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Network Source Coding

Network Source Coding. Lee Center Workshop 2006 Wei-Hsin Gu (EE, with Prof. Effros). Outline. Introduction Previously Solved Problems Our Results Summary. Problem Formulation (1). General network : A directed graph G = (V,E). Source inputs and reproduction demands .

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Network Source Coding

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  1. Network Source Coding Lee Center Workshop 2006Wei-Hsin Gu (EE, with Prof. Effros)

  2. Outline • Introduction • Previously Solved Problems • Our Results • Summary

  3. Problem Formulation (1) • General network : • A directed graph G = (V,E). • Source inputs and reproduction demands. • All links are directed and error-free. • Source sequences • Distortion measures are given.

  4. Problem Formulation (2) • Rate Distortion Region • Given a network. • : source pmf. • : distortion vector. • A rate vector is - achievable if there exists a sequence of length n codes of rate whose reproductions satisfy the distortion constraints asymptotically. • The closure of the set consisting of all achievable is called the rate distortion region, . • A rate vector is losslessly achievable if the error probability of reproductions can be arbitrarily small. Lossless rate region .

  5. Black : sources Red : reproductions • is achievable if and only if Example Encoder Encoder Decoder Decoder

  6. Outline • Introduction • Previously Solved Problems • Our Results • Summary

  7. Encoder1 Encoder Encoder Decoder Decoder Decoder 1 Encoder3 Decoder 2 Decoder Encoder Encoder1 Encoder2 • Lossless : Source Coding Theorem • Lossy : Rate-Distortion Theorem Black : sources Black : sources Black : sources Red : reproductions Red : reproductions Red : reproductions Encoder2 Known Results • Lossy : [Gray and Wyner ’74] • Lossless :Slepian-Wolf problem • Lossless : Solved [Ahlswede and Korner `75].

  8. Outline • Introduction • Previously Solved Problems • Our Results • Two Multi-hop Networks • Properties of Rate Distortion Regions • Summary

  9. Black : sources Red : reproductions Multi-Hop Network (1)Achievability Result • Source coding for the following multihop network Node 2 Node 1 Node 3 Encoder Encoder Decoder Decoder Achievability Result

  10. Black : sources Red : reproductions Multi-Hop Network (1)Converse Result • Source coding for the following multihop network Node 2 Node 1 Node 3 Encoder Encoder Decoder Decoder Converse Result

  11. Decoder1 Encoder2 Encoder1 Decoder3 Encoder3 Decoder2 Multi-Hop Network (2) • Diamond Network

  12. Is optimal Diamond NetworkSimpler Case • When are independent • Lossless (by Fano’s inequality) :

  13. Diamond NetworkAchievability Result

  14. Diamond NetworkConverse Result

  15. Outline • Introduction • Previously Solved Problems • Our Results • Two Multi-hop Networks • Properties of Rate Distortion Regions • Summary

  16. Properties of RD Regions • : Rate distortion region. • : Lossless rate region. • is continuous in for finite-alphabet sources. • Conjecture • is continuous in source pmf

  17. Continuity • Can allow small errors estimating source pmf. • Trivial for point-to-point networks – rate distortion function is continuous in probability distribution. • Two convex subsets of are if and only if • Continuity means that • Proved to be true for those networks whose one-letter characterizations have been found.

  18. Continuity - Example • Slepian-Wolf problem

  19. Continuity - Example • Coded side information problem • Since alphabet of is bounded, and are uniformly continuous in over all

  20. Outline • Introduction • Previously Solved Problems • Our Results • Summary

  21. Summary • Study the RD regions for two multi-hop networks. • Solved for independent sources. • Achievability and converse results are not yet known to be tight. • Study general properties of RD regions • RD regions are continuous in distortion vector for finite-alphabet sources. • Conjecture that RD regions are continuous in pmf for finite-alphabet sources.

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