Improving Robustness for Anisotropic Sparse Recovery using Matrix Extensions

Recovery guarantees in compressed sensing (CS) often require upper bounds on the noise level. The robustness with respect to additive errors of unknown power depends on quotient bounds of the measurement matrix. For isotropic random matrices like iid. Gaussian matrices these bounds are known to behave well. In this work we focus instead on explicitly given anisotropic sensing matrices which are more relevant for real world applications. We propose straightforward quotientmodifications of CS decoders using matrix extensions to improve robustness. We reformulate this as a recovery problem under partial off-support knowledge and discuss the implications. Finally, we present numerical results for measurement matrices taken from a particular radar application where our idea of matrix extensions shows substantial performance improvements.