Constrained maximum likelihood solution of linear equations

The total least squares (TLS) is used to solve a set of inconsistent linear equations Ax≈y when there are errors not only in the observations y but in the modeling matrix A as well. The TLS seeks the least squares perturbation of both y and A that leads to a consistent set of equations. When y and A have a defined structure, we usually want the perturbations to also have this structure. Unfortunately, standard TLS does not generally preserve the perturbation structure, so other methods are required. We examine this problem using a probabilistic framework and derive an approach to determining the most probable set of perturbations, given an a priori perturbation probability density function. While our approach is applicable to both Gaussian and non-Gaussian distributions, we show in the uncorrelated Gaussian case that our method is equivalent to several existing methods. Our approach is therefore more general and can be applied to a wider variety of signal processing problems