Interference Alignment as a Rank Constrained Rank Minimization Dimitris S. Papailiopoulos and...
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Interference Alignment as a Rank Constrained Rank Minimization
Dimitris S. Papailiopoulos and Alexandros G. DimakisUSC
Globecom 2010
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Overview
• K user MIMO Interference Channel
• Rewrite IA using Ranks
• Relax: Nuclear Norm Heuristic
• Compare with leakage minimization
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System ModelInterference AlignmentIA as a RCRMNuclear Norm RelaxationSimulations
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K-user MIMO interference Channel
• K users, MIMO, Gaussian noise• Users beamform and transmit symbols• Rx s zeroforce received superpositions
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Rx1
Rx2
Rx3
Q: what rates can we achieve?
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Signal Spaces5
Rx1
• 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?
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System ModelInterference AlignmentIA as a RCRMNuclear Norm RelaxationSimulations
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DoF
• Objective: Max. Rate• (high-SNR) rate = DoF*log(SNR)• Max. DoF: use IA (Select beamformers and ZF)
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Rx1
Rx2
Rx3
Theorem: Sum DoF=
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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]
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System ModelInterference AlignmentIA as a RCRMNuclear Norm RelaxationSimulations
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Rank Constrained Rank Minimization
• OK. Let’s reformulate our objective.• We want find s and s s.t.: 1) maximize useful dimensions
2) minimize interference • Max sum DoF:
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System ModelInterference AlignmentIA as a RCRMNuclear Norm RelaxationSimulations
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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
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Replace:
with:
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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.
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Close and “bound”
Convex sets
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Nuclear Norm Heuristic
• Now we have a convex relaxation.• Fix and solve
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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]
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VS
Low rank (high DoF) Low energy
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System ModelInterference AlignmentIA as a RCRMNuclear Norm RelaxationSimulations
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Simulations17
• 3 users• 5 transmit, 3 receive antennas• d = 1,2Leakage minimization and max-SINR run for 50 iterationsε = 0.01
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Simulations18
• 3 users• 8 transmit, 4 receive antennas• d = 2,3Leakage minimization and max-SINR run for 50 iterationsε = 0.01
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Conclusions19
• 3 users• 11 transmit, 5 receive antennas• d = 3,4Leakage minimization and max-SINR run for 50 iterationsε = 0.01
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Conclusions20
• 3 users• 21 transmit, 15 receive antennas• d = 9Leakage minimization and max-SINR run for 50 iterationsε = 0.01
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FIN