Nonlinear dimensionality reduction with q-Gaussian distribution

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.