Abstract:
|
Nonlinear dimensionality reduction with the manifold assumption, often called manifold learning, has proven its usefulness in a wide range of high-dimensional data analysis applications. However, despite the recent advances of manifold learning, current state-of-the-art techniques are focused on preserving only local or global structure information of the data. Moreover, the dimensionality reduction results cannot be generalized to unseen data. In this paper, we propose iGLoMAP, a novel inductive manifold learning method that has a trade-off between global and local preservation. iGLoMAP learns the embedding by preserving the fuzzy simplicial representations of the input space, which requires a quality geodesic distance estimation. We propose a geodesic distance estimator with theoretical justifications by careful analysis of the topological structure of the manifold. Furthermore, by utilizing a deep neural network as a mapper, iGLoMAP can provide the lower-dimensional embedding for an unseen, novel point without any additional optimization. We successfully apply iGLoMAP to the simulated and real-data settings, with comparative experiments against Isomap, t-SNE, UMAP, and PaCMAP.
|