  In this paper, we account for approaches of sparse recovery from large
underdetermined linear models with perturbation present in both the
measurements and the dictionary matrix. Existing methods have high computation
and low efficiency. The total least-squares (TLS) criterion has well-documented
merits in solving linear regression problems while FOCal Underdetermined System
Solver (FOCUSS) has low-computation complexity in sparse recovery. Based on TLS
and FOCUSS methods, the present paper develops more fast and robust algorithms,
TLS-FOCUSS and SD-FOCUSS. TLS-FOCUSS algorithm is not only near-optimum but
also fast in solving TLS optimization problems under sparsity constraints, and
thus fit for large scale computation. In order to reduce the complexity of
algorithm further, another suboptimal algorithm named D-FOCUSS is devised.
SD-FOCUSS can be applied in MMV (multiple-measurement-vectors) TLS model, which
fills the gap of solving linear regression problems under sparsity constraints.
The convergence of TLS-FOCUSS algorithm and SD-FOCUSS algorithm is established
with mathematical proof. The simulations illustrate the advantage of TLS-FOCUSS
and SD-FOCUSS in accuracy and stability, compared with other algorithms.
