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 2

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

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

5. Signal Spaces 5 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? All useful information is in All useless information is in Q: So, what rates can we achieve?

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

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

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] 8

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

10. 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: 10

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

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 12 Replace: with: 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. 13 Close and “bound” Convex sets

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

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] 15 VS Low rank (high DoF) Low energy VS Low rank (high DoF) Low energy

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

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

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

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

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

21. FIN