Inductive Global and Local Manifold Approximation and Projection

Nonlinear dimensional reduction with the manifold assumption, often calledmanifold learning, has proven its usefulness in a wide range ofhigh-dimensional data analysis. The significant impact of t-SNE and UMAP hascatalyzed intense research interest, seeking further innovations towardvisualizing not only the local but also the global structure information of thedata. Moreover, there have been consistent efforts toward generalizabledimensional reduction that handles unseen data. In this paper, we first proposeGLoMAP, a novel manifold learning method for dimensional reduction andhigh-dimensional data visualization. GLoMAP preserves locally and globallymeaningful distance estimates and displays a progression from global to localformation during the course of optimization. Furthermore, we extend GLoMAP toits inductive version, iGLoMAP, which utilizes a deep neural network to mapdata to its lower-dimensional representation. This allows iGLoMAP to providelower-dimensional embeddings for unseen points without needing to re-train thealgorithm. iGLoMAP is also well-suited for mini-batch learning, enablinglarge-scale, accelerated gradient calculations. We have successfully appliedboth GLoMAP and iGLoMAP to the simulated and real-data settings, withcompetitive experiments against the state-of-the-art methods.