Sparse estimation from sign measurements with general sensing matrix perturbation

In this paper, the problem of estimating a sparse deterministic parameter vector from its sign measurements with a general perturbed sensing matrix is considered. Firstly, the best achievable mean square error (MSE) performance is explored by deriving the sparsity constrained Cramér Rao lower bound (CRLB). Secondly, the maximum likelihood (ML) estimator is utilized to estimate the unknown parameter vector. Although the ML estimation problem is non-convex, we find it can be reformulated as a convex optimization problem by re-parametrization and relaxation, which guarantees numerical algorithms to converge to the optimal point. Thirdly, a fixed point continuation (FPC) algorithm is used to solve the relaxed ML estimation problem. Finally, numerical simulations are performed to show that this relaxed method works well, and the ML estimator asymptotically approaches the CRLB as the number of measurements increases.