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

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

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

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

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

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

Prior art H. Cox, R. M. Zeskind, and M. H. Owen, “Robust adaptive beamforming” IEEE Transactions on Acoustic, Speech, and Signal Processing, vol. 35, 1987. 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

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:

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 !

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), 2011 49th Annual Allerton Conference on. IEEE, 2011.

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

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

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

Numerical Results Error gain versus number of interferers:

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

thank you! Questions?