Sub-optimality of superposition coding for three or more receivers Chandra Nair, & Mehdi Yazdanpanah The Chinese University of Hong Kong CISS 2018 23 Mar, 2018
The story of superposition coding is the story of broadcast channels with degradation (receivers or message sets) Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 2 / 17
The story of superposition coding is the story of broadcast channels with degradation (receivers or message sets) It is also the story of auxiliary random variables Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 2 / 17
In the beginning ◮ Cover (1972) proposed the superposition coding achievable region for degraded broadcast channels He used an auxiliary variable to represent the message for the weaker of two receivers. ◮ Bergmans (1973, Gaussian) and Gallager (1974, discrete-memoryless) established the optimality of superposition coding for the degraded broadcast channel Gallager’s proof of optimality of superposition coding naturally extended to a sequence of degraded receivers Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 3 / 17
In the beginning ◮ Cover (1972) proposed the superposition coding achievable region for degraded broadcast channels He used an auxiliary variable to represent the message for the weaker of two receivers. ◮ Bergmans (1973, Gaussian) and Gallager (1974, discrete-memoryless) established the optimality of superposition coding for the degraded broadcast channel Gallager’s proof of optimality of superposition coding naturally extended to a sequence of degraded receivers The optimality of superposition coding region was then established for ◮ Weaker notions of weaker receiver in a two-receiver broadcast channel Less Noisy (Korner-Marton 75), More Capable (El Gamal 79) ◮ Degraded message sets (Korner-Marton 77) Comparison of the sizes of the images of a set via two channels Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 3 / 17
In the beginning ◮ Cover (1972) proposed the superposition coding achievable region for degraded broadcast channels He used an auxiliary variable to represent the message for the weaker of two receivers. ◮ Bergmans (1973, Gaussian) and Gallager (1974, discrete-memoryless) established the optimality of superposition coding for the degraded broadcast channel Gallager’s proof of optimality of superposition coding naturally extended to a sequence of degraded receivers The optimality of superposition coding region was then established for ◮ Weaker notions of weaker receiver in a two-receiver broadcast channel Less Noisy (Korner-Marton 75), More Capable (El Gamal 79) ◮ Degraded message sets (Korner-Marton 77) Comparison of the sizes of the images of a set via two channels A (seemingly) natural "Implication": Auxiliaries (in superposition coding) captured the coarser message. Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 3 / 17
More recent times .. The optimality of superposition coding was established for ◮ Three receiver less-noisy broadcast channel ( N -Wang 2010) Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 4 / 17
More recent times .. The optimality of superposition coding was established for ◮ Three receiver less-noisy broadcast channel ( N -Wang 2010) However for the following three-receiver broadcast channel setting: ◮ Receivers Y 2 , Y 3 wish to decode message M 0 ◮ Receiver Y 1 wishes to decode messages M 0 , M 1 superposition coding region was not optimal (El Gamal- N 2009) Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 4 / 17
More recent times .. The optimality of superposition coding was established for ◮ Three receiver less-noisy broadcast channel ( N -Wang 2010) However for the following three-receiver broadcast channel setting: ◮ Receivers Y 2 , Y 3 wish to decode message M 0 ◮ Receiver Y 1 wishes to decode messages M 0 , M 1 superposition coding region was not optimal (El Gamal- N 2009) Demonstrates that auxiliaries do not capture the coarser message. ◮ Associating an auxiliary with information decoded by groups of receivers improved the achievable region U 123 , U 12 , U 13 , and U 1 = X . The achievable region was no longer a superposition coding region, it also involved the other (old) idea: random binning A (seemingly) natural "Implication" (Take 2): Auxiliaries (in superposition coding) captured the information decoded by groups of receivers. ◮ Binning and Superposition both present Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 4 / 17
Consistency The revised intuition about auxiliaries is consistent with Marton’s achievable scheme for two-receiver broadcast channels with private messages ◮ The region employs three auxiliaries: U 1 , U 2 , U 12 ◮ It is known that this region is strictly better than the one with only U 1 , U 2 (even for private messages). Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 5 / 17
Consistency The revised intuition about auxiliaries is consistent with Marton’s achievable scheme for two-receiver broadcast channels with private messages ◮ The region employs three auxiliaries: U 1 , U 2 , U 12 ◮ It is known that this region is strictly better than the one with only U 1 , U 2 (even for private messages). Open Question Is Marton’s achievable scheme for two-receiver broadcast channels optimal? Three-or-more receivers A three-receiver broadcast channel with private messages would have seven auxiliaries U 123 , U 12 , U 13 , U 23 , U 1 , U 2 , U 3 and a natural extension of Marton’s scheme would have two layers of superposition coding and binning between random variables in each layer. A succinct clean representation of the rate constraints is not available Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 5 / 17
Consistency The revised intuition about auxiliaries is consistent with Marton’s achievable scheme for two-receiver broadcast channels with private messages ◮ The region employs three auxiliaries: U 1 , U 2 , U 12 ◮ It is known that this region is strictly better than the one with only U 1 , U 2 (even for private messages). Open Question Is Marton’s achievable scheme for two-receiver broadcast channels optimal? Three-or-more receivers A three-receiver broadcast channel with private messages would have seven auxiliaries U 123 , U 12 , U 13 , U 23 , U 1 , U 2 , U 3 and a natural extension of Marton’s scheme would have two layers of superposition coding and binning between random variables in each layer. A succinct clean representation of the rate constraints is not available However: this region is not optimal (Padlakanda and Pradhan (2015)) Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 5 / 17
Still one fundamental setting remained Consider the setting: ◮ Receivers Y 3 wish to decode message M 0 ◮ Receiver Y 1 , Y 2 wishes to decode messages M 0 , M 1 Interpretation of auxiliaries: U 123 and U 12 = X , and only superposition coding Question : Is superposition coding optimal? (Note: The first layer superposition coding of the previous private message setting) Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 6 / 17
Still one fundamental setting remained Consider the setting: ◮ Receivers Y 3 wish to decode message M 0 ◮ Receiver Y 1 , Y 2 wishes to decode messages M 0 , M 1 Interpretation of auxiliaries: U 123 and U 12 = X , and only superposition coding Question : Is superposition coding optimal? (Note: The first layer superposition coding of the previous private message setting) Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 6 / 17
Still one fundamental setting remained Consider the setting: ◮ Receivers Y 3 wish to decode message M 0 ◮ Receiver Y 1 , Y 2 wishes to decode messages M 0 , M 1 Interpretation of auxiliaries: U 123 and U 12 = X , and only superposition coding Question : Is superposition coding optimal? (Note: The first layer superposition coding of the previous private message setting) Csiszar’s open problem is very closely tied to finding the capacity region Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 6 / 17
A remark Körner had proposed a region (1984) for the image size characterization over three channels Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 7 / 17
Suboptimality of superposition coding Superposition coding is sub-optimal for the setting ( N -Yazdanpanah 2017) ◮ Constructed a channel whose 2-letter superposition-coding region was larger than the 1-letter one Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 8 / 17
Suboptimality of superposition coding Superposition coding is sub-optimal for the setting ( N -Yazdanpanah 2017) ◮ Constructed a channel whose 2-letter superposition-coding region was larger than the 1-letter one Note : The same (counter)-example showed that Korner’s region is a proper subset of H ( X ; Y ; Z | S ) . Chandra Nair and Mehdi Yazadanpanah GIC 23 Mar, 2018 8 / 17
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