Instance Optimality and Quotient Property
This chapter introduces the notion of instance optimality. While uniform ℓ 1-instance optimality is a practical concept in compressive sensing, uniform ℓ 2-instance optimality is shown not to be. A new property, called quotient property, is then developed to analyze measurement–reconstruction schemes. This property of the measurement matrix, coupled with equality-constrained ℓ 1-minimization, guarantees robustness of the scheme under measurement errors as well as nonuniform ℓ 2-instance optimality. The quotient property is proved to hold with high probability for Gaussian matrices and, modulo a slight modification, for subgaussian matrices.
Keywordsuniform instance optimality gaussian matrices subgaussian matrices stability robustness ℓ1-minimization quotient property nonuniform instance optimality
- 210.S. Foucart, Stability and robustness of ℓ 1-minimizations with Weibull matrices and redundant dictionaries. Lin. Algebra Appl. to appearGoogle Scholar
- 228.E. Gluskin, Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. Mat. Sb. (N.S.) 136(178)(1), 85–96 (1988)Google Scholar
- 510.P. Wojtaszczyk, ℓ 1 minimisation with noisy data. SIAM J. Numer. Anal. 50(2), 458–467 (2012)Google Scholar