧✐❡s t❤❡

❈♦♥❥❡❝t✉r❡ t❤❛t ❧♦❝❛❧ t❡♥s♦r✐③❛t✐♦♥ ✐♠♣❧✐❡s ❣❧♦❜❛❧ t❡♥s♦r✐③❛t✐♦♥

  • ❆❧❣♦r✐t❤♠s✿ ❙✉♣♣♦s❡ ♦♥❡ ✇❛♥ts t♦ ❛♣♣r♦①✐♠❛t❡ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ♣❛r❛♠❡t❡rs

⋆ ❙t❛rt ✇✐t❤ p → ∞ ⋆ ❆♣♣r♦①✐♠❛t❡ t❤❡ ♠❛①✐♠✐③✐♥❣ ❞✐str✐❜✉t✐♦♥ ❛t t❤✐s ❜♦✉♥❞❛r② ✈❛❧✉❡ ♦❢ ♥♦r♠✳ ⋆ ❉❡❝r❡❛s❡ p ❛♥❞ tr❛❝❦ t❤❡ ❣❧♦❜❛❧ ♠❛①✐♠✐③❡rs ❜② ❧♦❝❛❧ s❡❛r❝❤✳

✹✺

slide-119
SLIDE 119

❖♣t✐♠✐③❛t✐♦♥ ❜❛s❡❞ ❛♣♣r♦❛❝❤❡s

❖♣t✐♠✐③❛t✐♦♥ ❜❛s❡❞ ❛♣♣r♦❛❝❤❡s ❤❛✈❡ ❜❡❡♥ ❣❛♠❡ ❝❤❛♥❣❡rs ❋✐rst ❥✉♠♣✿ ▲✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ t♦ ❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❙❡♠✐✲❞❡✜♥✐t❡ ♣r♦❣r❛♠ ❜❛s❡❞ ❛❧❣♦r✐t❤♠ ❞❡s✐❣♥ ❛♥❞ ❛♥❛❧②s✐s

  • ❈♦♠♣r❡ss✐✈❡ s❡♥s✐♥❣
  • P❤❛s❡ r❡❝♦✈❡r②
  • ❈❧✉st❡r✐♥❣
  • ■♠❛❣❡ ♣r♦❝❡ss✐♥❣

◆❡✇ ❏✉♠♣✿ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ t♦ s♣❡❝✐✜❝ ❢❛♠✐❧✐❡s ♦❢ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❙t✉❞✐❡s ♦♥ t❤❡s❡ ❢❛♠✐❧✐❡s ❛r❡ ❛❧r❡❛❞② ♠❛❦✐♥❣ ✐♠♣❛❝t ✐♥ ▼❛❝❤✐♥❡ ❧❡❛r♥✐♥❣ ❛♥❞ ❆■ ✭❙✐♥❣✉❧❛r ❱❛❧✉❡ ❉❡❝♦♠♣♦s✐t✐♦♥✮

  • r❛♣❤✐❝❛❧ ♠♦❞❡❧s ❛♥❞ ❙t❛t✐st✐❝❛❧ P❤②s✐❝s ❜❛s❡❞ ❛♣♣r♦❛❝❤❡s ✭s✉♠ ♦❢ ❡♥❡r❣② ❛♥❞

❡♥tr♦♣② t❡r♠s✮ ❈♦♠♠✉♥✐❝❛t✐♦♥ ♥❡t✇♦r❦s ✭❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❡♥tr♦♣✐❡s ♦❢ s✉❜s❡ts✮

✹✻

slide-120
SLIDE 120

❖♣t✐♠✐③❛t✐♦♥ ❜❛s❡❞ ❛♣♣r♦❛❝❤❡s

❖♣t✐♠✐③❛t✐♦♥ ❜❛s❡❞ ❛♣♣r♦❛❝❤❡s ❤❛✈❡ ❜❡❡♥ ❣❛♠❡ ❝❤❛♥❣❡rs ❋✐rst ❥✉♠♣✿ ▲✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ t♦ ❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❙❡♠✐✲❞❡✜♥✐t❡ ♣r♦❣r❛♠ ❜❛s❡❞ ❛❧❣♦r✐t❤♠ ❞❡s✐❣♥ ❛♥❞ ❛♥❛❧②s✐s

  • ❈♦♠♣r❡ss✐✈❡ s❡♥s✐♥❣
  • P❤❛s❡ r❡❝♦✈❡r②
  • ❈❧✉st❡r✐♥❣
  • ■♠❛❣❡ ♣r♦❝❡ss✐♥❣

◆❡✇ ❏✉♠♣✿ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ t♦ s♣❡❝✐✜❝ ❢❛♠✐❧✐❡s ♦❢ ♥♦♥✲❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❙t✉❞✐❡s ♦♥ t❤❡s❡ ❢❛♠✐❧✐❡s ❛r❡ ❛❧r❡❛❞② ♠❛❦✐♥❣ ✐♠♣❛❝t ✐♥

  • ▼❛❝❤✐♥❡ ❧❡❛r♥✐♥❣ ❛♥❞ ❆■ ✭❙✐♥❣✉❧❛r ❱❛❧✉❡ ❉❡❝♦♠♣♦s✐t✐♦♥✮
  • ●r❛♣❤✐❝❛❧ ♠♦❞❡❧s ❛♥❞ ❙t❛t✐st✐❝❛❧ P❤②s✐❝s ❜❛s❡❞ ❛♣♣r♦❛❝❤❡s ✭s✉♠ ♦❢ ❡♥❡r❣② ❛♥❞

❡♥tr♦♣② t❡r♠s✮

  • ❈♦♠♠✉♥✐❝❛t✐♦♥ ♥❡t✇♦r❦s ✭❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❡♥tr♦♣✐❡s ♦❢ s✉❜s❡ts✮

✹✻

slide-121
SLIDE 121

❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts ✭❘♦❣✉❡s ❣❛❧❧❡r②✮

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ❨❛♥❧✐♥ ●❡♥❣ ❙✐❞❛ ▲✐✉ ❙❛❧♠❛♥ ❇❡✐❣✐ ❆♠✐♥ ●♦❤❛r✐ ❉❛✈✐❞ ◆❣ ▲✐♥❣①✐❛♦ ❳✐❛ ▼❛① ❈♦st❛ ❱❛r✉♥ ❏♦❣ ❱✐♥❝❡♥t ❲❛♥❣ ❇❛❜❛❦ ❨❛③❞❛♥♣❛♥❛❤ ❆❜❜❛s ❊❧ ●❛♠❛❧ ❏❛♥♦s ❑♦r♥❡r ❨❛♥ ◆❛♥ ❲❛♥❣

❚❍❆◆❑ ❨❖❯

slide-122
SLIDE 122

❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts ✭❘♦❣✉❡s ❣❛❧❧❡r②✮

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ❨❛♥❧✐♥ ●❡♥❣ ❙✐❞❛ ▲✐✉ ❙❛❧♠❛♥ ❇❡✐❣✐ ❆♠✐♥ ●♦❤❛r✐ ❉❛✈✐❞ ◆❣ ▲✐♥❣①✐❛♦ ❳✐❛ ▼❛① ❈♦st❛ ❱❛r✉♥ ❏♦❣ ❱✐♥❝❡♥t ❲❛♥❣ ❇❛❜❛❦ ❨❛③❞❛♥♣❛♥❛❤ ❆❜❜❛s ❊❧ ●❛♠❛❧ ❏❛♥♦s ❑♦r♥❡r ❨❛♥ ◆❛♥ ❲❛♥❣

❚❍❆◆❑ ❨❖❯