This paper proposes a novel approach to discrimina-tive training of probabilistic linear discriminant analysis (PLDA) for speaker verification with x-vectors. Model over-fitting is a well-known issue with discriminative PLDA (D-PLDA) forspeaker verification. As opposed to prior approaches which address this by limiting the number of trainable parameters, the proposed method parameterizes the discriminative PLDA (D-PLDA) model in a manner which allows for intuitive regularization, permitting the entire model to be optimized. Specifically, the within-class and across-class covariance matrices which comprise the PLDA model are expressed as products of orthonormal and diagonal matrices, and the structure of these matrices is enforced during model training. The proposed approach provides consistent performance improvements relative to previous D-PLDA methods when applied to a variety of speaker recognition evaluations, including the Speakers in the Wild Core-Core, SRE16, SRE18 CMN2, SRE19 CMN2, and VoxCeleb1 Tasks. Additionally, when implemented in Tensorflow using a modernGPU, D-PLDA optimization is highly efficient, requiring less than 20 minutes.