Exact error linearization uses nonlinear input-output injection to design observers with linear error dynamics in certain coordinates. This approach can only be applied nongenerically. We propose an observer for a wider class of multivariable systems which uniformly minimizes the nonlinear part of the system that cannot be canceled by nonlinear input-output injection. Our approach is numerical, constructive, and provides locally exponentially stable error dynamics. An example compares our design with a high-gain method.