Interference Alignment as a Rank Constrained Rank Minimization

Dimitris S. Papailiopoulos and Alexandros G. DimakisUSC

Globecom 2010

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?

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?

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=

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:

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

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

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

Simulations18

• 3 users• 8 transmit, 4 receive antennas• d = 2,3Leakage minimization and max-SINR run for 50 iterationsε = 0.01

Conclusions19

• 3 users• 11 transmit, 5 receive antennas• d = 3,4Leakage minimization and max-SINR run for 50 iterationsε = 0.01

Conclusions20

• 3 users• 21 transmit, 15 receive antennas• d = 9Leakage minimization and max-SINR run for 50 iterationsε = 0.01

FIN