The paper addresses the limitation of Denoising Score Matching (DSM), which struggles to estimate the score of a "noised" distribution at low noise levels. It introduces the Target Score Identity (TSI) and the Target Score Matching (TSM) regression loss, which leverage the known score of the "clean" distribution to improve the accuracy of score estimates at low noise levels. The TSI and TSM are derived for additive noise models and extended to more general scenarios, including non-additive noise and Lie groups. The authors demonstrate the benefits of these novel score estimators through experiments on 1-d mixture of Gaussian targets, showing improved variance and convergence compared to traditional methods. The paper also discusses extensions to bridge matching and provides proofs for the main results.The paper addresses the limitation of Denoising Score Matching (DSM), which struggles to estimate the score of a "noised" distribution at low noise levels. It introduces the Target Score Identity (TSI) and the Target Score Matching (TSM) regression loss, which leverage the known score of the "clean" distribution to improve the accuracy of score estimates at low noise levels. The TSI and TSM are derived for additive noise models and extended to more general scenarios, including non-additive noise and Lie groups. The authors demonstrate the benefits of these novel score estimators through experiments on 1-d mixture of Gaussian targets, showing improved variance and convergence compared to traditional methods. The paper also discusses extensions to bridge matching and provides proofs for the main results.