In recent years, the dimensionality reduction has become more important as the number of dimensions of data used in various tasks such as regression and classification has increased. As popular nonlinear dimensionality reduction methods, t-distributed stochastic neighbor embedding (t-SNE) and uniform manifold approximation and projection (UMAP) have been proposed. However, the former outputs only one low-dimensional space determined by the t-distribution and the latter is difficult to control the distribution of distance between each pair of samples in low-dimensional space. To tackle these issues, we propose novel t-SNE and UMAP extended by q-Gaussian distribution, called q-Gaussian-distributed stochastic neighbor embedding (q-SNE) and q-Gaussian-distributed uniform manifold approximation and projection (q-UMAP). The q-Gaussian distribution is a probability distribution derived by maximizing the tsallis entropy by escort distribution with mean and variance, and a generalized version of Gaussian distribution with a hyperparameter q. Since the shape of the q-Gaussian distribution can be tuned smoothly by the hyperparameter q, q-SNE and q-UMAP can in- tuitively derive different embedding spaces. To show the quality of the proposed method, we compared the visualization of the low-dimensional embedding space and the classification accuracy by k-NN in the low-dimensional space. Empirical results on MNIST, COIL-20, OliverttiFaces and FashionMNIST demonstrate that the q-SNE and q-UMAP can derive better embedding spaces than t-SNE and UMAP.

近年来，随着回归和分类等各种任务中使用的数据维数的增加，降维变得更加重要。作为流行的非线性降维方法，人们提出了t分布随机邻域嵌入（t-SNE）和均匀流形逼近和投影（UMAP）。然而，前者仅输出一个由t分布确定的低维空间，而后者难以控制低维空间中每对样本之间的距离分布。为了解决这些问题，我们提出了通过 q-高斯分布扩展的新型 t-SNE 和 UMAP，称为 q-高斯分布随机邻域嵌入（q-SNE）和 q-高斯分布均匀流形逼近和投影（q-UMAP） 。q-高斯分布是通过具有均值和方差的护航分布最大化 tsallis 熵而导出的概率分布，也是具有超参数 q 的高斯分布的广义版本。由于q-高斯分布的形状可以通过超参数q平滑调整，因此q-SNE和q-UMAP可以直观地推导出不同的嵌入空间。为了展示所提出方法的质量，我们比较了低维嵌入空间的可视化和低维空间中 k-NN 的分类精度。MNIST、COIL-20、OliverttiFaces 和 FashionMNIST 上的实证结果表明，q-SNE 和 q-UMAP 可以比 t-SNE 和 UMAP 获得更好的嵌入空间。