Maximum Likelihood Estimation From Sign Measurements With Sensing Matrix Perturbation

The problem of estimating an unknown deterministic parameter vector from sign measurements with a perturbed sensing matrix is studied in this paper. We analyze the best achievable mean-square error (MSE) performance by exploring the corresponding Cramér-Rao lower bound (CRLB). To estimate the parameter, the maximum likelihood (ML) estimator is utilized and its consistency is proved. We show that, compared with the perturbed-free setting, the perturbation on the sensing matrix exacerbates the performance of the ML estimator in most cases. However, suitable perturbation may improve the performance in some special cases. Then, we reformulate the original ML estimation problem as a convex optimization problem, which can be solved efficiently. Furthermore, theoretical analysis implies that the perturbation-ignored estimation is a scaled version with the same direction of the ML estimation. Finally, numerical simulations are performed to validate our theoretical analysis.

I. Introduction

The linear regression problem with perturbed sensing matrix has been extensively studied in recent years [1]–[3]. Mathematically, the vector is observed via a corrupted sensing matrix as $${\bf y}=({\bf H}+{\bf E})^{\rm T}{\bf w}+{\bf n},\eqno{\hbox{(1)}}$$ where is a deterministic known sensing matrix, and is a random matrix each of whose elements is i.i.d., , and denotes transposition. The additive noise vector is independent of and satisfies , where is viewed as the strength of perturbation. To estimate the unknown parameter vector , the perturbation is treated as a nuisance parameter and the maximum likelihood (ML) method is used. Several numerical methods have been proposed including minimax search, maximin search, and the classical expectation-maximization (EM) algorithm [3]. It can be seen that the sensing matrix is modeled as a random Gaussian matrix in (1). Another approach to describing the uncertainty is the standard errors-in-variables (EIV) model. In this setting, the sensing matrix is considered as a deterministic unknown matrix, and a noisy observation on this matrix is available. It is shown that the ML estimation of is the well-known total least squares estimator [4], [5].