Capacity Bounds for Diamond Networks Gerhard Kramer (TUM) joint - PowerPoint PPT Presentation

Technische Universität München Capacity Bounds for Diamond Networks Gerhard Kramer (TUM) joint work with Shirin Saeedi Bidokhti (TUM & Stanford) DIMACS Workshop on Network Coding Rutgers University, NJ December 15, 2015 Institute for Communications Engineering

Technische Universität München What is a “Diamond Network” ? • Cascade of a 2-receiver broadcast channel (BC) and a 2-transmitter multi- access channel (MAC) • Simplifications: (1) MAC is two bit-pipes; (2) BC is two bit-pipes R1 Y 1 X 1 W X Y Ŵ Src Enc BC MAC Dec Sink Y 2 X 2 R2

Technische Universität München What is a “Diamond Network” ? • Cascade of a 2-receiver broadcast channel (BC) and a 2-transmitter multi- access channel (MAC) • Simplifications: (1) MAC is two bit-pipes; (2) BC is two bit-pipes B bits n symbols R1 Y 1 X 1 W X Y Ŵ Src Enc BC MAC Dec Sink Y 2 X 2 R = B/n R2

Technische Universität München Background General Problem • B. E. Schein, Distributed coordination in network information theory. PhD Dissertation, MIT, 2001 MAC is 2 Bit Pipes • A. Sanderovich, S. Shamai, Y. Steinberg, G. Kramer, “Communication via decentralized processing,” IEEE Trans. IT, 2008 BC is 2 Bit Pipes • D. Traskov, G. Kramer, “Reliable communication in networks with multi- access interference,” ITW 2007 • W. Kang, N. Liu, and W. Chong, “The Gaussian multiple access diamond channel,” arxiv 2011 (v1) and 2015 (v2)

Technische Universität München Here: BC is two bit pipes • Capacity limitations C 1 and C 2 . Problem seems difficult! • Gaussian MAC partially solved by Kang-Liu (2011) using Ozarow’s trick (1980) • Contribution: new capacity upper bound for discrete MACs • Contribution: solved binary adder MAC capacity by extending Mrs. Gerber’s Lemma R1 X 1 V 1 W Y Ŵ Src Enc MAC Dec Sink V 2 X 2 R2

Outline The Problem Setup A Lower Bound An Upper-Bound Examples The Gaussian MAC The binary adder MAC 3 / 28

The Problem Setup X n 1 Relay 1 MAC Y n ˆ W W Source Encoder Decoder Sink p ( y | x 1 ,x 2 ) X n 2 Relay 2 I W message of rate R 4 / 28

The Problem Setup X n 1 Relay 1 MAC Y n ˆ W W Source Encoder Decoder Sink p ( y | x 1 ,x 2 ) X n 2 Relay 2 I W message of rate R I Bit-pipes of capacities C 1 , C 2 4 / 28

The Problem Setup X n 1 Relay 1 MAC Y n ˆ W W Source Encoder Decoder Sink p ( y | x 1 ,x 2 ) X n 2 Relay 2 I W message of rate R I Bit-pipes of capacities C 1 , C 2 I Goal: What is the highest rate R such that P r( W 6 = ˆ W ) ! 0? 4 / 28

A Lower Bound X n Relay 1 1 MAC Y n ˆ W W Source Encoder Decoder Sink p ( y | x 1 ,x 2 ) X n 2 Relay 2 I Rate splitting: W = ( W 12 , W 1 , W 2 ) I Superposition Coding: W 12 encoded in V n . X n 1 , X n 2 superposed on V n . I Marton’s Coding … a sophisticated superposition … a sophisticated superposition … a sophisticated superposition … a sophisticated superposition 5 / 28

Technische Universität München Rate Bounds • Rate-splitting bounds: • Now apply Fourier-Motzkin elimination

A Lower Bound (Cont.) Theorem (Lower Bound) The rate R is achievable if it satisfies the following condition for some pmf p ( v, x 1 , x 2 , y ) = p ( v, x 1 , x 2 ) p ( y | x 1 , x 2 ) : 8 C 1 + C 2 � I ( X 1 ; X 2 | V ) 9 > > > > C 2 + I ( X 1 ; Y | X 2 V ) > > > > < = R  min C 1 + I ( X 2 ; Y | X 1 V ) 1 2 ( C 1 + C 2 + I ( X 1 X 2 ; Y | V ) � I ( X 1 ; X 2 | V )) > > > > > > > > I ( X 1 X 2 ; Y ) : ; V 2 V , |V|  min { |X 1 | |X 2 | +2 , |Y| +4 } - S. Saeedi Bidokhti, G. Kramer, “Capacity bounds for a class of diamond networks,” ISIT 2014 - S. Saeedi Bidokhti, G. Kramer, “Capacity bounds for a class of diamond networks,” ISIT 2014 - S. Saeedi Bidokhti, G. Kramer, “Capacity bounds for a class of diamond networks,” ISIT 2014 - S. Saeedi Bidokhti, G. Kramer, “Capacity bounds for a class of diamond networks,” ISIT 2014 - W. Kang, N. Liu, W. Chong, “The Gaussian multiple access diamond channel,” arxiv 1104.3300, v2, 2015 - W. Kang, N. Liu, W. Chong, “The Gaussian multiple access diamond channel,” arxiv 1104.3300, v2, 2015 - W. Kang, N. Liu, W. Chong, “The Gaussian multiple access diamond channel,” arxiv 1104.3300, v2, 2015 - W. Kang, N. Liu, W. Chong, “The Gaussian multiple access diamond channel,” arxiv 1104.3300, v2, 2015 6 / 28

The Cut-Set Bound Cut-Set bound: R is achievable only if it satisfies the following bounds for some p ( x 1 , x 2 ): Four Cuts: Four Cuts: Four Cuts: Four Cuts: X 2 R C 1 + C 2  C 2 C 1 + I ( X 2 ; Y | X 1 ) R  source Y sink R C 2 + I ( X 1 ; Y | X 2 )  R  I ( X 1 X 2 ; Y ) . C 1 X 1 7 / 28

Example I: binary adder MAC I X 1 = X 2 = { 0 , 1 } , Y = { 0 , 1 , 2 } I Y = X 1 + X 2 1 . 58 1 . 56 1 . 54 Rate R 1 . 52 1 . 5 1 . 48 Cut-Set bound 1 . 46 Lower bound 0 . 74 0 . 76 0 . 78 0 . 8 0 . 82 0 . 84 0 . 86 Link Capacity C 8 / 28

Example I: binary adder MAC I X 1 = X 2 = { 0 , 1 } , Y = { 0 , 1 , 2 } I Y = X 1 + X 2 1 . 58 1 . 56 1 . 54 Rate R 1 . 52 1 . 5 1 . 48 Cut-Set bound 1 . 46 Lower bound and capacity 0 . 74 0 . 76 0 . 78 0 . 8 0 . 82 0 . 84 0 . 86 Link Capacity C 8 / 28

Example II: Gaussian MAC I Y = X 1 + X 2 + Z , Z ⇠ N (0 , 1) P n P n 1 i =1 E ( X 2 1 i =1 E ( X 2 1 ,i )  P 1 , 2 ,i )  P 2 , P 1 = P 2 = 1 I n n 1 . 2 1 . 15 1 . 1 1 . 05 1 Rate R 0 . 95 0 . 9 0 . 85 0 . 8 0 . 75 Cut-Set bound 0 . 7 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 1 . 3 1 . 4 Link Capacity C 9 / 28

Example II: Gaussian MAC I Y = X 1 + X 2 + Z , Z ⇠ N (0 , 1) P n P n 1 i =1 E ( X 2 1 i =1 E ( X 2 1 ,i )  P 1 , 2 ,i )  P 2 , P 1 = P 2 = 1 I n n 1 . 2 1 . 15 1 . 1 1 . 05 1 Rate R 0 . 95 0 . 9 0 . 85 0 . 8 Cut-Set bound 0 . 75 Lower bound (no superposition coding) 0 . 7 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 1 . 3 1 . 4 Link Capacity C 9 / 28

Example II: Gaussian MAC I Y = X 1 + X 2 + Z , Z ⇠ N (0 , 1) P n P n 1 i =1 E ( X 2 1 i =1 E ( X 2 1 ,i )  P 1 , 2 ,i )  P 2 , P 1 = P 2 = 1 I n n 1 . 2 1 . 15 1 . 1 1 . 05 1 Rate R 0 . 95 0 . 9 0 . 85 0 . 8 Cut-Set bound Lower bound (no superposition coding) 0 . 75 Lower bound (Joint Gaussian dist.) 0 . 7 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 1 . 3 1 . 4 Link Capacity C 9 / 28

Example II: Gaussian MAC I Y = X 1 + X 2 + Z , Z ⇠ N (0 , 1) P n P n 1 i =1 E ( X 2 1 i =1 E ( X 2 1 ,i )  P 1 , 2 ,i )  P 2 , P 1 = P 2 = 1 I n n 1 . 2 1 . 15 1 . 1 1 . 05 1 Rate R 0 . 95 0 . 9 0 . 85 Cut-Set bound 0 . 8 Lower bound (no superposition coding) Lower bound (Joint Gaussian dist.) 0 . 75 Lower bound (Mixture of two Gaussian dist.) 0 . 7 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 1 . 3 1 . 4 Link Capacity C 9 / 28

Example II: Gaussian MAC I Y = X 1 + X 2 + Z , Z ⇠ N (0 , 1) P n P n 1 i =1 E ( X 2 1 i =1 E ( X 2 1 ,i )  P 1 , 2 ,i )  P 2 , P 1 = P 2 = 1 I n n 1 . 2 1 . 15 1 . 1 Lower bound Lower bound is tight is tight � � � � � � � � � � � � � � � � � � ! 1 . 05 1 Rate R 0 . 95 0 . 9 0 . 85 Cut-Set bound 0 . 8 Lower bound (no superposition coding) Lower bound (Joint Gaussian dist.) 0 . 75 Lower bound (Mixture of two Gaussian dist.) 0 . 7 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 1 . 3 1 . 4 Link Capacity C 9 / 28

Is the Cut-Set Bound Tight? Cut-Set bound: R  C 1 + C 2 X 2 R C 1 + I ( X 2 ; Y | X 1 )  C 2 R  C 2 + I ( X 1 ; Y | X 2 ) source Y sink R I ( X 1 X 2 ; Y ) .  C 1 Maximize over p ( x 1 , x 2 ). X 1 10 / 28

Is the Cut-Set Bound Tight? Cut-Set bound: R  C 1 + C 2 X 2 R C 1 + I ( X 2 ; Y | X 1 )  C 2 R  C 2 + I ( X 1 ; Y | X 2 ) source Y sink R I ( X 1 X 2 ; Y ) .  C 1 Maximize over p ( x 1 , x 2 ). X 1 It turns out that the cut-set bound is not tight. One culprit is the cut ({source}, {R1,R2,sink}) One culprit is the cut ({source}, {R1,R2,sink}) One culprit is the cut ({source}, {R1,R2,sink}) One culprit is the cut ({source}, {R1,R2,sink}) One culprit is the cut ({source}, {R1,R2,sink}) One culprit is the cut ({source}, {R1,R2,sink}) One culprit is the cut ({source}, {R1,R2,sink}) One culprit is the cut ({source}, {R1,R2,sink}) 10 / 28

Refining the Cut-Set Bound (cf. [TraskovKramer’07]) (cf. [TraskovKramer’07]) (cf. [TraskovKramer’07]) (cf. [TraskovKramer’07]) (cf. [TraskovKramer’07]) (cf. [TraskovKramer’07]) (cf. [TraskovKramer’07]) (cf. [TraskovKramer’07]) I Motivated by [Ozarow’80, KangLiu’11] 11 / 28

Refining the Cut-Set Bound I Motivated by [Ozarow’80, KangLiu’11] nR  nC 1 + nC 2 � I ( X n 1 ; X n 2 ) 11 / 28