spatial coupling -- what is it, why does it work, and …...spatial coupling -- what is it, why does...
TRANSCRIPT
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Spatial Coupling -- What is it, why does it work, and what are the open challenges?
Ruediger Urbanke EPFL
!April 17th, 2014
Urbanke
Based on slides jointly developed with
Shrinivas Kudekar.
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Part IV: Spatial coupling - A General Phenomenon
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Outline Spatial Coupling - A General Phenomenon
• General one-dimensional systems • General BMS channels - Threshold Saturation and Universality • Multi-user communications and ISI channels
- Multi-access channels - Noisy Slepian-Wolf - Finite state channels - Many more...
• Problems beyond Communications - Compressive sensing - K-SAT
Practical Aspects and Open Questions • Universality • Windowed decoding • Rate loss • Scaling • Decoding Speed • Complexity and choice of parameters
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General one-dimensional systems
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General Coupled One-Dimensional Analysis
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
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General Coupled One-Dimensional Analysis
Irregular Ensembles
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
�(x) =3x + 3x2 + 14x50
20�(x) = x15
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General Coupled One-Dimensional Analysis
Irregular Ensembles
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
�(x) =3x + 3x2 + 14x50
20�(x) = x15
�BPuncoupled = 0.3531
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General Coupled One-Dimensional Analysis
Irregular Ensembles
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
�(x) =3x + 3x2 + 14x50
20�(x) = x15
� = 0.4032
Balance of areas
�BPuncoupled = 0.3531
![Page 9: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/9.jpg)
General Coupled One-Dimensional Analysis
Irregular Ensembles
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
�(x) =3x + 3x2 + 14x50
20�(x) = x15
� = 0.4032
Balance of areas
�BPuncoupled = 0.3531
�BPcoupled = 0.4032
![Page 10: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/10.jpg)
General Coupled One-Dimensional Analysis
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
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General Coupled One-Dimensional Analysis
Gaussian Approximation
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
�(x) = x2
�(x) = x5
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General Coupled One-Dimensional Analysis
Gaussian Approximation
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
�(x) = x2
�(x) = x5
BAWGN
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General Coupled One-Dimensional Analysis
Gaussian Approximation
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
hBPuncoupled = 0.4293
�(x) = x2
�(x) = x5
BAWGN
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General Coupled One-Dimensional Analysis
Gaussian Approximation
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
hBPuncoupled = 0.4293
�(x) = x2
�(x) = x5
v
f(u)g(v)
u
h = 0.42915
BAWGN
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General Coupled One-Dimensional Analysis
Gaussian Approximation
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
hBPuncoupled = 0.4293
�(x) = x2
�(x) = x5
v
f(u)g(v)
u u
v
g(v)f(u)
h = 0.42915 h = 0.4758
Balance of areasBAWGN
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General Coupled One-Dimensional Analysis
Gaussian Approximation
Balance of areas in the EXIT chart of uncoupled ensembles gives the BP threshold of coupled systems
hBPcoupled = 0.4794
hBPuncoupled = 0.4293
�(x) = x2
�(x) = x5
v
f(u)g(v)
u u
v
g(v)f(u)
h = 0.42915 h = 0.4758
Balance of areasBAWGN
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General BMS channels
Coupled Codes are Provably Capacity-Achieving under BP Decoding and they are universal with respect to all BMS channels
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General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
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General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
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General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 21: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/21.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 22: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/22.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 23: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/23.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 24: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/24.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 25: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/25.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 26: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/26.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 27: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/27.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 28: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/28.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 29: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/29.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 30: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/30.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 31: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/31.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 32: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/32.jpg)
General BMS Channels - Threshold Saturation
BAWGN Channel
(3,6,L) coupled code ensemble with increasing L≈ 0.4789 ⇡ hArea(l, r)
![Page 33: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/33.jpg)
General BMS Channels - Main Statement
where the bounds are independent of the channel.
For a fixed spatially coupled ensemble with parameters (dl, dr, w, L) and a given BMS channel,(dl, dr, w, L)
hArea �O(1/⇤
w) ⇥ hBPcoupled ⇥ hMAP
coupled ⇥ hArea + O(w/L)
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General BMS Channels - Main Statement
where the bounds are independent of the channel.
For a fixed spatially coupled ensemble with parameters (dl, dr, w, L) and a given BMS channel,(dl, dr, w, L)
hArea �O(1/⇤
w) ⇥ hBPcoupled ⇥ hMAP
coupled ⇥ hArea + O(w/L)
(hA ,2006 Cyril Measson)hArea
uniformly over all channelshArea deg.�⇥� hShannon
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General BMS Channels - Main Statement
where the bounds are independent of the channel.
Concentration: Almost all elements of the ensemble are good for all channels.
For a fixed spatially coupled ensemble with parameters (dl, dr, w, L) and a given BMS channel,(dl, dr, w, L)
hArea �O(1/⇤
w) ⇥ hBPcoupled ⇥ hMAP
coupled ⇥ hArea + O(w/L)
(hA ,2006 Cyril Measson)hArea
uniformly over all channelshArea deg.�⇥� hShannon
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Good means that we can transmit using belief propagation decoding with (block/bit) error probability at most e.�
Let C(c) denote the set of all BMS channels with capacity c and let e > 0. Then there exists a fixed spatially coupled code ensemble of rate at least c- e such that almost every code in the ensemble is good for all channels in C(c).
C(c)�
C(c)c� �
c
Most Codes are Universal
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General BMS Channels - Proofs of Threshold Saturation
Potential Function Analysis:
Generalized EXIT Analysis:
http://arxiv.org/pdf/1201.2999.pdf
http://arxiv.org/pdf/1301.6111.pdf
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General BMS Channels - Proofs of Threshold Saturation
Generalized EXIT Analysis:
http://arxiv.org/pdf/1201.2999.pdf
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General BMS Channels - Proofs of Threshold Saturation
Generalized EXIT Analysis:
http://arxiv.org/pdf/1201.2999.pdf
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General BMS Channels - Proofs of Threshold Saturation
Generalized EXIT Analysis:
http://arxiv.org/pdf/1201.2999.pdf
special FP
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General BMS Channels - Proofs of Threshold Saturation
Generalized EXIT Analysis:
http://arxiv.org/pdf/1201.2999.pdf
(generalized) EXIT special FP
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Multi-user Communications Finite-state channels, Noisy
Slepian-Wolf
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Multi-Access Channels
Y = h1X1 + h2X2 + N
Gaussian MAC Channel
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Multi-Access Channels
Y = h1X1 + h2X2 + N
Gaussian MAC Channel
[Figures from Yedla et al. ]
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Multi-Access Channels
Y = h1X1 + h2X2 + N
Gaussian MAC Channel
[Figures from Yedla et al. ]
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Multi-Access Channels
References
Y = h1X1 + h2X2 + N
Gaussian MAC Channel
[Figures from Yedla et al. ]
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Multi-Access Channels
References
Y = h1X1 + h2X2 + N
Gaussian MAC Channel
[Figures from Yedla et al. ]
![Page 48: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/48.jpg)
Multi-Access Channels
References
Y = h1X1 + h2X2 + N
Gaussian MAC Channel
[Figures from Yedla et al. ]
![Page 49: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/49.jpg)
Noisy Slepian-Wolf
Correlated sources (BSC) transmitted over
BAWGNC(h) !
Symmetric channel conditions, joint iterative
decoding[Figure from Yedla et al. ]
![Page 50: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/50.jpg)
Noisy Slepian-Wolf
Correlated sources (BSC) transmitted over
BAWGNC(h) !
Symmetric channel conditions, joint iterative
decoding
References[Figure from Yedla et al. ]
![Page 51: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/51.jpg)
Finite-State Channels
Generalized Erasure Channel
Output of a binary input linear filter (1-D) transmitted
over erasure channel
![Page 52: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/52.jpg)
Finite-State Channels
[Figure from Phong et al ]
Generalized Erasure Channel
Output of a binary input linear filter (1-D) transmitted
over erasure channel
![Page 53: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/53.jpg)
Finite-State Channels
[Figure from Phong et al ]
Generalized AWGN Channel
Output of a binary input linear filter (1-D) transmitted
over AWGN channel
![Page 54: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/54.jpg)
Finite-State Channels
References
[Figure from Phong et al ]
Generalized AWGN Channel
Output of a binary input linear filter (1-D) transmitted
over AWGN channel
![Page 55: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/55.jpg)
Finite-State Channels
References
[Figure from Phong et al ]
Generalized AWGN Channel
Output of a binary input linear filter (1-D) transmitted
over AWGN channel
![Page 56: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/56.jpg)
Finite-State Channels
References
[Figure from Phong et al ]
Generalized AWGN Channel
Output of a binary input linear filter (1-D) transmitted
over AWGN channel
![Page 57: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/57.jpg)
Many more...
Weight Distribution Minimum Distance
Trapping set, Pseudocodeword
analysis
CDMA
Relay Channel
![Page 58: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/58.jpg)
Many more...
Wiretap Channel
Rateless Coding
Lattice codes
Lossy Source Coding
Quantum Error Correction
Linear Programming Decoding
Bounded Complexity
![Page 59: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/59.jpg)
Problems beyond Communications - Compressive Sensing and K-Satisfiability
![Page 60: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/60.jpg)
Compressive Sensing
y = Ax + w
![Page 61: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/61.jpg)
Compressive Sensing
y = Ax + w
=( (y
m� 1( (A
m� n( (x
n� 1
+ w ((m� 1
![Page 62: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/62.jpg)
Compressive Sensing
y = Ax + w
=( (y
m� 1( (A
m� n( (x
n� 1
+ w ((m� 1
m < n cont. dist. withmass at zero
fX(x)1� �
x is sparse
![Page 63: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/63.jpg)
Compressive Sensing
y = Ax + w
=( (y
m� 1( (A
m� n( (x
n� 1
+ w ((m� 1
� = k/n� = m/nRegime: m, n�⇥
![Page 64: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/64.jpg)
(Bayesian) Compressive Sensing
![Page 65: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/65.jpg)
(Bayesian) Compressive Sensing
m � nd(fX) + o(n)
d(fX) is the Renyi Information Dimension
![Page 66: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/66.jpg)
(Bayesian) Compressive Sensing
m � nd(fX) + o(n)
d(fX) is the Renyi Information Dimension
d(fX) is less than for signal with sparsity equal to e� �
![Page 67: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/67.jpg)
(Bayesian) Compressive Sensing
0 1
1
�
�
![Page 68: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/68.jpg)
(Bayesian) Compressive Sensing
0 1
1
�
�
Optimal trade-off for sparsity and undersampling
![Page 69: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/69.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
Using coupled measurement matrices with message-passing decoder
![Page 70: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/70.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
Using coupled measurement matrices with message-passing decoder
...
![Page 71: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/71.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
Using coupled measurement matrices with message-passing decoder
�
⇧⇧⇧⇧⇧⇧⇧⇤
1 1 0 0 0 0 0 . . . 01 1 1 1 0 0 0 . . . 01 1 1 1 1 1 0 . . . 00 1 1 1 1 1 0 . . . 0...
......
......
......
. . . 00 0 0 0 0 0 0 . . . 1
⇥
⌃⌃⌃⌃⌃⌃⌃⌅
Band Diagonal Adjacency Matrix(Base) Measurement Matrix
![Page 72: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/72.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
![Page 73: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/73.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0
base measurement
matrix representing
heteroscedasticity
![Page 74: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/74.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0
![Page 75: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/75.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
does not affect overall
undersampling{�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0
![Page 76: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/76.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
{�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0
![Page 77: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/77.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
extra measurements
to help “kickstart” {
�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0
![Page 78: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/78.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
{�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0
![Page 79: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/79.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
coupling structure will
then take over{�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0
![Page 80: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/80.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0
![Page 81: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/81.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0
![Page 82: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/82.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0+
![Page 83: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/83.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
Approximate Message Passing (AMP) decoder
�
⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇧⇤
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
⇥
⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌃⌅
1...1
1...1
0
0
0...
0
0+
![Page 84: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/84.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
![Page 85: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/85.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
[Figure from Javanmard et al ]
![Page 86: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/86.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
References
[Figure from Javanmard et al ]
![Page 87: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/87.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
References
[Figure from Javanmard et al ]
![Page 88: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/88.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
References
[Figure from Javanmard et al ]
![Page 89: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/89.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
References
[Figure from Javanmard et al ]
![Page 90: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/90.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
References
[Figure from Javanmard et al ]
![Page 91: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/91.jpg)
Compressive Sensing and Coupling [Kudekar, Pfister, ‘10] [Krzakala, Mezard, Sausset, Sun, Zdeborova, ‘12] [Donoho, Javanmard, Montanari, ‘12]
References
[Figure from Javanmard et al ]
![Page 92: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/92.jpg)
Compressive Sensing - Proof of Threshold Saturation
Proof based on:
State evolution (evolution of the MSE under AMP) + Continuum analysis + Potential function analysis
![Page 93: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/93.jpg)
K-Satisfiability -- Setup
![Page 94: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/94.jpg)
K-Satisfiability -- Effect of Spatial Coupling
As for coding we can construct spatially coupled K-SAT formulas and we can show that for many algorithms the threshold of M/n up to which one can find satisfiable assignments is improved.
Combining this with the interpolation technique this can be used to prove better lower bounds on the SAT/UNSAT thresholds of uncoupled formulas.
![Page 95: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/95.jpg)
Some more ...
![Page 96: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/96.jpg)
Part III: Practical Aspects and Open Questions
![Page 97: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/97.jpg)
Windowed Decoding
is the size of the sliding window
...
(dl, dr, L,W ) W
⇤ ⇥� ⌅W
![Page 98: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/98.jpg)
Windowed Decoding
![Page 99: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/99.jpg)
Windowed Decoding
[Figure reproduced from
Iyengar et al ]
(3,6,64)
![Page 100: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/100.jpg)
Windowed Decoding
Latency equals W/L fraction of
BP decoderW/L
[Figure reproduced from
Iyengar et al ]Complexity in the waterfall region scales
like O(MW2L) as opposed to O(ML2)
(3,6,64)
![Page 101: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/101.jpg)
Windowed Decoding
Latency equals W/L fraction of
BP decoderW/L
[Figure reproduced from
Iyengar et al ]Complexity in the waterfall region scales
like O(MW2L) as opposed to O(ML2)
Threshold goes exponentially fast in W
to BP threshold
(3,6,64)
![Page 102: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/102.jpg)
Windowed Decoding
Latency equals W/L fraction of
BP decoderW/L
[Figure reproduced from
Iyengar et al ]
Reference
Complexity in the waterfall region scales
like O(MW2L) as opposed to O(ML2)
Threshold goes exponentially fast in W
to BP threshold
(3,6,64)
![Page 103: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/103.jpg)
Windowed Decoding
Latency equals W/L fraction of
BP decoderW/L
[Figure reproduced from
Iyengar et al ]
Reference
Complexity in the waterfall region scales
like O(MW2L) as opposed to O(ML2)
Threshold goes exponentially fast in W
to BP threshold
(3,6,64)
![Page 104: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/104.jpg)
Windowed Decoding
Latency equals W/L fraction of
BP decoderW/L
[Figure reproduced from
Iyengar et al ]
Reference
Complexity in the waterfall region scales
like O(MW2L) as opposed to O(ML2)
Threshold goes exponentially fast in W
to BP threshold
Vol. 56, No. 10, pp. 5274
(3,6,64)
![Page 105: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/105.jpg)
Windowed Decoding
Latency equals W/L fraction of
BP decoderW/L
[Figure reproduced from
Iyengar et al ]
Reference
Complexity in the waterfall region scales
like O(MW2L) as opposed to O(ML2)
Threshold goes exponentially fast in W
to BP threshold
(3,6,64)
![Page 106: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/106.jpg)
Rate-loss
![Page 107: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/107.jpg)
Rate Loss Due to Boundary
L
dl=3 dr=6
![Page 108: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/108.jpg)
Rate Loss Due to Boundary
L
dl=3 dr=6
![Page 109: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/109.jpg)
Ldl=4 dr=8
Rate Loss Due to Boundary
![Page 110: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/110.jpg)
Ldl=4 dr=8
drdl
L variable nodes
Rate Loss Due to Boundary
![Page 111: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/111.jpg)
Ldl=4 dr=8
drdl
L variable nodes
L+ dl � 1 check nodes
Rate Loss Due to Boundary
![Page 112: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/112.jpg)
Ldl=4 dr=8
r = 1� dldr
� dl(dl � 1)
dr
1
L
drdl
L variable nodes
L+ dl � 1 check nodes
Rate Loss Due to Boundary
![Page 113: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/113.jpg)
Can we do better?
Rate Loss Due to Boundary
![Page 114: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/114.jpg)
Can we do better?
Rate Loss Due to Boundary
‣ Mitigating rate-loss
![Page 115: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/115.jpg)
Can we do better?
Rate Loss Due to Boundary
‣ Mitigating rate-loss‣ One-Sided Termination
![Page 116: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/116.jpg)
Can we do better?
Rate Loss Due to Boundary
‣ Mitigating rate-loss‣ One-Sided Termination‣ Deletion
![Page 117: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/117.jpg)
Ldl=4 dr=8
Mitigating rate-loss: One-sided termination [ Kudekar, Richardson, Urbanke, 2012]
![Page 118: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/118.jpg)
Ldl=4 dr=8
combine
Mitigating rate-loss: One-sided termination [ Kudekar, Richardson, Urbanke, 2012]
![Page 119: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/119.jpg)
Ldl=4 dr=8
dldr
L variable nodes
combine
Mitigating rate-loss: One-sided termination [ Kudekar, Richardson, Urbanke, 2012]
![Page 120: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/120.jpg)
Ldl=4 dr=8
dldr
L variable nodes
combine
L+
dl � 1
2
check nodes
Mitigating rate-loss: One-sided termination [ Kudekar, Richardson, Urbanke, 2012]
![Page 121: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/121.jpg)
Ldl=4 dr=8
dldr
L variable nodes
combine
L+
dl � 1
2
check nodes
r = 1� dldr
� dl(dl � 1)
2dr
1
L
Mitigating rate-loss: One-sided termination [ Kudekar, Richardson, Urbanke, 2012]
![Page 122: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/122.jpg)
Ldl=4 dr=8
dldr
L variable nodes
combine
L+
dl � 1
2
check nodes
r = 1� dldr
� dl(dl � 1)
2dr
1
L
Mitigating rate-loss: One-sided termination [ Kudekar, Richardson, Urbanke, 2012]
![Page 123: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/123.jpg)
Mitigation of rate-loss: Deletion [Kasai et. al. ITW 2012]
Ldl=4 dr=8
![Page 124: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/124.jpg)
Mitigation of rate-loss: Deletion [Kasai et. al. ITW 2012]
Ldl=4 dr=8
delete!dl � 2
![Page 125: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/125.jpg)
Mitigation of rate-loss: Deletion [Kasai et. al. ITW 2012]
Ldl=4 dr=8
delete!dl � 2
![Page 126: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/126.jpg)
Mitigation of rate-loss: Deletion [Kasai et. al. ITW 2012]
Ldl=4 dr=8
dldr
L variable nodes
delete!dl � 2
![Page 127: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/127.jpg)
Mitigation of rate-loss: Deletion [Kasai et. al. ITW 2012]
Ldl=4 dr=8
dldr
L variable nodes
delete!
L+ 1 check nodes
dl � 2
![Page 128: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/128.jpg)
Mitigation of rate-loss: Deletion [Kasai et. al. ITW 2012]
Ldl=4 dr=8
dldr
L variable nodes
delete!
r = 1� dldr
� dldr
1
L
L+ 1 check nodes
dl � 2
![Page 129: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/129.jpg)
Finite-length Scaling
![Page 130: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/130.jpg)
Finite-Length Scaling (BEC)
gap to capacity = � +�
L
=
↵pn
pL+
�
L
= �n� 13
n = LM �R =�
L
⇠ n�0.275
optimal codes
polar codes
⇠ n� 12
PB ⇠ LpM�e�M�2
L = n13
![Page 131: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/131.jpg)
Finite-Length Scaling (BEC) [Olmos, Urbanke, 2012]
gap to capacity = � +�
L
=
↵pn
pL+
�
L
= �n� 13
n = LM �R =�
L
⇠ n�0.275
optimal codes
polar codes
⇠ n� 12
PB ⇠ LpM�e�M�2
L = n13
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Many more...
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Open Questions
1. Simplify, simplify, simplify, ..., 2. Spatial coupling as a proof technique (Maxwell conjecture, better bounds on K-SAT threshold, etc.) 3. Rate loss mitigation, other ways of introducing “boundary effect”? Derive fundamental lower bounds on the rate-loss. 4. Find systematic ways of designing codes (finite-length scaling, determine wave speed, ...). 5. Further applications.
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Main Message
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Main Message
Coupled ensembles under BP decoding behave like uncoupled ensembles under MAP decoding.
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Main Message
Coupled ensembles under BP decoding behave like uncoupled ensembles under MAP decoding.
Since coupled ensemble achieve the highest threshold they can achieve (namely the MAP threshold) under BP we speak of the threshold saturation phenomenon.
![Page 137: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/137.jpg)
Main Message
Coupled ensembles under BP decoding behave like uncoupled ensembles under MAP decoding.
Since coupled ensemble achieve the highest threshold they can achieve (namely the MAP threshold) under BP we speak of the threshold saturation phenomenon.
By using spatial coupling we can construct codes which are capacity-achieving universally across the whole set of BMS channels.
![Page 138: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/138.jpg)
Main Message
Coupled ensembles under BP decoding behave like uncoupled ensembles under MAP decoding.
Since coupled ensemble achieve the highest threshold they can achieve (namely the MAP threshold) under BP we speak of the threshold saturation phenomenon.
By using spatial coupling we can construct codes which are capacity-achieving universally across the whole set of BMS channels.
The basic principle is applicable to a wide range of graphical models.
![Page 139: Spatial Coupling -- What is it, why does it work, and …...Spatial Coupling -- What is it, why does it work, and what are the open challenges? Ruediger Urbanke EPFL! April 17th, 2014](https://reader035.vdocuments.mx/reader035/viewer/2022081523/5fd9d64b5c6ae21d0d2efd9d/html5/thumbnails/139.jpg)
Main Message
Coupled ensembles under BP decoding behave like uncoupled ensembles under MAP decoding.
Since coupled ensemble achieve the highest threshold they can achieve (namely the MAP threshold) under BP we speak of the threshold saturation phenomenon.
By using spatial coupling we can construct codes which are capacity-achieving universally across the whole set of BMS channels.
The basic principle is applicable to a wide range of graphical models.
The phenomenon of threshold saturation is closely connected to the way of how crystals grow.