Stability analysis of a class of sparse optimization problems

Abstract

The sparse optimization problems arise in many areas of science and engineering, such as compressed sensing, image processing, statistical and machine learning. The ℓ0-minimization problem is one of such optimization problems, which is typically used to deal with signal recovery. The ℓ1-minimization method is one of the plausible approaches for solving the ℓ0-minimization problems, and thus the stability of such a numerical method is vital for signal recovery. In this paper, we establish a stability result for the ℓ1-minimization problems associated with a general class of ℓ0-minimization problems. To this goal, we introduce the concept of restricted weak range space property (RSP) of a transposed sensing matrix, which is a generalized version of the weak RSP of the transposed sensing matrix introduced in [Zhao et al., Math. Oper. Res., 44(2019), 175-193]. The stability result established in this paper includes several existing ones as special cases.

Dual-density-based reweighted $$\ell _{1}$$-algorithms for a class of $$\ell _{0}$$-minimization problems

“…Although the experiments have shown that DRAtyped algorithms outperform 1 -minimization and some classic reweighted 1 -algorithms, there still exist some future work to do. For example, the convergence and the stability of DRA-typed algorithms are worthwhile future work, which might be investigated under certain assumptions such as the so-called restricted weak range space property (see, e.g., [34]).…”

“…In this case, the difficulty for solving the problems ( 33) and ( 34) is that ε (λ 6 ) might attain an infinite value when w i → ∞. We may introduce a bounded merit function ε ∈ F into (33) and (34) so that the value of ε (λ 6 ) is finite. Moreover, to avoid the infinite optimal value in the model ( 33), w ∈ ζ can be relaxed to −λ 1 − λ T 2 b + λ T 3 y ≤ 1 due to the weak duality.…”

“…To this need, we introduce a bounded convex set W for w to approximate the set ζ . By replacing ζ with W in the models (33), (34) and (37), we obtain the following three types of convex relaxation models of ( 22):…”

Stability of 1-Bit Compressed Sensing in Sparse Data Reconstruction

“…e stability of recovery means that recovery errors stay under control even if the measurements are slightly inaccurate and the data are not exactly sparse. Recent stability study for CS can be found in [21][22][23][24][25]. However, few theoretical results are available on the stability of 1-bit CS.…”

“…) * , where T * α and (T P 0 α ) * are the solution of (18) and (33) (cf (24). and (34)) and T P 0 α is given as(25) with P ≔ P 0 .…”