  Dimension reduction and visualization of high-dimensional data have become
very important research topics because of the rapid growth of large databases
in data science. In this paper, we propose using a generalized sigmoid function
to model the distance similarity in both high- and low-dimensional spaces. In
particular, the parameter b is introduced to the generalized sigmoid function
in low-dimensional space, so that we can adjust the heaviness of the function
tail by changing the value of b. Using both simulated and real-world data sets,
we show that our proposed method can generate visualization results comparable
to those of uniform manifold approximation and projection (UMAP), which is a
newly developed manifold learning technique with fast running speed, better
global structure, and scalability to massive data sets. In addition, according
to the purpose of the study and the data structure, we can decrease or increase
the value of b to either reveal the finer cluster structure of the data or
maintain the neighborhood continuity of the embedding for better visualization.
Finally, we use domain knowledge to demonstrate that the finer subclusters
revealed with small values of b are meaningful.
