Stability Analysis of $\ell _{0,\infty }$-Norm Based Convolutional Sparse Coding Using Stripe Coherence

Theoretical guarantees for the <inline-formula><tex-math notation="LaTeX">$\ell _{0,\infty }$</tex-math></inline-formula>-pseudo-norm based convolutional sparse coding have been established in a recent work. However, the stability analysis in the noisy case via the stripe coherence is absent. This coherence is a stronger characterization of the convolutional dictionary, and a considerably more informative measure than the standard global mutual coherence. The present paper supplements this missing part. Formally, three main results together with their proofs are given. The first one is for the stability of the solution to the <inline-formula><tex-math notation="LaTeX">$P_{0,\infty }^{\epsilon }$</tex-math></inline-formula> problem, the second one and the third one are for the stable recovery of orthogonal matching pursuit (OMP) algorithm in the presence of noise. Under a reasonable assumption, the first two results are compared with the corresponding two that use the global mutual coherence in the previous work, respectively, showing the advantages of our results: 1) the stability guarantee conditions are at least as strong as the existing ones; 2) the upper-bound of the distance between the true sparse vector and the solution to the <inline-formula><tex-math notation="LaTeX">$P_{0,\infty }^{\epsilon }$</tex-math></inline-formula> problem is tighter than the counterpart, so is the one between the true sparse vector and the solution obtained by OMP. Moreover, the second result is compared with the third one, stressing the difference in the restriction on noise energy. Also, some experiments are presented to intuitively compare the aforesaid upper-bounds.