The effect of perturbation and noise folding on the recovery performance of low-rank matrix via the nuclear norm minimization

Abstract

• The completely perturbed low rank matrix recovery model takes in consideration the perturbations in the measurement map and the measurement matrix. • The perturbation on the measurement map result in a multiplicative noise. • The completely perturbed model is equivalent to the partially perturbed model. • The impact of the multiplicative noise on the Signal-to-Noise Ratio (SNR) is different than the additive noise. Previous works in low-rank matrix recovery literature focused on the study of the partially perturbed low-rank matrix recovery model y = A ( X ) + ω , where y ∈ R M is the observed vector, A : R m × n → R M is the measurement operator, X ∈ R m × n is the matrix to be recovered, and ω ∈ R M is the additive noise. However in practice A may only represent a perturbed version of the measurement operator since it models a physical phenomena and there is no guarantee that the desired conditions such as the restricted isometry property (RIP) hold. Similar to the measurement matrix X ∈ R m × n , in a variety of application scenarios, it may also be corrupted by noise. In this paper we introduce the completely perturbed low-rank matrix model by incorporating to the partially perturbed model a non zero perturbation E : R m × n → R M into the measurement operator A which results in a multiplicative noise, and a noise Z ∈ R m × n into the measurement matrix X . First by extending the results of Zhou et al. (2016) we explore the RIP constants of our new proposed model. Then based on the RIP we establish a sufficient condition for robust and stable recovery of low-rank matrices and give an upper bound estimation of the recovery error in the completely perturbed context. The analysis has shown that our model after whitening is equivalent to the partially perturbed model the only distinction is noise variance increase which has a sever impact on the Signal-to-Noise Ratio (SNR).