论文标题
高斯变换
The Gaussian Transform
论文作者
论文摘要
我们介绍了高斯变换（GT），这是一种最佳运输启发的迭代方法，用于降生和增强数据集中的潜在结构。在引擎盖下，GT通过计算$ \ ell^2 $ -wasserstein距离通过将数据集定位到单个点获得的某些高斯密度估计之间的$ \ ell^2 $ -Wasserstein距离，从而在给定数据集上生成了新的距离函数（GT距离）。我们的贡献是双重的：（1）从理论上讲，我们首先确定GT在扰动下是稳定的，其次，在连续情况下，每个点在GT距离方面都具有渐近的椭圆形邻域； （2）在计算上，我们通过确定减少矩阵平方根计算数量的策略来加速GT，通过增强的邻里机制，通过$ \ ell^2 $ -WASSERSTEIN距离之间的距离以及避免冗余GT之间的GT距离计算。我们还观察到GT既是平均移位（MS）方法的概括，也是加强的概括，也是对最近提出的Wasserstein Transform（WT）方法的计算有效的专业化。我们进行了广泛的实验，以比较它们在不同情况下的性能。
We introduce the Gaussian transform (GT), an optimal transport inspired iterative method for denoising and enhancing latent structures in datasets. Under the hood, GT generates a new distance function (GT distance) on a given dataset by computing the $\ell^2$-Wasserstein distance between certain Gaussian density estimates obtained by localizing the dataset to individual points. Our contribution is twofold: (1) theoretically, we establish firstly that GT is stable under perturbations and secondly that in the continuous case, each point possesses an asymptotically ellipsoidal neighborhood with respect to the GT distance; (2) computationally, we accelerate GT both by identifying a strategy for reducing the number of matrix square root computations inherent to the $\ell^2$-Wasserstein distance between Gaussian measures, and by avoiding redundant computations of GT distances between points via enhanced neighborhood mechanisms. We also observe that GT is both a generalization and a strengthening of the mean shift (MS) method, and it is also a computationally efficient specialization of the recently proposed Wasserstein Transform (WT) method. We perform extensive experimentation comparing their performance in different scenarios.
