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

2. 2 Outline Introduction Previously Solved Problems Our Results Summary

3. 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. 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. 5 Example Decoder Encoder Decoder Black : sources Red : reproductions is achievable if and only if

6. 6 Outline Introduction Previously Solved Problems Our Results Summary

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

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

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

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

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

12. 12 Diamond Network Simpler Case When are independent  Lossless (by Fano’s inequality) : Is optimal

13. 13 Diamond Network Achievability Result

14. 14 Diamond Network Converse Result

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

16. 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. 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. 18 Continuity - Example Slepian-Wolf problem

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

20. 20 Outline Introduction Previously Solved Problems Our Results Summary

21. 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.