2 research outputs found
On -ratio CMSV for sparse recovery
Sparse recovery aims to reconstruct an unknown spare or approximately sparse
signal from significantly few noisy incoherent linear measurements. As a kind
of computable incoherence measure of the measurement matrix, -ratio
constrained minimal singular values (CMSV) was proposed in Zhou and Yu
\cite{zhou2018sparse} to derive the performance bounds for sparse recovery. In
this paper, we study the geometrical property of the -ratio CMSV, based on
which we establish new sufficient conditions for signal recovery involving both
sparsity defect and measurement error. The -truncated set -width of
the measurement matrix is developed as the geometrical characterization of
-ratio CMSV. In addition, we show that the -ratio CMSVs of a class of
structured random matrices are bounded away from zero with high probability as
long as the number of measurements is large enough, therefore satisfy those
established sufficient conditions. Overall, our results generalize the results
in Zhang and Cheng \cite{zc} from to any and complement
the arguments of -ratio CMSV from a geometrical view
Sparse recovery based on q-ratio constrained minimal singular values
We study verifiable sufficient conditions and computable performance bounds
for sparse recovery algorithms such as the Basis Pursuit, the Dantzig selector
and the Lasso estimator, in terms of a newly defined family of quality measures
for the measurement matrices. With high probability, the developed measures for
subgaussian random matrices are bounded away from zero as long as the number of
measurements is reasonably large. Comparing to the restricted isotropic
constant based performance analysis, the arguments in this paper are much more
concise and the obtained bounds are tighter. Numerical experiments are
presented to illustrate our theoretical results