1. Manifold Sparse Beamforming
Volkan cevher
Joint work with: baran gözcü, afsaneh asaei

2. outline Array acquisition model Spatial linear prediction
Minimum variance distortion-less response (MVDR)
Regularization
Manifold Sparse Beamforming
Atomic Norm Minimization
Numerical results
Concluding remarks

3. Acquisition model where Sensor array acquisition forward model for
input signal
where θs
Δ=λ/2 xM x1 x2 x. x. x.

4. Spatial filtering
Objective: Prediction of s given the observation x and through spatial linear filter The steering vectors are obtained via minimization of the expected prediction risk: θs Δ xM x1 x2 x. x. x. w? WM W. W. W. W2 W1 Σ

5. Optimum weights
Assumptions: Signal is uncorrelated with the the interferers and noise Prediction risk minimization
This is MVDR !
(Minimum variance distortionless (c=1) response)

6. DIAGONAL LOADING
MVDR in practice: Sample covariance matrix: ISSUE: Array covariance matrix is rank-deficient  diagonal loading This corresponds to Tikhonov regularization:
Unique solution !

7. Prior art
H. Cox, R. M. Zeskind, and M. H. Owen, “Robust adaptive beamforming” IEEE Transactions on Acoustic, Speech, and Signal Processing, vol. 35,
Diagonal Loading …
S. A. Vorobyov, “Principles of minimum variance robust adaptive beamforming design” Signal Processing, Special Issue: Advances in Sensor Array Processing, 2013.
Eigen-space projection
Worst case optimization over an uncertainty bound
Iterative refinement using sequential quadratic programming

8. A New regularization: Manifold sparsity
We claim that the optimum weights accept
an S-sparse linear sum of manifold vectors:
A heuristic justification:
If interferer angles are random,
optimum weights lies in S=M-K+1
dimensional subspace.
Hence this set of equations has a solution
when K+1≤M-K+1 i.e. when K≤M/2.
Therefore if
then error is minimized. Plugging in
the S-sparse linear combination above,
we have the following linear system:

9. ATOMIC NORM
To enforce a sparse linear combination of manifold vectors, we introduce the following norm:
where A is the infinite set of manifold vectors :
grid-free !

10. atomic norm when atoms are sinusoids
Application to line spectral estimation:
Dual problem
Bhaskar, Badri Narayan, and Benjamin Recht. "Atomic norm denoising with applications to line spectral estimation." Communication, Control, and Computing (Allerton), th Annual Allerton Conference on. IEEE, 2011.

11. Manifold sparse beamforming
Proposed regularization
is equivalent to
SDP!
where t is a real number and T is the map that makes a Hermitian Toeplitz
matrix out of its input vector

12. NUMERICAL EXPERIMENTS
We would like to compare the two optimization problems: versus Parameters:
(Diagonal Loading)
(Manifold Sparse)
Number of snaphots: T=80
Number of sensors: M=8
SIR levels= -10, 0, 10, and 20 dB
SNR levels= 20db and no noise

13. SIMULATION procedure FOR 8 sources ✕ 4 SIRs ✕ 2 SNRs ✕ 2500 runs
Choose random DOAs and take T snapshots on the sensor array
Select a lambda in [1, 2] for oracle tuning
1- Find a good initial point by grid search
2- Use fmincon to find a finer optimum lambda
Solve the two optimization problem by CVX using that lambda and evaluate the errors
END

14. Numerical Results
Error gain
versus
number of interferers:

15. conclusions Novel sparsity regularized beamforming
Solution obtained by semi-definite optimization of atomic norm
Manifold sparsity assumption yields up to 2-dB gain in signal estimation over diagonal loading
Semidefinite formulation enables grid-free estimation
Further extensions
Regularization parameter selection
Learning theoretic study of regularized beamforming
Gridless source localization

16. thank you!
Questions?