For a linear least squares problem, it is possible to define a resolution matrix, relating the estimated model parameters to the true model parameters. If we are solving a regularized problem, $$ \min_x || y - A x ||^2 + x^T \Lambda x $$ then $$ x_{est} = (A^TA + \Lambda)^{-1} A^T y $$ and if we assume that $y = A x_{true}$, then we have $$ x_{est} = R x_{true} $$ with the resolution matrix $R = (A^TA + \Lambda)^{-1} A^TA$. Resolution matrices are often used to study biases introduced into least squares by a priori regularization (the $\Lambda$ matrix).
However, what about nonlinear least squares? Is it possible to define an analogous resolution operator to study this? Here we have a residual vector $$ r = y - f(x) $$ where $f(x)$ is a nonlinear function of $x$. We can assume (as in the linear case) that $y = f(x_{true})$, but we cannot write down a closed form expression for $x_{est}$, nor can we write $f(x) = Ax$ for some matrix $A$.
Does anyone know what can be done here? I have not had luck finding literature on this topic.