Scoring matching (SM), and its related counterpart, Stein discrepancy (SD) have achieved great success in model training and evaluations. However, recent research shows their limitations when dealing with certain types of distributions. One possible fix is incorporating the original score matching (or Stein discrepancy) with a diffusion matrix, which is called diffusion score matching (DSM) (or diffusion Stein discrepancy (DSD)). However, the lack of interpretation of the diffusion limits its usage within simple distributions and manually chosen matrix. In this work, we plan to fill this gap by interpreting the diffusion matrix using normalizing flows. Specifically, we theoretically prove that DSM (or DSD) is equivalent to the original score matching (or Stein discrepancy) evaluated in the transformed space defined by the normalizing flow, where the diffusion matrix is the inverse of the flow's Jacobian matrix. In addition, we also build its connection to Riemannian manifolds and further extend it to continuous flows, where the change of DSM is characterized by an ODE.

翻译：Scorizing匹配(SM)及其相关对应方Stein差异(SD)在模式培训和评价方面取得了巨大成功。然而,最近的研究表明,在处理某些类型的分布时,它们存在局限性。一种可能的解决方案是将最初的得分匹配(或Stein差异)与扩散矩阵(即传播分匹配(DSM)(或扩散Stein差异(DSD))。然而,对扩散的解读不足限制了其在简单分布和人工选择矩阵中的使用。在这项工作中,我们计划通过使用正常化流程解释传播矩阵来填补这一差距。具体地说,我们理论上证明,DSM(或DSD)相当于在正常化流程所定义的变换空间所评估的原得分匹配(或Stein差异),而扩散矩阵是流的雅各矩阵的反面。此外,我们还建立其与里曼元的连接,并将它进一步扩展到持续流中,而DSDSM的特征是源值变化。