用多项式层次结构改进VAE，提升生成质量与表征可解释性。

The variational autoencoder (VAE) is a simple and efficient generative artificial intelligence method for modeling complex probability distributions of various types of data, such as images and texts. However, it suffers some main shortcomings, such as lack of interpretability in the latent variables, difficulties in tuning hyperparameters while training, producing blurry, unrealistic downstream outputs or loss of information due to how it calculates loss functions and recovers data distributions, overfitting, and origin gravity effect for small data sets, among other issues. These and other limitations have caused unsatisfactory generation effects for the data with complex distributions. In this work, we proposed and developed a polynomial hierarchical variational autoencoder (PH-VAE), in which we used a polynomial hierarchical date format to generate or to reconstruct the data distributions. In doing so, we also proposed a novel Polynomial Divergence in the loss function to replace or generalize the Kullback-Leibler (KL) divergence, which results in systematic and drastic improvements in both accuracy and reproducibility of the re-constructed distribution function as well as the quality of re-constructed data images while keeping the dataset size the same but capturing fine resolution of the data. Moreover, we showed that the proposed PH-VAE has some form of disentangled representation learning ability.