(untitled) (Journal of Computational Mathematics, 2024)
In this paper, we establish the oracle inequalities of highly corrupted linear observations b = Ax_0 + f_0 + e ∈ R^m. Here the vector x_0 ∈ R^n with n ≫ m is a (approximately) sparse signal and f_0 ∈ R^m is a sparse error vector with nonzero entries that can be possible infinitely large, e ∼ N (0, σ^2 I_m) represents the Gaussian random noise vector. We extend the oracle inequality ||x-x_0||^2_2_imin{|x_0(i)|^2,σ^2} for Dantzig selector and Lasso models in [E.J. Candès and T. Tao, Ann. Statist., 35 (2007), 2313–2351] and [T.T. Cai, L. Wang, and G. Xu, IEEE Trans. Inf. Theory, 56 (2010), 3516–3522] to ||x-x_0||^2_2+||f-f_0||^2_2_imin{|x_0(i)|^2,σ^2}+_jmin{|λ f_0(j)|^2,σ^2} for the extended Dantzig selector and Lasso models. Here (x, f) is the solution of the extended model, and λ > 0 is the balance parameter between ||x||_1 and ||f||_1 i.e. ||x||_1+λ||f||_1.