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Interference Alignment as a Rank Constrained Rank Minimization

Interference Alignment as a Rank Constrained Rank Minimization. Dimitris S. Papailiopoulos and Alexandros G. Dimakis USC Globecom 2010. Overview. K user MIMO Interference Channel Rewrite IA using Ranks Relax: Nuclear Norm Heuristic Compare with leakage minimization. System Model

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Interference Alignment as a Rank Constrained Rank Minimization

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  1. Interference Alignment as a Rank Constrained Rank Minimization Dimitris S. Papailiopoulos and Alexandros G. Dimakis USC Globecom 2010

  2. Overview • K user MIMO Interference Channel • Rewrite IA using Ranks • Relax: Nuclear Norm Heuristic • Compare with leakage minimization

  3. System Model Interference Alignment IA as a RCRM Nuclear Norm Relaxation Simulations

  4. K-user MIMO interference Channel • K users, MIMO, Gaussian noise • Users beamform and transmit symbols • Rx szeroforce received superpositions Q: what rates can we achieve? Rx1 Rx2 Rx3

  5. Signal Spaces • Rx “observes” a vector in a given space. • Observed space = useful space + interference All useful information is in All useless information is in Q: So, what rates can we achieve? Rx1

  6. System Model Interference Alignment IA as a RCRM Nuclear Norm Relaxation Simulations

  7. DoF • Objective: Max. Rate • (high-SNR) rate = DoF*log(SNR) • Max. DoF: use IA (Select beamformers and ZF) Theorem: Sum DoF= Rx1 Rx2 Rx3

  8. Feasibility of IA (DoF Achievability) • Theorem [Yetis,Gou,Jafar,Kayran]: (w.h.p.) i.e. We can find s and s such that • NP-hard [Razaviyayn,Sanjabi,Luo]

  9. System Model Interference Alignment IA as a RCRM Nuclear Norm Relaxation Simulations

  10. Rank Constrained Rank Minimization • OK. Let’s reformulate our objective. • We want find s and ss.t.: 1) maximize useful dimensions 2) minimize interference • Max sum DoF:

  11. System Model Interference Alignment IA as a RCRM Nuclear Norm Relaxation Simulations

  12. Relax the ranks • Find good relaxation for cost function • Cues: [Recht,Fazel,Parrilo], [Candes,Tao]… • sum of singular values ( -norm) a.k.a. the nuclear norm Replace: with:

  13. Relax the ranks • Find good relaxation for the rank constraints • For any BF and ZF matrices • new BF & ZF with same “rank properties” s.t. Close and “bound” Convex sets

  14. Nuclear Norm Heuristic • Now we have a convex relaxation. • Fix and solve

  15. What is leakage minimization • When perfect IA is possible the “interference leakage” will be zero. • Alternating minimization of • = -norm of singular values. [Gomadam,Cadambe,Jafar] [Peters,Heath] VS Low rank (high DoF) Low energy

  16. System Model Interference Alignment IA as a RCRM Nuclear Norm Relaxation Simulations

  17. Simulations • 3 users • 5 transmit, 3 receive antennas • d = 1,2 • Leakage minimization and max-SINR run for 50 iterations • ε = 0.01

  18. Simulations • 3 users • 8 transmit, 4 receive antennas • d = 2,3 • Leakage minimization and max-SINR run for 50 iterations • ε = 0.01

  19. Conclusions • 3 users • 11 transmit, 5 receive antennas • d = 3,4 • Leakage minimization and max-SINR run for 50 iterations • ε = 0.01

  20. Conclusions • 3 users • 21 transmit, 15 receive antennas • d = 9 • Leakage minimization and max-SINR run for 50 iterations • ε = 0.01

  21. FIN

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