We propose a Likelihood Matching approach for training diffusion models by first establishing an equivalence between the likelihood of the target data distribution and a likelihood along the sample path of the reverse diffusion. To efficiently compute the reverse sample likelihood, a quasi-likelihood is considered to approximate each reverse transition density by a Gaussian distribution with matched conditional mean and covariance, respectively. The score and Hessian functions for the diffusion generation are estimated by maximizing the quasi-likelihood, ensuring a consistent matching of both the first two transitional moments between every two time points. A stochastic sampler is introduced to facilitate computation that leverages both the estimated score and Hessian information. We establish consistency of the quasi-maximum likelihood estimation, and provide non-asymptotic convergence guarantees for the proposed sampler, quantifying the rates of the approximation errors due to the score and Hessian estimation, dimensionality, and the number of diffusion steps. Empirical and simulation evaluations demonstrate the effectiveness of the proposed Likelihood Matching and validate the theoretical results.

翻译：我们提出一种用于训练扩散模型的似然匹配方法，首先建立目标数据分布的似然与反向扩散采样路径上似然之间的等价关系。为高效计算反向采样似然，引入拟似然方法，通过分别匹配条件均值与协方差的 Gaussian 分布来近似每个反向转移密度。扩散生成的得分函数与 Hessian 函数通过最大化拟似然进行估计，确保任意两个时间点之间前两个转移矩的一致匹配。引入一种随机采样器以促进计算，该采样器同时利用估计的得分与 Hessian 信息。我们建立了拟极大似然估计的一致性，并为所提出的采样器提供了非渐近收敛保证，量化了由得分与 Hessian 估计、维度以及扩散步数引起的近似误差率。实证与仿真评估证明了所提似然匹配方法的有效性，并验证了理论结果。