In the process of training a generative model, it becomes essential to
measure the discrepancy between two high-dimensional probability distributions:
the generative distribution and the ground-truth distribution of the observed
dataset. Recently, there has been growing interest in an approach that involves
slicing high-dimensional distributions, with the Cramer-Wold distance emerging
as a promising method. However, we have identified that the Cramer-Wold
distance primarily focuses on joint distributional learning, whereas
understanding marginal distributional patterns is crucial for effective
synthetic data generation. In this paper, we introduce a novel measure of
dissimilarity, the mixture Cramer-Wold distance. This measure enables us to
capture both marginal and joint distributional information simultaneously, as
it incorporates a mixture measure with point masses on standard basis vectors.
Building upon the mixture Cramer-Wold distance, we propose a new generative
model called CWDAE (Cramer-Wold Distributional AutoEncoder), which shows
remarkable performance in generating synthetic data when applied to real
tabular datasets. Furthermore, our model offers the flexibility to adjust the
level of data privacy with ease.

利用混合 Cramer-Wold 距离提出了 CWDAE（Cramer-Wold Distributional AutoEncoder）生成模型，在真实表格数据集上表现出了出色的合成数据生成性能，并且具备灵活调整数据隐私级别的特性。

通过混合 Cramer-Wold 距离实现均衡的边际和联合分布学习

Balanced Marginal and Joint Distributional Learning via Mixture  Cramer-Wold Distance

We propose a semi-supervised generative model, SeGMA, which learns a joint
probability distribution of data and their classes and which is implemented in
a typical Wasserstein auto-encoder framework. We choose a mixture of Gaussians
as a target distribution in latent space, which provides a natural splitting of
data into clusters. To connect Gaussian components with correct classes, we use
a small amount of labeled data and a Gaussian classifier induced by the target
distribution. SeGMA is optimized efficiently due to the use of Cramer-Wold
distance as a maximum mean discrepancy penalty, which yields a closed-form
expression for a mixture of spherical Gaussian components and thus obviates the
need of sampling. While SeGMA preserves all properties of its semi-supervised
predecessors and achieves at least as good generative performance on standard
benchmark data sets, it presents additional features: (a) interpolation between
any pair of points in the latent space produces realistically-looking samples;
(b) combining the interpolation property with disentangled class and style
variables, SeGMA is able to perform a continuous style transfer from one class
to another; (c) it is possible to change the intensity of class characteristics
in a data point by moving the latent representation of the data point away from
specific Gaussian components.

提出了一种半监督生成模型 SeGMA，它是在典型的 Wasserstein 自编码器框架中实现的，可以有效地进行优化，并在标准基准数据集上达到至少与其半监督前身同样好的生成性能。