Interference Alignment as a Rank Constrained Rank Minimization
[Papailiopoulos, Dimakis]
Where? partial results in Globecom 2010, submitted to IEEE TSP.
What we do? We prove that the DoF maximization in a K user MIMO system can be put in the form of a rank constrained, rank minimization (RCRM) problem. We can do that because we rewrite the per user DoF as
DoF = rank(useful space)-rank(interference space)
So to maximize the sum-DoF we minimize the sum of interference ranks subject to the constraint that the useful spaces span all dimensions.
This variational characterization is hard to solve, but we can approximate it. We provide a convex relaxation of the problem: the rank cost function is replaced by the nuclear norm, and the rank constraints by positivity constraints on minimum eigenvalues of nonnegative definite matrices.
Download: globecom, TSP, slides
We also run some simulations in MATLAB. Here is the MATLAB code we used for our simulations.
It simulates a K-user interference channel. We implement our nuclear norm approach, the leakage minimization, and the max-SINR algorithm (both normalized and orthogonalized versions).
The files included in the .zip are:
K_user_IC.m: is the main file simulates a K-user interference channel. We implement our nuclear norm approach, the leakage minimization, and the max-SINR algorithm (both the simple and the orthogonalized version).
At each Monte Carlo run, it computes the sum-rate and the interference free dimensions of the system.
The user can select all system parameter: # Tx or Rx antennas, iterations that each scheme runs for, input powers, pursued DoF, etc.
nuclear_IA_K_user.m: runs the nuclear norm heuristic for a number of iterations and returns the set of K beamforming and the set of K zeroforcing matrices V and U.
leakage_minimization_K_user.m: runs the leakage minimization algorithm for a number of iterations and returns the set of K beamforming V and the set of K zeroforcing U matrices.
maxSINR_K_user.m: runs the max-SINR algorithm for a number of iterations and returns the set of beamforming and zeroforcing matrices V and U.
block_svd.m: calculates the singular values of the signal and interference space matrices.
rate_K_user_MIMO.m: computes the data sum-rate of a real K-user IC.
orthogonalize.m: orthogonalizes beamforming or zerofocring matrices.
normalize.m: normalizes beamforming or zerofocring input matrices.
In our code we use routines of the CVX toolbox, which can be found here.
Feel free to use the code. If you have any comments, find bugs or mistakes, if you use it in your research, or make it run faster (it’s kind of slow), or prove that after all it solves 3-SAT in O(n), please send me an email :).
Distributed Storage Codes Meet Multiple-Access Wiretap Channels
[Papailiopoulos, Dimakis]
Where? Submitted IEEE Trans. IT
What we do? We prove that the code repair and DoF maximization problems are connected. We use this connection to characterize the secure DoF of the multiple access compound wiretap channel. Interestingly, no matter how many eavesdroppers are in the system, the users can still get the same sum DoF as if it was one, (or none, asymptotically in the number of users). So we prove that the per user S-DoF is
1-1/L
where L is the number of users. This coincides with the S-DoF outerbound.
Download: allerton, IT, slides
