SPARSE SUPERPOSITION CODESQuest for Provably Practical and Reliable High Rate Communication The...

SPARSE SUPERPOSITION CODES

Andrew BarronDepartment of Statistics

Yale University

Joint work with Sanghee Cho, Antony Joseph

Workshop on Modern Error Correcting CodesTokyo, Aug 30, 2013

Quest for Provably Practical and ReliableHigh Rate Communication

• The Channel Communication Problem

• Shannon’s Formulation and Solution

• Gaussian Channel

• History of Methods

• Communication by Regression

• Sparse Superposition Coding

• Adaptive Successive Decoding

• Rate, Reliability, and Computational Complexity

• Distributional Analysis

Shannon Formulation• Input bits: U = (U1,U2, . . . . . . ,UK ) indep Bern(1/2)

↓• Encoded: x = (x1, x2, . . . , xn)

↓• Channel: p(y |x)

↓• Received: Y = (Y1,Y2, . . . ,Yn)

↓• Decoded: U = (U1, U2, . . . . . . , UK )

• Rate: R = Kn Capacity C = maxPX I(X ; Y )

• Reliability: Want small Prob{U 6= U}and small Prob{Fraction mistakes ≥ α}

Gaussian Noise Channel• Input bits: U = (U1,U2, . . . . . . ,UK )

↓• Encoded: x = (x1, x2, . . . , xn) 1

i=1 x2i∼= P

↓• Channel: p(y |x) Y = x(U) + ε , ε ∼ N(0, σ2I)

↓ snr = P/σ2

• Received: Y = (Y1,Y2, . . . ,Yn)

↓• Decoded: U = (U1, U2, . . . . . . , UK )

• Rate: R = Kn Capacity C = 1

2 log(1 + snr)

• Reliability: Want small Prob{U 6= U}and small Prob{Fraction mistakes ≥ α}

Shannon Theory

• Channel Capacity:Supremum of rates R such that reliable communication ispossible, with arbitrarily small error probability

• Information Capacity: C = maxPX I(X ; Y )

Where I(X ; Y ) is the Shannon information, also known asthe Kullback divergence between PX ,Y and PX × PY

• Shannon Channel Capacity Theorem:The supremum of achievable communication rates Requals the information capacity C

• Books:Shannon (49), Gallager (68), Cover & Thomas (06)

The Gaussian Noise Model

The Gaussian noise channel is the basic model for

• wireless communicationradio, cell phones, television, satellite, space

• wired communicationinternet, telephone, cable

Shannon Theory meets Coding Practice• Forney and Ungerboeck 1998 review:

• modulation, shaping and coding for the Gaussian channel• Richardson & Urbanke 2008 cover coding state of the art:

• Empirically good LDPC and turbo codes with fast encodingand decoding based on Bayesian belief networks

• Some tools for theoretical analysis, yet obstacles remain forproof of rates up to capacity for the Gaussian channel

• Arikan 2009, Arikan and Teletar 2009 polar codes:• Adapted to Gaussian channel (Abbe and Barron 2011)

• Tropp 2006,2008 codes from compressed sensing andrelated sparse signal recovery work:

• Wainwright; Fletcher, Rangan, Goyal; Zhang; others• `1-constrained least squares practical, has positive rate• but not capacity achieving

• Method here different from those above. Prior knowledgeof such is not necessary to follow what we present.

Sparse Superposition Code

• Input bits: U = (U1 . . . . . . . . . . . .UK )

• Coefficients: β = (00 ∗ 0000000000 ∗ 00 . . . 0 ∗ 000000)T

• Sparsity: L entries non-zero out of N• Matrix: X , n by N, all entries indep Normal(0,1)

• Codeword: Xβ, superposition of a subset of columns• Receive: Y = Xβ + ε, a statistical linear model• Decode: β and U from X ,Y

• Input bits: U = (U1 . . . . . . . . . . . .UK )

• Coefficients: β = (00 ∗ 0000000000 ∗ 00 . . . 0 ∗ 000000)T

• Codeword: Xβ, superposition of a subset of columns• Receive: Y = Xβ + ε, a statistical linear model• Decode: β and U from X ,Y

• Input bits: U = (U1 . . . . . . . . . . . .UK )

• Coefficients: β = (00 ∗ 0000000000 ∗ 00 . . . 0 ∗ 000000)T

• Codeword: Xβ, superposition of a subset of columns• Receive: Y = Xβ + ε

• Decode: β and U from X ,Y• Rate: R = K

n from K = log(N

), near L log

(NL e)

• Input bits: U = (U1 . . . . . . . . . . . .UK )

• Coefficients: β = (00 ∗ 0000000000 ∗ 00 . . . 0 ∗ 000000)T

n from K = log(N

)• Reliability: small Prob{Fraction βmistakes ≥ α}, small α

• Input bits: U = (U1 . . . . . . . . . . . .UK )

• Coefficients: β = (00 ∗ 0000000000 ∗ 00 . . . 0 ∗ 000000)T

n from K = log(N

)• Reliability: small Prob{Fraction βmistakes ≥ α}, small α

Want reliability with rate up to capacity

Partitioned Superposition Code• Input bits: U = (U1 . . . , . . . , . . . , . . .UK )

• Coefficients: β=(00 ∗ 00000, 00000 ∗ 00, . . . , 0 ∗ 000000)

• Sparsity: L sections, each of size M =N/L, a power of 21 non-zero entry in each section

• Indices of nonzeros: (j1, j2, . . . , jL) specified by U segments• Matrix: X , n by N, splits into L sections• Codeword: Xβ, superposition of columns, one from each• Receive: Y = Xβ + ε

• Decode: β and U• Rate: R = K

n from K = L log NL = L log M

• Coefficients: β=(00 ∗ 00000, 00000 ∗ 00, . . . , 0 ∗ 000000)

• Reliability: small Prob{Fraction βmistakes ≥ α}, small α

Is it reliable with rate up to capacity?

• Coefficients: β=(00 ∗ 00000, 00000 ∗ 00, . . . , 0 ∗ 000000)

• Ultra-sparse case: Impractical M = 2nR/L with L constant(reliable at all rates R < C: Cover 1972,1980)

• Moderately-sparse: Practical M = n with L = nR/ log n(Still reliable at all R < C)

• Coefficients: β=(00 ∗ 00000, 00000 ∗ 00, . . . , 0 ∗ 000000)

• Reliability: small Prob{Fraction mistakes ≥ α} small α• Outer RS code: rate 1−α, corrects remaining mistakes• Overall rate: Rtot = (1−α)R• Overall rate: up to capacity

Power Allocation

• Coefficients: β=(00∗00000, 00000∗00, . . . ,0∗000000)

• Indices of nonzeros: sent = (j1, j2, . . . , jL)

• Coeff. values: βj` =√

P` for ` = 1,2, . . . ,L

• Power control:∑L

`=1 P` = P

• Codewords: Xβ, have average power P

• Power Allocations

• Constant power: P` = P/L

• Variable power: P` proportional to e−2C `/L

Variable Power Allocation

`=1 P` = P ‖β‖2 = P

• Variable power: P` proportional to e−2C`/L for ` = 1, . . . ,L

0 20 40 60 80 100

section index

Variable Power Allocation

`=1 P` = P ‖β‖2 = P

• Variable power: P` proportional to e−2C`/L for ` = 1, . . . ,L

• Successive decoding motivation• Incremental capacity

σ2 + P`+1 + · · ·+ PL

matching the section rate

=log M

Adaptive Successive Decoder

Decoding Steps (with thresholding)

• Start: [Step 1]• Compute the inner product of Y with each column of X• See which are above a threshold• Form initial fit as weighted sum of columns above threshold

• Iterate: [Step k ≥ 2]• Compute the inner product of residuals Y − Fitk−1 with

each remaining column of X• Standardize by dividing by ‖Y − Fitk−1‖• See which are above the threshold

√2 log M + a

• Add these columns to the fit

• Stop:• At Step k = 1 + snr log M, or• if there are no additional inner products above threshold

Complexity of Adaptive Successive Decoder

Complexity in parallel pipelined implementation

• Space: (use k = snr log M copies of the n by N dictionary)

• knN = snr C M n2 memory positions• kN multiplier/accumulators and comparators

• Time: O(1) per received Y symbol

Adaptive Successive DecoderDecoding Steps (with iteratively optimal statistics)

• Start: [Step 1]• Compute the inner product of Y with each column of X• Form initial fit

• Iterate: [Step k ≥ 2]• Compute inner product of residuals Y − Fitk−1 with each Xj .• Adjusted form: statk,j equals (Y − X βk−1,−j )

• Standardize by dividing it by ‖Y − X βk−1‖.• Form the new fit

βk,j =√

P` wj (α) =√

P`eα statk,j∑

j∈sec` eα statk,j

• Stop:• At Step k = O(log M), sufficient if R < C.

Iteratively Bayes optimal βkWith prior j`∼Unif on sec`, the Bayes estimate based on statk

βk = E[β|statk ]

has representation βk ,j =√

P` wk ,j with

wk ,j = Prob{j` = j |statk}.

Here, when the statk ,j are independent N(α`,k1{j=j`},1), wehave the logit representation wk ,j = w(α`,k ) where

wk ,j(α) =eα statk,j∑

j∈sec` eα statk,j.

βk = E[β|statk ]

P` wk ,j with

βk = E[β|statk ]

P` wk ,j with

Recall statk ,j is standardized inner product of residuals with Xj

statk ,j =(Y − X βk−1,−j)

‖Y − X βk−1‖

Distributional Analysis• Approximate distribution of statk .

statk ,j =

√n βj√

σ2 + E‖βk−1 − β‖2+ Zk ,j

independent standard normal, shifted for terms sent

statk ,j = α`,xk−11{j sent} + Zk ,j

α = α`,x =

√nP`

σ2 + P(1− x)

• HereE‖βk−1 − β‖2 = P (1− xk−1)

• Update rule xk = g(xk−1) where

g(x) =L∑`=1

(P`/P)E [wj`(α`,x )].

Distributional Analysis• Approximate distribution of statk .

statk ,j =

√n βj√

σ2 + E‖βk−1 − β‖2+ Zk ,j

independent standard normal, shifted for terms sent

statk ,j = α`,xk−11{j sent} + Zk ,j

α = α`,x =

√nP`

σ2 + P(1− x)

• HereE‖βk−1 − β‖2 = P (1− xk−1)

g(x) =L∑`=1

(P`/P)E [wj`(α`,x )].

Success Rate Update Rule

g(x) =L∑`=1

(P`/P)E [wj`(α`,x )]

• this is the success rate update function expressed as aweighted sum of the posterior prob of the term sent

Decoding progression

0.0 0.2 0.4 0.6 0.8 1.0

M = 29 , L =Msnr=7C=1.5 bitsR=1.2 bits(0.8C)

Figure : Plot of g(x) and the sequence xk .

Update fuctions

0.0 0.2 0.4 0.6 0.8 1.0

M = 29 , L =Msnr=7C=1.5 bitsR=1.2 bits(0.8C)

g(x)Lower bounda=0a=0.5

Figure : Comparison of update functions. Blue and light blue linesindicates {0,1} decision using the threshold τ =

√2logM + a with

respect to the value a as indicated.

Transition plots0.0

soft decisionhard decision with a=1/2

0.0 0.2 0.4 0.6 0.8

Figure : Transition plots : M = 29, L = M, C = 1.5 bits and R = 0.8C.We used Monte Carlo simulation with replicate size 10000. Thehorizontal axis depicts u(`) = 1− e−2C`/L which is an increasingfunction of `. Area under curve equals g(x).

Rate and Reliability

Result for Optimal ML Decoder [Joseph and B. 2012],with outer RS decoder, and with equal power allowed acrossthe sections

• Prob error exponentially small in n for all R < C

Prob{Error} ≤ e−n (C−R)2/2V

• In agreement with the Shannon-Gallager exponent ofoptimal code, though with a suboptimal constant Vdepending on the snr

Rate and Reliability of Fast Superposition Decoder

Practical: Adaptive Successive Decoder, with outer RS code.

• prob error exponentially small in n/(log M) for R < C

• Value CM approaching capacity

1 + c1/ log M

• Probability error exponentially small in L for R < CM

Error}≤ e−L(CM−R)2c2

• Improves to e−c3L(CM−R)2(log M)0.5using a Bernstein bound.

Summary

Sparse superposition codes with adaptive successive decoding

• Simplicity of the code permits:• distributional analysis of the decoding progression• low complexity decoder• exponentially small error probability for any fixed R < C

• Asymptotics superior to polar code bounds for such rates

Sparse Superposition Code for the Gaussian Channel

uInput bits(length K )

βSparse

coeff. vector(length N)L non-zero‖β‖2 = P

XDictionary

n by Nindep N(0,1)

XβCodeword(length n)

Channel

εNoise ∼ N(0, σ2I)

Yreceived(length n)

Decoder u

βSparse

XDictionary

n by Nindep N(0,1)

Channel

Yreceived(length n)

Decoder u

Linear Model Y = Xβ + ε

βSparse

XDictionary

n by Nindep N(0,1)

Channel

Yreceived(length n)

Decoder u

• Partitioned Coef.: β = (00∗0000, 000∗000, . . . ,0∗00000)

• L sections of size M = N/L, one non-zero in each

• Rate R = Kn = L log M

n , Capacity C = 12 log(1 + snr)

Maximum likelihood decoder (Joseph & Barron 2010a ISIT, 2012a IT)

Adaptive successive decoder with threshold (J&B 2010b ISIT, 2012b)

Adaptive successive decoder with soft decision (B&C ISIT 2012)

βSparse

XDictionary

n by Nindep N(0,1)

Channel

Yreceived(length n)

Decoder u

snr = Pσ2

βSparse

XDictionary

n by Nindep N(0,1)

Channel

Yreceived(length n)

Decoder u

snr = Pσ2

Iterative Estimation

For k ≥ 1

• Codeword fits: Fk = X βk

• Vector of statistics: statk = function of (X ,Y ,F1, . . . ,Fk )

• e.g. statk ,j proportional to X Tj (Y − Fk )

• Update βk+1 as a function of statk

• Thresholding: Adaptive Successive Decoderβk+1,j =

√P` 1{statk,j>thres}

• Soft decision:βk+1,j = E[βj |statk ] =

√P` wk,j

with thresholding on the last step

Iterative Estimation

For k ≥ 1

• Vector of statistics: statk = function of (X ,Y ,F1, . . . ,Fk )

• e.g. statk ,j proportional to X Tj (Y − Fk )

• Update βk+1 as a function of statk

• Thresholding: Adaptive Successive Decoderβk+1,j =

√P` 1{statk,j>thres}

• Soft decision:βk+1,j = E[βj |statk ] =

√P` wk,j

with thresholding on the last step

Statistics

• Orthogonalization : Let G0 = Y and for k ≥ 1

Gk = part of Fk orthogonal to G0,G1, . . . ,Gk−1

• Components of statistics

Zk ,j =X T

‖Gk‖

• Statistics such as statk built from X Tj (Y − Fk ,−j) are linear

combinations of these Zk ,j

Distributional AnalysisLemma 1: shifted normal conditional distributionGiven Fk−1 = (‖G0‖, . . . , ‖Gk−1‖,Z0,Z1, . . . ,Zk−1), the Zk hasthe distributional representation

Zk =‖Gk‖σk

bk + Zk

• ‖Gk‖2/σ2k ∼ Chi-square(n − k)

• b0,b1, . . . ,bk the successive orthonormal components of[βσ

], . . . ,

](∗)

• Zk ∼ N(0,Σk ) indep of ‖Gk‖

• Σk = I − b0bT0 − b1bT

1 − . . .− bkbTk

= projection onto space orthogonal to (∗)

• σ2k = βT

k Σk−1βk

Idealized statisticsClass of statistics statk formed by combining Z0, . . . ,Zk

statk ,j = Zcombk ,j +

√n√ckβk ,j

with Zcombk = λ0,k Z0 +λ1,k Z1 + . . .+λk ,k Zk ,

∑kk ′=0 λ

2k ′,k = 1

Desired form of the statistics

stat idealk =

√n√ckβ + Z comb

for ck = σ2 + (1− xk )P

For the terms sent the shift α`,k has an effective snrinterpretation

α`,k =

P`σ2 + Premaining,k

Idealized statisticsClass of statistics statk formed by combining Z0, . . . ,Zk

statk ,j = Zcombk ,j +

√n√ckβk ,j

with Zcombk = λ0,k Z0 +λ1,k Z1 + . . .+λk ,k Zk ,

∑kk ′=0 λ

2k ′,k = 1

Desired form of the statistics

stat idealk =

√n√ckβ + Z comb

for ck = σ2 + (1− xk )P

For the terms sent the shift α`,k has an effective snrinterpretation

α`,k =

P`σ2 + Premaining,k

Example of statisticsWeights of combination: based on λk proportional to(

(σY − bT0 βk ), −(bT

1 βk ), . . . , −(bTk βk )

)If we combine Zk , replacing χn−k with

√n, it produces the

desired distributional representation

statk =

√n√

σ2 + ‖β − βk‖2β + Z comb

with Z combk ∼ N(0, I) and σ2

Y = σ2 + P.

• We cannot calculate the weights not knowing βe.g. b0 = β/σ

• ‖β − βk‖2 is close to its known expectation• This provides approximation of the distribution of the statk ,j

as independent shifted normals.

1 βk ), . . . , −(bTk βk )

)If we combine Zk , replacing χn−k with

√n, it produces the

desired distributional representation

statk =

√n√

with Z combk ∼ N(0, I) and σ2

Y = σ2 + P.• We cannot calculate the weights not knowing β

e.g. b0 = β/σ

• ‖β − βk‖2 is close to its known expectation• This provides approximation of the distribution of the statk ,j

as independent shifted normals.

1 βk ), . . . , −(bTk βk )

)Estimated weights of combination proportional to(

(‖Y‖ − ZT0 βk ), −(ZT

1 βk ), . . . , −(ZTk βk )

)These estimated weights produce the residual based statisticspreviously discussed

Using idealized weights of combinationsFor given β1, . . . , βk and c0 > . . . > ck , ck = σ2 + (1− xk )PWe combine Zk ′ =

√n bk ′ + Zk ′ with

λ∗k =√

(√1c0,−

√1c1− 1

c0, . . . ,−

√1ck− 1

ck−1

)Approximately optimal estimate is based on

statk =k∑

k ′=0

λ∗k ′,kZk ′ +

√n√ckβk

Lemma: For 1 ≤ k ≤ k∗, for any δ > 0,

|βT βk − xkP| ≤ ak (n/L)k−1/2δ n/L = (log M)/R

except an event of probability of

k∗exp{−Lδ2/16}+ 6k∗exp{− 2c2 Lδ2}

Cholesky decomposition based estimates

Recall if we combine√

n bk + Zk with weights λ∗k proportional to(

1 βk ), . . . , −(bTk βk )

)producing the desired distributional representation

ˆstat∗k =

√n√

we cannot calculate λ∗k not knowing β

Estimate weight of combinationEstimation of(

1 βk ), . . . , −(bTk βk )

)These bT

k ′ βk arise inthe QR-decomposition for B = [β, β1, . . . , βk ]

Estimate weight of combination

These bTk ′ βk arise in the Cholesky decomposition BT B =RT R,

The ZTk βk/

√n provides estimates for the diagonals of R using

ZTk βk/

√n = (bT

k βk )χn−k/√

The error of this estimate is controlled by χn−k/√

n − 1.Based on the estimates we can recover the rest of thecomponents of the matrix.

Estimate weights of combinations

Lemma: For 1 ≤ k ′ < k ≤ k∗,

|bTk ′ βk − rk ′,k | ≤ FR,k ′,k

√L/n δ

except an event of probability of 6k∗e−Lδ2.

• FR,k ′,k is a function of elements from the upper k ′ × k partof the matrix R.

• We believe each bTk ′ βk is close to deterministic value.

• If it is true, FR,k ′,k is close to FR∗,k ′,k where R∗ is replacingbT

k ′ βk with deterministic constants.

Estimate weights of combinations

Lemma: For 1 ≤ k ′ < k ≤ k∗,

|bTk ′ βk − rk ′,k | ≤ FR,k ′,k

√L/n δ

except an event of probability of 6k∗e−Lδ2.

• FR,k ′,k is a function of elements from the upper k ′ × k partof the matrix R.

• We believe each bTk ′ βk is close to deterministic value.

• If it is true, FR,k ′,k is close to FR∗,k ′,k where R∗ is replacingbT

k ′ βk with deterministic constants.

Strategy

Current investigation, show inductively, for k ≥ 1,

(a)∣∣∣βT βk − xkP

∣∣∣ ≤√n/L δ

(b)∣∣∣‖βk‖2 − xkP

∣∣∣ ≤√n/L δ

(c)∣∣∣βT

k ′ βk − xk ′P∣∣∣ ≤√n/L δ for k ′ = 1, . . . , k − 1.

(d)∣∣∣bT

k ′ βk − ck

ck′− 1

ck′−1

∣∣∣ ≤√n/L δ for k ′ = 1, . . . , k − 1.

(e)∣∣∣bT

k ′ βk − rk ′,k

∣∣∣ ≤√L/n δ for k ′ = 1, . . . , k − 1.

Conclusion

• Iterative estimation for sparse superposition codes foradditive white Gaussian noise channel.

• Iteratively obtained statistics Zk ,j are approximately shiftednormal distributed.

• The update function shows the progression of theestimation approximately

• There are different choices of function of statistics statk weuse for the estimation

• Using residuals• Using idealized weights of combinations• Cholesky decomposition based estimates

Thank you!

Rate versus Section Size

● ●●

26 28 210 212 214 216

detailed envelopecurve when L == Bcurve using simulation runscapacitysnr == 15

Figure : Rate as a function of B for snr =15 and error probability10−3.

Rate versus Section Size

26 28 210 212 214 216

detailed envelopecurve when L == Bcurve using simulation runscapacitysnr == 1

Figure : Rate as a function of B for snr = 1 and error probability 10−3.

SPARSE SUPERPOSITION CODESQuest for Provably Practical and Reliable High Rate Communication The...

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