[PPT] - Graceful degradation over the BEC via non-linear codes Hajir PowerPoint Presentation

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Graceful degradation over the BEC via non-linear codes

Hajir Roozbehani, Yury Polyanskiy

LIDS Massachusetts Institute of Technology

June 2020 2020 IEEE International Symposium on Information Theory

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 1

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Graceful degradation over BEC

Source generates iid bits Sk ∼ Ber(1/2)k

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 2

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Graceful degradation over BEC

Source generates iid bits Sk ∼ Ber(1/2)k Channel is Yi(ǫ) = BECǫ(Xi)

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 2

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Graceful degradation over BEC

Source generates iid bits Sk ∼ Ber(1/2)k Channel is Yi(ǫ) = BECǫ(Xi) Interested in BER for systematic bits BERf,g(ǫ) = 1

kE[dH(Sk, ˆ

Sk(ǫ))]

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 2

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Graceful degradation over BEC

Source generates iid bits Sk ∼ Ber(1/2)k Channel is Yi(ǫ) = BECǫ(Xi) Interested in BER for systematic bits BERf,g(ǫ) = 1

kE[dH(Sk, ˆ

Sk(ǫ))]

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 2

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Graceful degradation over BEC

Source generates iid bits Sk ∼ Ber(1/2)k Channel is Yi(ǫ) = BECǫ(Xi) Interested in BER for systematic bits BERf,g(ǫ) = 1

kE[dH(Sk, ˆ

Sk(ǫ))] Both encoder/decoder are fixed

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 2

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Graceful degradation over BEC

Source generates iid bits Sk ∼ Ber(1/2)k Channel is Yi(ǫ) = BECǫ(Xi) Interested in BER for systematic bits BERf,g(ǫ) = 1

kE[dH(Sk, ˆ

Sk(ǫ))] Both encoder/decoder are fixed Want an encoder/decoder pair with a smooth BER curve as ǫ varies

BER

C/R 1

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 2

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Graceful degradation over BEC

Source generates iid bits Sk ∼ Ber(1/2)k Channel is Yi(ǫ) = BECǫ(Xi) Interested in BER for systematic bits BERf,g(ǫ) = 1

kE[dH(Sk, ˆ

Sk(ǫ))] Both encoder/decoder are fixed Want an encoder/decoder pair with a smooth BER curve as ǫ varies

BER

C/R 1

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 2

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Graceful degradation over BEC

Source generates iid bits Sk ∼ Ber(1/2)k Channel is Yi(ǫ) = BECǫ(Xi) Interested in BER for systematic bits BERf,g(ǫ) = 1

kE[dH(Sk, ˆ

Sk(ǫ))] Both encoder/decoder are fixed Want an encoder/decoder pair with a smooth BER curve as ǫ varies

BER

C/R 1

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 2

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Graceful degradation over BEC

Source generates iid bits Sk ∼ Ber(1/2)k Channel is Yi(ǫ) = BECǫ(Xi) Interested in BER for systematic bits BERf,g(ǫ) = 1

kE[dH(Sk, ˆ

Sk(ǫ))] Both encoder/decoder are fixed Want an encoder/decoder pair with a smooth BER curve as ǫ varies Need two-point converses

BER

C/R 1

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 2

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Two point converses

1 C/R BER

Shannon converse

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 3

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Two point converses

1 C/R BER

Shannon converse Anchor point

A

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 3

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Two point converses

1 C/R BER

Shannon converse Anchor point

A

Exists?

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 3

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Two point converses

1 C/R BER

Shannon converse

B

Separation scheme Anchor point

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 3

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Two point converses

1 C/R BER

Shannon converse

B

Separation scheme Anchor point Exists?

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 3

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Two point converses

1 C/R BER

Shannon converse

B

Separation scheme Anchor point Exists?

We will show that non-trivial codes exist = ⇒

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 3

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Two point converses

1 C/R BER

Shannon converse

B

Separation scheme Anchor point Exists?

We will show that non-trivial codes exist = ⇒ Need to understand the two-point fundamental limits

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 3

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Motivation: channel transforms

Goal: transform a channel with high BER into one with low BER

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 4

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Motivation: channel transforms

Goal: transform a channel with high BER into one with low BER

Channel

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 4

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Motivation: channel transforms

Goal: transform a channel with high BER into one with low BER

Channel

1 C/R BER

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 4

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Motivation: channel transforms

Goal: transform a channel with high BER into one with low BER

Applications: coding for optical channels [ZK17, BK18], tornado-raptor codes, multi-user information theory, delay-sensitive applications (control, short-packet communication)

Channel

1 C/R BER

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 4

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Motivation: channel transforms

Goal: transform a channel with high BER into one with low BER

Applications: coding for optical channels [ZK17, BK18], tornado-raptor codes, multi-user information theory, delay-sensitive applications (control, short-packet communication)

Our main result today: proof that non-linear codes (namely, LDMCs) can provably outperform any linear-systematic code

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 4

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Motivation: channel transforms

Goal: transform a channel with high BER into one with low BER

Applications: coding for optical channels [ZK17, BK18], tornado-raptor codes, multi-user information theory, delay-sensitive applications (control, short-packet communication)

Our main result today: proof that non-linear codes (namely, LDMCs) can provably outperform any linear-systematic code How? new two-point converse bounds for linear-systematic codes

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 4

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Low Density Majority Codes (LDMCs)

Sparse graph codes:

1 2 3 4

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 5

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Low Density Majority Codes (LDMCs)

Sparse graph codes:

1 2 3 4

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 5

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Low Density Majority Codes (LDMCs)

Sparse graph codes:

1 2 3 4

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 5

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Low Density Majority Codes (LDMCs)

Sparse graph codes:

1 2 3 4

Raptor codes[Sho06], non-linear compression [CMZ05], [MM08],[GV09] , Gallager codes [Mac05]

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 5

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Low Density Majority Codes (LDMCs)

Sparse graph codes:

1 2 3 4

Raptor codes[Sho06], non-linear compression [CMZ05], [MM08],[GV09] , Gallager codes [Mac05] Here f = majority

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 5

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Low density majority codes

Sample a few indices randomly+ take majorities: x → (maj(x1, x2, x3), maj(x1, x3, x5), · · · )

Example

Suppose we have 1 equation left in three variables. Best possible distortion with XOR? y = x1 + x2 = ⇒ P(x1 = 1) = P(x1 = 0) = 1/2 Best linear code is x → x1. Average distortion is 1/3. But P(x1 = maj(x1, x2, x3)) = 3/4 Average distortion with majority is 1/4!

H. Roozbehani, Y. Polyanskiy

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Animation

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 7

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Classic codes vs non-linear codes

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 C/R 0.0 0.1 0.2 0.3 0.4 0.5 BER

Regular systematic LDMC(5)-5 BP steps Regular LDMC(5)-5 BP steps LDPC-50 peeling steps Repetition

H. Roozbehani, Y. Polyanskiy

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Classic codes vs non-linear codes

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 C/R 0.0 0.1 0.2 0.3 0.4 0.5 BER

Regular systematic LDMC(5)-5 BP steps Regular LDMC(5)-5 BP steps LDPC-50 peeling steps Repetition

Did we pick a bad LDPC? Can we do better than LDMCs with linear codes? = ⇒ need converse bounds.

H. Roozbehani, Y. Polyanskiy

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Two point converses

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 C/R 0.0 0.1 0.2 0.3 0.4 0.5 BER A

Single-point Shannon converse (achievable) Two-point converse Performance of a generic separation scheme LDMC simulations

Orange line: converse for any code whose BER curve passes through

r is below point A [KOP20, TKS13, KC16]

Not strong enough! We need a better converse for linear codes

H. Roozbehani, Y. Polyanskiy

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Main result (linear codes)

Theorem

Let f : s → sG be a systematic linear code of rate 1/ρ with generator matrix G = [I A] over F2. Fix ǫ1 > ǫ2 and δ1 ≤ ǫ1

2 . If BERf(ǫ1) ≤ δ1,

then BERf(ǫ2) ≥ ǫ2 − 1−ǫ2

1−ǫ1

ǫ2

ǫ1 γ + (ρ − 1)(1 − ǫ1) − γ

2

with γ = ǫ1 − 2δ1. If ǫ2 > ǫ1 then BERf(ǫ2) ≥ ǫ2 2 − ǫ2 ǫ2 − ǫ1 1 1 − ǫ1

δ1 − 1

2(1 − ρ(1 − ǫ1))

.
H. Roozbehani, Y. Polyanskiy

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Two point converse for linear codes

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 C/R 0.0 0.1 0.2 0.3 0.4 0.5 BER B

Single-point converse for linear codes (achievable) Two-point converse for systematic linear codes LDMC simulations

Orange line is a lower bound on BER for any linear-systematic code whose BER curve passes through or is below point B

H. Roozbehani, Y. Polyanskiy

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Two point converse for linear codes

Other consequences: = ⇒ optimal (w.r.t single point bound) systematic linear codes are not graceful = ⇒ no linear (systematic) code can dominate repetition The latter does not follow from the known conservation laws (area theorem):

Example

Consider f1 : x → (x, x) vs f2 : xi → (xi, xi, xi) for odd i and xi → xi for even i. Both codes have a rate of 1/2 but f1 dominates f2 for all erasure probabilities.

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 12

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Proof sketch

Solving xG = y uniquely determines xj iff ker(G) ⊂ {x : xj = 0}

H. Roozbehani, Y. Polyanskiy

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Proof sketch

Solving xG = y uniquely determines xj iff ker(G) ⊂ {x : xj = 0} Otherwise xj remains unbiased

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 13

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Proof sketch

Solving xG = y uniquely determines xj iff ker(G) ⊂ {x : xj = 0} Otherwise xj remains unbiased We need the concept of hrank Given a matrix A define hrank(A) |{j : ker(A) ⊂ {x : xj = 0}}|

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 13

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Proof sketch

Solving xG = y uniquely determines xj iff ker(G) ⊂ {x : xj = 0} Otherwise xj remains unbiased We need the concept of hrank Given a matrix A define hrank(A) |{j : ker(A) ⊂ {x : xj = 0}}| Need to understand the behavior of hrank under column sub-sampling

H. Roozbehani, Y. Polyanskiy

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Proof sketch (continued)

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits

Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits 3 Consider two erasure probabilities ǫ2 < ǫ1

Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits 3 Consider two erasure probabilities ǫ2 < ǫ1

Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits 3 Consider two erasure probabilities ǫ2 < ǫ1 4 Question: how is the hrank of A1 related to hrank A2?

Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits 3 Consider two erasure probabilities ǫ2 < ǫ1 4 Question: how is the hrank of A1 related to hrank A2? 5 Couple A1 and A2

Coupling Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits 3 Consider two erasure probabilities ǫ2 < ǫ1 4 Question: how is the hrank of A1 related to hrank A2? 5 Couple A1 and A2 6 Large hrank ⇔ large rank+small redundancy in the basis

Coupling Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits 3 Consider two erasure probabilities ǫ2 < ǫ1 4 Question: how is the hrank of A1 related to hrank A2? 5 Couple A1 and A2 6 Large hrank ⇔ large rank+small redundancy in the basis 7 Code is good for ǫ1 =

⇒ A1 has large hrank

Coupling Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits 3 Consider two erasure probabilities ǫ2 < ǫ1 4 Question: how is the hrank of A1 related to hrank A2? 5 Couple A1 and A2 6 Large hrank ⇔ large rank+small redundancy in the basis 7 Code is good for ǫ1 =

⇒ A1 has large hrank

(6)

= ⇒ A1 is sparse

Coupling Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits 3 Consider two erasure probabilities ǫ2 < ǫ1 4 Question: how is the hrank of A1 related to hrank A2? 5 Couple A1 and A2 6 Large hrank ⇔ large rank+small redundancy in the basis 7 Code is good for ǫ1 =

⇒ A1 has large hrank

(6)

= ⇒ A1 is sparse = ⇒ A1 ∩ A2 has low rank

Coupling Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits 3 Consider two erasure probabilities ǫ2 < ǫ1 4 Question: how is the hrank of A1 related to hrank A2? 5 Couple A1 and A2 6 Large hrank ⇔ large rank+small redundancy in the basis 7 Code is good for ǫ1 =

⇒ A1 has large hrank

(6)

= ⇒ A1 is sparse = ⇒ A1 ∩ A2 has low rank = ⇒ A2 has low rank

Coupling Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Proof sketch (continued)

1 Random columns of A correspond to unerased non-systematic bits

returned by the channel

2 Random rows correspond to erased systematic bits 3 Consider two erasure probabilities ǫ2 < ǫ1 4 Question: how is the hrank of A1 related to hrank A2? 5 Couple A1 and A2 6 Large hrank ⇔ large rank+small redundancy in the basis 7 Code is good for ǫ1 =

⇒ A1 has large hrank

(6)

= ⇒ A1 is sparse = ⇒ A1 ∩ A2 has low rank = ⇒ A2 has low rank = ⇒ A2 has low hrank

Coupling Sample rows and columns

H. Roozbehani, Y. Polyanskiy

Graceful degradation over the BEC via non-linear codes 14

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Main result (non-linear codes)

Theorem

Let ǫ2 < ǫ1. Let f be a binary code of rate R with BER(ǫ2) ≤ δ2. Define ζ(x, ǫ2, ǫ1) sup

{ǫ0:ǫ0<ǫ2}

1 R

1

(ǫ1 − ǫ0)(R − (1 − ǫ1) − ǫ0 xR ǫ2 − ǫ0 ) − 1 + R

.

If f is systematic (but possibly non-linear), then BERf(ǫ1) ≥ ǫ1h−1

b

(ζ(hb(δ2), ǫ2, ǫ1)) .

H. Roozbehani, Y. Polyanskiy

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Main result (non-linear codes)

Theorem

Let ǫ2 < ǫ1. Let f be a binary code of rate R with BER(ǫ2) ≤ δ2. Define ζ(x, ǫ2, ǫ1) sup

{ǫ0:ǫ0<ǫ2}

1 R

1

(ǫ1 − ǫ0)(R − (1 − ǫ1) − ǫ0 xR ǫ2 − ǫ0 ) − 1 + R

.

If f is systematic (but possibly non-linear), then BERf(ǫ1) ≥ ǫ1h−1

b

(ζ(hb(δ2), ǫ2, ǫ1)) . Follows from the area theorem and a new lemma that relates BER of coded bits to BER of information bits Improves on [KOP20, TKS13, KC16] in certain regimes

H. Roozbehani, Y. Polyanskiy

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Two point converse for general systematic codes

0.0 0.2 0.4 0.6 0.8 1.0 C/R 0.0 0.1 0.2 0.3 0.4 0.5 BER A

Two-point converse for non-linear systematic codes Two-point converse Single-point Shannon converse (achievable)

H. Roozbehani, Y. Polyanskiy

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Two point converse for general systematic codes

0.0 0.2 0.4 0.6 0.8 1.0 C/R 0.0 0.1 0.2 0.3 0.4 0.5 BER A

Two-point converse for non-linear systematic codes Two-point converse Single-point Shannon converse (achievable)

Blue curve is a lower bound on BER of any systematic code passing through A. Orange curve is from [KOP20, TKS13, KC16] = ⇒ Optimal (w.r.t Shannon limit) systematic codes of high rate are

H. Roozbehani, Y. Polyanskiy

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More on LDMCs and non-linear codes

Can improve performance of LDGMs with LDMCs (see the full arxiv version [RP19]) Can analyze LDMCs in certain cases using IT tools (see the channel comparison lemmas [RP19]) Open problem: do there exist graceful (non-systematic, non-linear) codes at high rates that are almost capacity achieving?

H. Roozbehani, Y. Polyanskiy

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Masoud Barakatain and Frank R Kschischang. Low-complexity concatenated ldpc-staircase codes. Journal of Lightwave Technology, 36(12):2443–2449, 2018. Stefano Ciliberti, Marc Mézard, and Riccardo Zecchina. Lossy data compression with random gates. Physical review letters, 95(3):038701, 2005. Ankit Gupta and Sergio Verdú. Nonlinear sparse-graph codes for lossy compression. IEEE Transactions on Information Theory, 55(5):1961–1975, 2009. Kia Khezeli and Jun Chen. A source-channel separation theorem with application to the source broadcast problem. IEEE Transactions on Information Theory, 62(4):1764–1781, 2016. Yuval Kochman, Or Ordentlich, and Yury Polyanskiy. A lower bound on the expected distortion of joint source-channel coding.

H. Roozbehani, Y. Polyanskiy

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IEEE Transactions on Information Theory, 2020. David JC MacKay. Encyclopedia of sparse graph codes, 2005. Andrea Montanari and Elchanan Mossel. Smooth compression, gallager bound and nonlinear sparse-graph codes. In 2008 IEEE International Symposium on Information Theory, pages 2474–2478. IEEE, 2008.

H. Roozbehani and Y. Polyanskiy.

Low density majority codes and the problem of graceful degradation. arXiv preprint arXiv:1911.12263, 2019. Amin Shokrollahi. Raptor codes. IEEE/ACM Transactions on Networking (TON), 14(SI):2551–2567, 2006. Louis Tan, Ashish Khisti, and Emina Soljanin.

H. Roozbehani, Y. Polyanskiy

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Distortion bounds for broadcasting a binary source over binary erasure channels. In 2013 13th Canadian Workshop on Information Theory, pages 49–54. IEEE, 2013. Lei M Zhang and Frank R Kschischang. Low-complexity soft-decision concatenated ldgm-staircase fec for high-bit-rate fiber-optic communication. Journal of Lightwave Technology, 35(18):3991–3999, 2017.

H. Roozbehani, Y. Polyanskiy

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