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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 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 Interference Alignment IA as a RCRM Nuclear Norm Relaxation Simulations
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
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
System Model Interference Alignment IA as a RCRM Nuclear Norm Relaxation Simulations
DoF • Objective: Max. Rate • (high-SNR) rate = DoF*log(SNR) • Max. DoF: use IA (Select beamformers and ZF) Theorem: Sum DoF= Rx1 Rx2 Rx3
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]
System Model Interference Alignment IA as a RCRM Nuclear Norm Relaxation Simulations
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:
System Model Interference Alignment IA as a RCRM Nuclear Norm Relaxation Simulations
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:
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
Nuclear Norm Heuristic • Now we have a convex relaxation. • Fix and solve
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
System Model Interference Alignment IA as a RCRM Nuclear Norm Relaxation Simulations
Simulations • 3 users • 5 transmit, 3 receive antennas • d = 1,2 • Leakage minimization and max-SINR run for 50 iterations • ε = 0.01
Simulations • 3 users • 8 transmit, 4 receive antennas • d = 2,3 • Leakage minimization and max-SINR run for 50 iterations • ε = 0.01
Conclusions • 3 users • 11 transmit, 5 receive antennas • d = 3,4 • Leakage minimization and max-SINR run for 50 iterations • ε = 0.01
Conclusions • 3 users • 21 transmit, 15 receive antennas • d = 9 • Leakage minimization and max-SINR run for 50 iterations • ε = 0.01