  Nonlinear dimensional reduction with the manifold assumption, often called
manifold learning, has proven its usefulness in a wide range of
high-dimensional data analysis. The significant impact of t-SNE and UMAP has
catalyzed intense research interest, seeking further innovations toward
visualizing not only the local but also the global structure information of the
data. Moreover, there have been consistent efforts toward generalizable
dimensional reduction that handles unseen data. In this paper, we first propose
GLoMAP, a novel manifold learning method for dimensional reduction and
high-dimensional data visualization. GLoMAP preserves locally and globally
meaningful distance estimates and displays a progression from global to local
formation during the course of optimization. Furthermore, we extend GLoMAP to
its inductive version, iGLoMAP, which utilizes a deep neural network to map
data to its lower-dimensional representation. This allows iGLoMAP to provide
lower-dimensional embeddings for unseen points without needing to re-train the
algorithm. iGLoMAP is also well-suited for mini-batch learning, enabling
large-scale, accelerated gradient calculations. We have successfully applied
both GLoMAP and iGLoMAP to the simulated and real-data settings, with
competitive experiments against the state-of-the-art methods.
