Generalized sliced Wasserstein distance is a variant of sliced Wasserstein distance that exploits the power of non-linear projection through a given defining function to better capture the complex structures of the probability distributions. Similar to sliced Wasserstein distance, generalized sliced Wasserstein is defined as an expectation over random projections which can be approximated by the Monte Carlo method. However, the complexity of that approximation can be expensive in high-dimensional settings. To that end, we propose to form deterministic and fast approximations of the generalized sliced Wasserstein distance by using the concentration of random projections when the defining functions are polynomial function, circular function, and neural network type function. Our approximations hinge upon an important result that one-dimensional projections of a high-dimensional random vector are approximately Gaussian.

翻译：普通切片瓦西斯坦距离是切片瓦西斯坦距离的一种变体,它利用非线性投影的力量,通过给定的确定功能更好地捕捉概率分布的复杂结构。与切片瓦西斯坦距离相似,一般切片瓦西斯坦被定义为对随机预测的预期,而蒙特卡洛方法可以近似于随机预测。然而,在高维环境中,这种近似的复杂性可能非常昂贵。为此,我们提议利用随机预测的集中度,在确定函数是多球函数、圆函数和神经网络类型函数时,形成非线性投影的确定性和快速近似值。我们的近似值取决于一个重要结果,即对高维随机矢量的一维预测约为高空。