On the Scalar Gaussian Interference Channel Chandra Nair, & - - PowerPoint PPT Presentation

SLIDE 1

On the Scalar Gaussian Interference Channel

Chandra Nair, & David Ng

The Chinese University of Hong Kong ITA 2018 13 Feb, 2018

SLIDE 2

Question Does Han-Kobayashi achievable region with Gaussian signaling exhaust the capacity region of the scalar Gaussian interference channel?

Chandra Nair and David Ng GIC 13 Feb, 2018 2 / 15

SLIDE 3

Question Does Han-Kobayashi achievable region with Gaussian signaling exhaust the capacity region of the scalar Gaussian interference channel? This talk Perhaps it may

◮ We establish some evidence towards this end ◮ Conjecture an information inequality, which if true, would establish the

• ptimality for the Z-interference channel

Chandra Nair and David Ng GIC 13 Feb, 2018 2 / 15

SLIDE 4

Scalar Gaussian Interference Channel

M1 M2 Encoder 1 Encoder 2 Xn

1

Xn

2

+ +

Zn

1

Zn

2

a b Y n

1

Y n

2

Decoder 1 Decoder 2 ˆ M1 ˆ M2

(Some) known results about the capacity region

◮ Determined a ≥ 1, b ≥ 1 (Sato ’79) ◮ Corner Points (Sato ’81, Costa ’85, Sason ’02, Polyanskiy-Wu ’15) ◮ Maximum rate-sum a(1 + b2P2) + b(1 + a2P1) ≤ 1 (3 groups ’09) ◮ Han–Kobayashi region within 0.5 bits per dimension (Etkin, Tse, Wang ’07)

Chandra Nair and David Ng GIC 13 Feb, 2018 3 / 15

SLIDE 5

As a side note

Investigations on this problem have led to

◮ Costa’s discovery: concavity of entropy power ◮ Use of HWI to establish converses (Polyanskiy-Wu ’15) ◮ Use of "genies"

To establish converses/bounds (Kramer, Etkin-Tse-Wang, ...) As a tool for proving sub-additivity/tensorization

Chandra Nair and David Ng GIC 13 Feb, 2018 4 / 15

SLIDE 6

On the Han–Kobayashi achievable region

Background

◮ 1981: Han and Kobayashi proposed an achievable region (HK-IB) for memoryless

interference channels

◮ 2015: HK-IB was shown to be strictly sub-optimal for some channels (with: Xia,

Yazdanpanah)

Result: 2-letter extension of HK-IB outperformed HK-IB Difficulty: Evaluating HK-IB (1-letter and 2-letter) Channels: Clean Z-interference channels

Chandra Nair and David Ng GIC 13 Feb, 2018 5 / 15

SLIDE 7

On the Han–Kobayashi achievable region

Background

◮ 1981: Han and Kobayashi proposed an achievable region (HK-IB) for memoryless

interference channels

◮ 2015: HK-IB was shown to be strictly sub-optimal for some channels (with: Xia,

Yazdanpanah)

Result: 2-letter extension of HK-IB outperformed HK-IB Difficulty: Evaluating HK-IB (1-letter and 2-letter) Channels: Clean Z-interference channels

Natural Questions How about if one restricts to the special case: scalar Gaussian interference channels?

◮ Is HK-IB (with Gaussian signaling) optimal? ◮ Or does k-letter extensions (with Gaussian signaling), or in other words do

correlated Gaussian input vectors improve the region?

Remark: There is a paper (2016) that claims such an improvement but it ignores the role of "power control" (which was known to improve on naive region since 1985; see also Costa - ITA 2010)

Chandra Nair and David Ng GIC 13 Feb, 2018 5 / 15

SLIDE 8

On the Han–Kobayashi achievable region

Background

◮ 1981: Han and Kobayashi proposed an achievable region (HK-IB) for memoryless

interference channels

◮ 2015: HK-IB was shown to be strictly sub-optimal for some channels (with: Xia,

Yazdanpanah)

Result: 2-letter extension of HK-IB outperformed HK-IB Difficulty: Evaluating HK-IB (1-letter and 2-letter) Channels: Clean Z-interference channels

Natural Questions How about if one restricts to the special case: scalar Gaussian interference channels?

◮ Is HK-IB (with Gaussian signaling) optimal? ◮ Or does k-letter extensions (with Gaussian signaling), or in other words do

correlated Gaussian input vectors improve the region?

Remark: There is a paper (2016) that claims such an improvement but it ignores the role of "power control" (which was known to improve on naive region since 1985; see also Costa - ITA 2010)

◮ Main Result: No improvement in going to correlated Gaussians

Cheng and Verdu had such a result for αI(Xk

1 ; Y k 1 ) + I(Xk 2 ; Y k 2 ) (1993)

We had a similar result for Z-interference (b = 0) last year.

Chandra Nair and David Ng GIC 13 Feb, 2018 5 / 15

SLIDE 9

H–K IB with Gaussian signaling (k-letter)

Non-negative rate pairs R1, R2 satisfying

R1 ≤ 1 2k EQ  log

• I + (KQ

U1 + KQ V1 ) + b2KQ V2

• I + b2KQ

V2

• 

 R2 ≤ 1 2k EQ  log

• I + (KQ

U2 + KQ V2 ) + a2KQ V1

• I + a2KQ

V1

• 

 R1 + R2 ≤ 1 2k EQ  log

• I + (KQ

U1 + KQ V1 ) + b2(KQ U2 + KQ V2 )

• I + b2KQ

V2

• + log
• I + KQ

V2 + a2KQ V1

• I + a2KQ

V1

• 

 R1 + R2 ≤ 1 2k EQ   1 2k log

• I + (KQ

U2 + KQ V2 ) + a2(KQ U1 + KQ V1 )

• I + a2KQ

V1

• + log
• I + KQ

V1 + b2KQ V2

• I + b2KQ

V2

• 

 R1 + R2 ≤ 1 2k EQ  log

• I + KQ

V1 + b2(KQ U2 + KQ V2 )

• I + b2KQ

V2

• + log
• I + KQ

V2 + a2(KQ U1 + KQ V1 )

• I + a2KQ

V1

• 

 2R1 + R2 ≤ 1 2k EQ  log

• I + (KQ

U1 + KQ V1 ) + b2(KQ U2 + KQ V2 )

• I + b2KQ

V2

• + log
• I + KQ

V1 + b2KQ V2

• I + b2KQ

V2

• + log
• I + KQ

V2 + a2(KQ U1 + KQ V1 )

• I + a2KQ

V1

• 

 R1 + 2R2 ≤ 1 2k EQ  log

• I + (KQ

U2 + KQ V2 ) + a2(KQ U1 + KQ V1 )

• I + a2KQ

V1

• + log
• I + KQ

V2 + a2KQ V1

• I + a2KQ

V1

• + log
• I + KQ

V1 + b2(KQ U2 + KQ V2 )

• I + b2KQ

V2

• 



for some Kq

U1, Kq V1, Kq U2, Kq V2 0 satisfying EQ

• tr
• KQ

U1 + KQ V1

• ≤ kP1 and

EQ

• tr
• KQ

U2 + KQ V2

• ≤ kP2, and some “time-sharing" variable Q.

Chandra Nair and David Ng GIC 13 Feb, 2018 6 / 15

SLIDE 10

Result: k-letter region is identical to 1-letter region

Note: Dealing with optimizers of a non-convex optimization problem

Chandra Nair and David Ng GIC 13 Feb, 2018 7 / 15

SLIDE 11

Result: k-letter region is identical to 1-letter region

Note: Dealing with optimizers of a non-convex optimization problem Proof: Define

ˆ Kq

V1 := diag

• {λi(Kq

V1)}

• ˆ

Kq

U1 := diag

• {λi(Kq

U1 + Kq V1) − λi(Kq V1)}

• ˆ

Kq

V2 := diag

• {λn+1−i(Kq

V2)}

• ˆ

Kq

U2 := diag

• {λn+1−i(Kq

U2 + Kq V2) − λn+1−i(Kq V1)}

• .

where λ1(A) ≤ · · · ≤ λk(A) denote the eigenvalues of a k × k Hermitian matrix A, and diag({ai}) indicates a diagonal matrix with diagonal entries a1, .., ak. These choices dominate the inequalities term-by-term. This "observation" and feasibility of these choices relies on two well-known results. Difficulty: Making this guess (came after a few months of failed other approaches) There were multiple solutions to KKT conditions, for instance.

Chandra Nair and David Ng GIC 13 Feb, 2018 7 / 15

SLIDE 12

Two results

Theorem (Courant-Fischer min-max theorem) Let A be a k × k Hermitian matrix. Then we have λi(A) = inf

V ⊆Rk dim V =i

sup

x∈V x=1

xT Ax = sup

V ⊆Rk dim V =n−i+1

inf

x∈V x=1

xT Ax, where V denotes subspaces of the indicated dimension. Corollary Let A, B be k × k Hermitian matrices with B 0. Then λi(A + B) ≥ λi(A) for i = 1, · · · , k.

Chandra Nair and David Ng GIC 13 Feb, 2018 8 / 15

SLIDE 13

Two results

Theorem (Courant-Fischer min-max theorem) Let A be a k × k Hermitian matrix. Then we have λi(A) = inf

V ⊆Rk dim V =i

sup

x∈V x=1

xT Ax = sup

V ⊆Rk dim V =n−i+1

inf

x∈V x=1

xT Ax, where V denotes subspaces of the indicated dimension. Corollary Let A, B be k × k Hermitian matrices with B 0. Then λi(A + B) ≥ λi(A) for i = 1, · · · , k. Theorem (Fiedler ’71) Let A, B be k × k Hermitian matrices. Suppose λk(A) + λk(B) ≥ 0. Then

k

• i=1

(λi(A) + λi(B)) ≤ |A + B| ≤

k

• i=1

(λi(A) + λk+1−i(B))

Chandra Nair and David Ng GIC 13 Feb, 2018 8 / 15

SLIDE 14

What next?

Obvious: Do Gaussian inputs optimize HK-IB? Observations

◮ Timesharing variable Q is a cause of trouble ◮ Without Q, there are P1, P2 for which non-Gaussian distributions outperform

Gaussian distribution

Using perturbations based on Hermite Polynomials (Abbe-Zhang 09)

Chandra Nair and David Ng GIC 13 Feb, 2018 9 / 15

SLIDE 15

What next?

Obvious: Do Gaussian inputs optimize HK-IB? Observations

◮ Timesharing variable Q is a cause of trouble ◮ Without Q, there are P1, P2 for which non-Gaussian distributions outperform

Gaussian distribution

Using perturbations based on Hermite Polynomials (Abbe-Zhang 09)

◮ What is Q doing?

Answer: Q is used to compute the upper concave envelope of a functional defined

• n (P1, P2)

Observation: Since the dual of the dual (in the sense of Fenchel) yields the concave envelope, we just need to check that Gaussians optimize the dual functional

Chandra Nair and David Ng GIC 13 Feb, 2018 9 / 15

SLIDE 16

A conjecture

Let α, β ≥ 0, and λ ≥ 1 be constants. Conjecture (main) The maximum of (λ − 1)h(X2 + aX1 + Z) + h(X1 + Z) − λh(aX1 + Z) − α E(X12) − β E(X22)

• ver independent variables X1 and X2 taking values in Rk is attained by Gaussians

X1 ∼ N(0, aI), X2 ∼ N(0, bI).

Chandra Nair and David Ng GIC 13 Feb, 2018 10 / 15

SLIDE 17

A conjecture

Let α, β ≥ 0, and λ ≥ 1 be constants. Conjecture (main) The maximum of (λ − 1)h(X2 + aX1 + Z) + h(X1 + Z) − λh(aX1 + Z) − α E(X12) − β E(X22)

• ver independent variables X1 and X2 taking values in Rk is attained by Gaussians

X1 ∼ N(0, aI), X2 ∼ N(0, bI). Why should one care about this

◮ If true, this establishes the capacity region of the Gaussian Z-interference channel ◮ Let α = 0. Suppose you show that, ∀β > 0, ∃λ∗ < ∞ such that the conjecture is

true ∀λ ≥ λ∗, then we improve on the outer bound obtained using HWI

◮ Flavor of a reverse-entropy-power inequality

Chandra Nair and David Ng GIC 13 Feb, 2018 10 / 15

SLIDE 18

Remarks

Easy regimes The conjecture is true when either

◮ β ≥ λ−1 2 ◮ α ≥ 1−a2 2

Proof: Consequence of data-processing and Entropy-Power-Inequality (or doubling trick)

Chandra Nair and David Ng GIC 13 Feb, 2018 11 / 15

SLIDE 19

Observation (numerical)

Appears that Stam’s path (O-U semigroup) may work Let X1, X2 be independent random variables. Suppose Q∗

1, Q∗ 2 maximizes

λ − 1 2 log(1 + a2Q1 + Q2) + 1 2 log(1 + Q1) − λ 2 log(1 + a2Q1) − αQ1 − βQ2. For t ∈ [0, 1] define f(t) := (λ − 1)h(X2t + aX1t + Z) + h(X1t + Z) − λh(aX1t + Z) − α E(X2

1t) − β E(X2 2t)

where X1t := √ 1 − tX1 + √ tN(0, Q∗

1)

X2t := √ 1 − tX2 + √ tN(0, Q∗

2).

Then f(t) is increasing and concave.

Chandra Nair and David Ng GIC 13 Feb, 2018 12 / 15

SLIDE 20

Increasing along the path

Conjecture Let X1, X2 be independent random variables. Suppose Q∗

1, Q∗ 2 maximizes

λ − 1 2 log(1 + a2Q1 + Q2) + 1 2 log(1 + Q1) − λ 2 log(1 + a2Q1) − αQ1 − βQ2 Then (λ − 1)(Q∗

2 + a2Q∗ 1 + 1)I(X2 + aX1 + Z) + (Q∗ 1 + 1)I(X1 + Z)

− λ(a2Q∗

1 + 1)I(aX1 + Z) − 2α(Q∗ 1 − E

• X2

1

• ) − 2β(Q∗

2 − E

• X2

2

• )

≥ 0, where I(X) is the Fisher information of X.

Chandra Nair and David Ng GIC 13 Feb, 2018 13 / 15

SLIDE 21

Increasing along the path

Conjecture Let X1, X2 be independent random variables. Suppose Q∗

1, Q∗ 2 maximizes

λ − 1 2 log(1 + a2Q1 + Q2) + 1 2 log(1 + Q1) − λ 2 log(1 + a2Q1) − αQ1 − βQ2 Then (λ − 1)(Q∗

2 + a2Q∗ 1 + 1)I(X2 + aX1 + Z) + (Q∗ 1 + 1)I(X1 + Z)

− λ(a2Q∗

1 + 1)I(aX1 + Z) − 2α(Q∗ 1 − E

• X2

1

• ) − 2β(Q∗

2 − E

• X2

2

• )

≥ 0, where I(X) is the Fisher information of X. Can establish this for some subset of the parameter space (involving E(X2

1), E(X2 2)).

Chandra Nair and David Ng GIC 13 Feb, 2018 13 / 15

SLIDE 22

Remarks about doubling trick

Does the "doubling trick" work to show Gaussian optimality? Remarks

◮ There are interference channels for which

CX1⊥X2[(λ − 1)H(Y2) + H(Y1) − λH(Y2|X2)] is not sub-additive

◮ To make this approach work, one needs to show the sub-additivity of the above

functional for Gaussian interference channel

The proof of subadditivity needs to use the channel structure

Chandra Nair and David Ng GIC 13 Feb, 2018 14 / 15

SLIDE 23

Remarks about doubling trick

Does the "doubling trick" work to show Gaussian optimality? Remarks

◮ There are interference channels for which

CX1⊥X2[(λ − 1)H(Y2) + H(Y1) − λH(Y2|X2)] is not sub-additive

◮ To make this approach work, one needs to show the sub-additivity of the above

functional for Gaussian interference channel

The proof of subadditivity needs to use the channel structure

◮ Of course, there are various arguments in literature that does rely on channel

structure

Genie based converses Injective deterministic interference channel

Chandra Nair and David Ng GIC 13 Feb, 2018 14 / 15

SLIDE 24

Recap

◮ Correlated Gaussians do not improve the HK-IB ◮ Conjectured an entropy-variance inequality

Motivated by considering the Fenchel dual form of a functional arising in the H–K region for the Z-interference channel Presented possible attack strategies

Chandra Nair and David Ng GIC 13 Feb, 2018 15 / 15

SLIDE 25

Recap

◮ Correlated Gaussians do not improve the HK-IB ◮ Conjectured an entropy-variance inequality

Motivated by considering the Fenchel dual form of a functional arising in the H–K region for the Z-interference channel Presented possible attack strategies Hope it will be resolved one-way or the other soon by someone (perhaps one of you)

Thank You

Chandra Nair and David Ng GIC 13 Feb, 2018 15 / 15