Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and...
Transcript of Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and...
![Page 1: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/1.jpg)
Andrea Montanari and Ruediger UrbankeTIFR
Tuesday, January 6th, 2008
Phase Transitions in Coding, Communications, and Inference
![Page 2: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/2.jpg)
Outline
![Page 3: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/3.jpg)
Outline
1) Thresholds in coding, the large size limit (definition and density evolution characterization)
2) The inversion of limits (length to infty vs size to infty)
![Page 4: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/4.jpg)
Outline
1) Thresholds in coding, the large size limit (definition and density evolution characterization)
2) The inversion of limits (length to infty vs size to infty)
3) Phase transitions in measurements (compressed sensing versus message passing, dense versus sparse matrices)
4) Phase transitions in collaborative filtering (the low-rank matrix model)
![Page 5: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/5.jpg)
Model
Shannon ’48
![Page 6: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/6.jpg)
Model
Shannon ’48
binary erasures channelcapacity: R≤1-ε
![Page 7: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/7.jpg)
Model
Shannon ’48
binary symmetric channelcapacity: R≤1-h(ε)
![Page 8: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/8.jpg)
Channel Coding
![Page 9: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/9.jpg)
Channel Coding
codeC={000, 010, 101, 111}
![Page 10: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/10.jpg)
Channel Coding
codeC={000, 010, 101, 111}
n ... blocklength
![Page 11: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/11.jpg)
Channel Coding
codeC={000, 010, 101, 111}
n ... blocklength
![Page 12: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/12.jpg)
Channel Coding
codeC={000, 010, 101, 111}
n ... blocklength
![Page 13: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/13.jpg)
Channel Coding
code
decoding
C={000, 010, 101, 111}
n ... blocklength
xMAP(y)=argmaxX in C p(x | y)
xiMAP(y)=argmaxXi p(xi |y)
![Page 14: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/14.jpg)
Factor Graph Representation of Linear Codes
![Page 15: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/15.jpg)
Factor Graph Representation of Linear Codes
every linear codeparity-check matrix
![Page 16: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/16.jpg)
Factor Graph Representation of Linear Codes
(7, 4) Hamming code
every linear codeparity-check matrix
![Page 17: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/17.jpg)
Factor Graph Representation of Linear Codes
(7, 4) Hamming code
every linear code
Tanner, Wiberg, Koetter, Loeliger, Frey
parity-check matrix
![Page 18: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/18.jpg)
Low-Density Parity Check Codes
Gallager ‘60
![Page 19: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/19.jpg)
Low-Density Parity Check Codes
(3, 4)-regular codes
Gallager ‘60
![Page 20: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/20.jpg)
Low-Density Parity Check Codes
(3, 4)-regular codes
Gallager ‘60
![Page 21: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/21.jpg)
Low-Density Parity Check Codes
(3, 4)-regular codes
Gallager ‘60
![Page 22: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/22.jpg)
Low-Density Parity Check Codes
(3, 4)-regular codes
Gallager ‘60
![Page 23: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/23.jpg)
Low-Density Parity Check Codes
(3, 4)-regular codes
Gallager ‘60
number of edges is linear in n
![Page 24: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/24.jpg)
Ensemble
![Page 25: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/25.jpg)
Ensemble
![Page 26: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/26.jpg)
Ensemble
![Page 27: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/27.jpg)
Ensemble
![Page 28: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/28.jpg)
Ensemble
![Page 29: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/29.jpg)
Variations on the Theme
![Page 30: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/30.jpg)
Variations on the Theme
degree distributions as well as structure
![Page 31: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/31.jpg)
Variations on the Theme
irregular LDPC ensemble
(Luby, Mitzenmacher, Shokrollahi, Spielman, and Stehman)
![Page 32: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/32.jpg)
Variations on the Theme
regular RA ensemble
Divsalar, Jin, and McEliece
![Page 33: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/33.jpg)
Variations on the Theme
irregular RA ensemble
Jin, Khandekar, and McEliece
![Page 34: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/34.jpg)
Variations on the Theme
irregular MN ensemble
Davey, MacKay
![Page 35: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/35.jpg)
Variations on the Theme
ARA ensemble
Abbasfar, Divsalar, Kung
![Page 36: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/36.jpg)
Variations on the Theme
irregular LDGM ensemble
![Page 37: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/37.jpg)
Variations on the Theme
turbo code
Berrou and Glavieux
![Page 38: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/38.jpg)
Variations on the Theme
protograph
Thorpe, Andrews, Dolinar
![Page 39: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/39.jpg)
Message-Passing Decoding -- BEC
?
?
0
![Page 40: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/40.jpg)
Message-Passing Decoding -- BEC
?
?
00
0
?
?
?
![Page 41: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/41.jpg)
Message-Passing Decoding -- BEC
?
?
0 0
0
?
?
?
![Page 42: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/42.jpg)
Message-Passing Decoding -- BEC
?
?
0
0+?
0
0?
?
?
![Page 43: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/43.jpg)
Message-Passing Decoding -- BEC
?
?
0
0+? =??
0
0?
?
?
![Page 44: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/44.jpg)
Message-Passing Decoding -- BEC
?
?
0
?
0
0
?
?
?
![Page 45: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/45.jpg)
Message-Passing Decoding -- BEC
?
?
0 0
?
??
?
![Page 46: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/46.jpg)
Message-Passing Decoding -- BEC
?
?
0 0?
??
0=00
?
![Page 47: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/47.jpg)
Message-Passing Decoding -- BEC
?
?
0 0
?
??
0
?
![Page 48: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/48.jpg)
Message-Passing Decoding -- BEC
?
?
0
??
?
?
0
![Page 49: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/49.jpg)
Message-Passing Decoding -- BEC
?
?
0
?
?
?
?
0
![Page 50: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/50.jpg)
Message-Passing Decoding -- BEC
?
?
0
?
?
?
?
0 0decoded
![Page 51: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/51.jpg)
Message-Passing Decoding -- BEC
?
?
00
0
?0
?
decoded
![Page 52: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/52.jpg)
Message-Passing Decoding -- BEC
?
?
0 0
0
?
0
?
decoded
![Page 53: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/53.jpg)
Message-Passing Decoding -- BEC
?
?
0 0
0
?
0?
decoded
0+0
![Page 54: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/54.jpg)
Message-Passing Decoding -- BEC
?
?
0 0
0
?
0?
decoded
0+0 =00
![Page 55: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/55.jpg)
Message-Passing Decoding -- BEC
?
?
0 0
0
?
0
?
decoded
0
![Page 56: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/56.jpg)
Message-Passing Decoding -- BEC
?
?
0 0
0
?
0
?
decoded
decoded 0
![Page 57: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/57.jpg)
Message-Passing Decoding -- BSCGallager Algorithm
![Page 58: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/58.jpg)
Asymptotic Analysis: Computation Graph
![Page 59: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/59.jpg)
Asymptotic Analysis: Computation Graph
![Page 60: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/60.jpg)
Asymptotic Analysis: Computation Graph
![Page 61: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/61.jpg)
Asymptotic Analysis: Computation Graph
![Page 62: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/62.jpg)
Asymptotic Analysis: Computation Graph
![Page 63: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/63.jpg)
Asymptotic Analysis: Computation Graph
![Page 64: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/64.jpg)
Asymptotic Analysis: Computation Graph
![Page 65: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/65.jpg)
Asymptotic Analysis: Computation Graph
![Page 66: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/66.jpg)
Asymptotic Analysis: Computation Graph
probability that computation graphof fixed depth becomes tree
tends to 1 as n tends to infinity
![Page 67: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/67.jpg)
Asymptotic Analysis: Computation Graph
![Page 68: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/68.jpg)
Asymptotic Analysis: Computation Graph
![Page 69: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/69.jpg)
Asymptotic Analysis: Density Evolution -- BEC
Luby,Mitzenmacher, Shokrollahi, Spielman,
and Steman ‘97
![Page 70: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/70.jpg)
Asymptotic Analysis: Density Evolution -- BEC
x x x
Luby,Mitzenmacher, Shokrollahi, Spielman,
and Steman ‘97
![Page 71: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/71.jpg)
Asymptotic Analysis: Density Evolution -- BEC
x
1-(1-x)r-1
x x
Luby,Mitzenmacher, Shokrollahi, Spielman,
and Steman ‘97
![Page 72: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/72.jpg)
Asymptotic Analysis: Density Evolution -- BEC
x
1-(1-x)r-1
x x
ε (1-(1-x)r-1)l-1
Luby,Mitzenmacher, Shokrollahi, Spielman,
and Steman ‘97
![Page 73: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/73.jpg)
Asymptotic Analysis: Density Evolution -- BEC
x
1-(1-x)r-1
x x
ε (1-(1-x)r-1)l-1
ε
Luby,Mitzenmacher, Shokrollahi, Spielman,
and Steman ‘97
![Page 74: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/74.jpg)
Asymptotic Analysis: Density Evolution -- BEC
ε
![Page 75: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/75.jpg)
Asymptotic Analysis: Density Evolution -- BEC
ε
phase transition: εBP so that xt → 0 for ε< εBP
xt → x∞>0 for ε> εBP
![Page 76: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/76.jpg)
Asymptotic Analysis: Density Evolution -- BEC
ε
phase transition: εBP so that xt → 0 for ε< εBP
xt → x∞>0 for ε> εBP
![Page 77: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/77.jpg)
Asymptotic Analysis: Density Evolution -- BSC, Gallager Algorithm
xt =ε (1-p+(xt-1))+(1-ε) p-(xt-1)
p+(x)=((1+(1-2x)r-1)/2) l-1 p-(x)=((1-(1-2x)r-1)/2) l-1
![Page 78: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/78.jpg)
Asymptotic Analysis: Density Evolution -- BSC, Gallager Algorithm
phase transition: εBP so that xt → 0 for ε< εBP
xt → x∞>0 for ε> εBP
xt =ε (1-p+(xt-1))+(1-ε) p-(xt-1)
p+(x)=((1+(1-2x)r-1)/2) l-1 p-(x)=((1-(1-2x)r-1)/2) l-1
![Page 79: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/79.jpg)
Asymptotic Analysis: Density Evolution -- BSC, Gallager Algorithm
xt =ε (1-p+(xt-1))+(1-ε) p-(xt-1)
p+(x)=((1+(1-2x)r-1)/2) l-1 p-(x)=((1-(1-2x)r-1)/2) l-1
![Page 80: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/80.jpg)
Asymptotic Analysis: Density Evolution -- BP
![Page 81: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/81.jpg)
Inversion of Limits
![Page 82: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/82.jpg)
Inversion of Limits
size versus number of iterations
![Page 83: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/83.jpg)
Density Evolution Limit
![Page 84: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/84.jpg)
Density Evolution Limit
![Page 85: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/85.jpg)
Density Evolution Limit
![Page 86: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/86.jpg)
Density Evolution Limit
![Page 87: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/87.jpg)
Density Evolution Limit
![Page 88: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/88.jpg)
Density Evolution Limit
![Page 89: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/89.jpg)
Density Evolution Limit
![Page 90: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/90.jpg)
Density Evolution Limit
![Page 91: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/91.jpg)
Density Evolution Limit
![Page 92: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/92.jpg)
Density Evolution Limit
![Page 93: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/93.jpg)
Density Evolution Limit
![Page 94: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/94.jpg)
“Practical” Limit
![Page 95: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/95.jpg)
“Practical” Limit
![Page 96: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/96.jpg)
“Practical” Limit
![Page 97: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/97.jpg)
“Practical” Limit
![Page 98: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/98.jpg)
“Practical” Limit
![Page 99: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/99.jpg)
“Practical” Limit
![Page 100: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/100.jpg)
“Practical” Limit
![Page 101: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/101.jpg)
The Two Limits
Easy: (Density Evolution Limit)
Hard(er): (“Practical Limit”)
![Page 102: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/102.jpg)
Binary Erasure Channel
![Page 103: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/103.jpg)
Binary Erasure Channel
DE Limit
![Page 104: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/104.jpg)
Binary Erasure Channel
DE Limit
![Page 105: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/105.jpg)
Binary Erasure Channel
DE Limit
![Page 106: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/106.jpg)
Binary Erasure Channel
DE Limit
“Practical” Limit
implies
![Page 107: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/107.jpg)
What about “General” Case
expansion
probabilistic methods
Korada and U.
![Page 108: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/108.jpg)
Expansion
expansion ~ 1-1/l
![Page 109: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/109.jpg)
Expansion
Miller and Burshtein: Random element of LDPC(l, r, n) ensemble is expander with expansion close to 1-1/l
with high probability
expansion ~ 1-1/l
![Page 110: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/110.jpg)
Why is Expansion Useful?
![Page 111: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/111.jpg)
Setting: Channel
![Page 112: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/112.jpg)
Setting: Ensemble
![Page 113: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/113.jpg)
Setting: Algorithm
![Page 114: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/114.jpg)
Aim: Show for this setting that ...
DE Limit
“Practical” Limit
implies
![Page 115: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/115.jpg)
Proof Outline
linearize the algorithm combine with density evolution correlation and interaction witness randomizing noise outside the witness sub-critical birth and death process
![Page 116: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/116.jpg)
Linearized Decoding Algorithm
![Page 117: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/117.jpg)
Proof Outline
linearize the algorithm combine with density evolution correlation and interaction witness randomizing noise outside the witness sub-critical birth and death process
![Page 118: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/118.jpg)
Combine with Density Evolution
![Page 119: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/119.jpg)
Combine with Density Evolution
![Page 120: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/120.jpg)
Combine with Density Evolution
![Page 121: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/121.jpg)
Combine with Density Evolution
![Page 122: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/122.jpg)
Proof Outline
linearize the algorithm combine with density evolution correlation and interaction witness randomizing noise outside the witness sub-critical birth and death process
![Page 123: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/123.jpg)
Correlation and Interaction
![Page 124: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/124.jpg)
Correlation and Interaction
![Page 125: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/125.jpg)
Correlation and Interaction
![Page 126: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/126.jpg)
Correlation and Interaction
0
![Page 127: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/127.jpg)
Correlation and Interaction
0 1
![Page 128: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/128.jpg)
Correlation and Interaction
0 1
![Page 129: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/129.jpg)
Correlation and Interaction
0 1
0
![Page 130: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/130.jpg)
Correlation and Interaction
0 1
00
![Page 131: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/131.jpg)
Correlation and Interaction
0 1
000
![Page 132: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/132.jpg)
Correlation and Interaction
0 1
1 000
![Page 133: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/133.jpg)
Correlation and Interaction
0 1
1 000
![Page 134: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/134.jpg)
Correlation and Interaction
0 1
1 000Expected growth:
![Page 135: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/135.jpg)
Correlation and Interaction
0 1
1 000Expected growth:
(r-1)(r-1)
![Page 136: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/136.jpg)
Correlation and Interaction
0 1
1 000Expected growth:
(r-1) 2 ε 2 ε
![Page 137: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/137.jpg)
Correlation and Interaction
0 1
1 000Expected growth:
(r-1) 2 ε ?< 1
![Page 138: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/138.jpg)
Correlation and Interaction
0 1
1 000Expected growth:
(r-1) 2 ε ?< 1
Problem: interaction correlation
![Page 139: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/139.jpg)
Correlation and Interaction
![Page 140: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/140.jpg)
Correlation and Interaction
![Page 141: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/141.jpg)
Correlation and Interaction
![Page 142: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/142.jpg)
Correlation and Interaction
![Page 143: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/143.jpg)
Proof Outline
linearize the algorithm combine with density evolution correlation and interaction witness randomizing noise outside the witness sub-critical birth and death process
![Page 144: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/144.jpg)
Witness
![Page 145: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/145.jpg)
Witness
![Page 146: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/146.jpg)
Witness
![Page 147: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/147.jpg)
Witness
![Page 148: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/148.jpg)
Witness
![Page 149: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/149.jpg)
Witness
![Page 150: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/150.jpg)
Witness
![Page 151: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/151.jpg)
Witness
![Page 152: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/152.jpg)
Witness
![Page 153: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/153.jpg)
Witness
![Page 154: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/154.jpg)
Proof Outline
linearize the algorithm combine with density evolution correlation and interaction witness randomizing noise outside the witness sub-critical birth and death process
![Page 155: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/155.jpg)
Monotonicity
![Page 156: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/156.jpg)
Monotonicity
![Page 157: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/157.jpg)
Monotonicity
![Page 158: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/158.jpg)
Monotonicity
![Page 159: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/159.jpg)
Monotonicity
![Page 160: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/160.jpg)
Monotonicity
![Page 161: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/161.jpg)
Monotonicity
![Page 162: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/162.jpg)
Randomizing the Noise Outside
![Page 163: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/163.jpg)
Randomizing the Noise Outside
![Page 164: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/164.jpg)
Randomizing the Noise Outside
→
![Page 165: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/165.jpg)
Randomizing the Noise Outside
→
←⁄
![Page 166: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/166.jpg)
Randomizing the Noise Outside
randomizing noise outside the witness increases the probability of error
FKG≤
![Page 167: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/167.jpg)
Proof Outline
linearize the algorithm combine with density evolution correlation and interaction witness randomizing noise outside the witness sub-critical birth and death process
![Page 168: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/168.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
![Page 169: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/169.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
![Page 170: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/170.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
![Page 171: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/171.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
![Page 172: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/172.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
![Page 173: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/173.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
0
![Page 174: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/174.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
0 1
![Page 175: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/175.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
0 1
![Page 176: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/176.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
0 1
0
![Page 177: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/177.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
0 1
00
![Page 178: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/178.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
0 1
000
![Page 179: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/179.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
0 1
1 000
![Page 180: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/180.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
0 1
1 000
![Page 181: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/181.jpg)
Expansion
random graph has expansion close to expansion of a treewith high probability
⇒this limits interaction
0 1
1 000
![Page 182: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/182.jpg)
References
For a list of references see:http://ipg.epfl.ch/doku.php?id=en:courses:2007-2208:mct
![Page 183: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/183.jpg)
Results
![Page 184: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/184.jpg)
Open Problems
0.0
0.4
0.3
0.2
0.1
0.2 0.4 0.6 0.8
Pb
channel entropy
![Page 185: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/185.jpg)
Open Problems
0.0
0.4
0.3
0.2
0.1
0.2 0.4 0.6 0.8
Pb
channel entropy
![Page 186: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/186.jpg)
Open Problems
0.0
0.4
0.3
0.2
0.1
0.2 0.4 0.6 0.8
Pb
channel entropy
![Page 187: Andrea Montanari and Ruediger Urbanke TIFR Tuesday ...montanar/OTHER/TALKS/mumbai09_1.pdf · and Steman ‘97. Asymptotic Analysis: Density Evolution -- BEC x x x Luby,Mitzenmacher,](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20e8aead0c1642554dced2/html5/thumbnails/187.jpg)
Open Problems
0.0
0.4
0.3
0.2
0.1
0.2 0.4 0.6 0.8
Pb
channel entropy