A parallel to the least squares for positive inverse problems

A new method is proposed to solve systems of linear approximate equations X theta approximate to y where the unknowns theta and the data y are positive and the matrix X consists of nonnegative elements. Writing the i-th near-equality X-i.theta/y(i) approximate to 1 the assumed model is X-i.theta/y(i) = zeta(i) with mutually independent positive errors zeta(i). The loss function is defined by Sigma w(i) (zeta(i) - 1) log zeta(i) in which w(i) is the importance weight for the i-th near-equality. A reparameterization reduces the method to unconstrained minimization of a smooth strictly convex function implying the unique existence of positive solution and the applicability of Newton's method that converges quadratically. The solution stability is controlled by weighting prior guesses of the unknowns theta. The method matches the maximum likelihood estimation if all weights w(i) are equal and zeta(i) independently follow the probability density function proportional to t(w(1-t)), 0 < w.