Nuisance attribute projection (NAP) has become a common method for compensation of channel effects, session variation, speaker variation, and general mismatch in speaker recognition. NAP uses an orthogonal projection to remove a nuisance subspace from a larger expansion space that contains the speaker information. Training the NAP subspace is based on optimizing pairwise distances to reduce intraspeaker variability and retain interspeaker variability. In this paper, we introduce a novel form of NAP called weighted NAP (WNAP) which significantly extends the current methodology. For WNAP, we propose a training criterion that incorporates two critical extensions to NAP variable metrics and instance-weighted training. Both an eigenvector and iterative method are proposed for solving the resulting optimization problem. The effectiveness of WNAP is shown on a NIST speaker recognition evaluation task where error rates are reduced by over 20%.