  As a pivotal approach in machine learning and data science, manifold learning
aims to uncover the intrinsic low-dimensional structure within complex
nonlinear manifolds in high-dimensional space. By exploiting the manifold
hypothesis, various techniques for nonlinear dimension reduction have been
developed to facilitate visualization, classification, clustering, and gaining
key insights. Although existing manifold learning methods have achieved
remarkable successes, they still suffer from extensive distortions incurred in
the global structure, which hinders the understanding of underlying patterns.
Scalability issues also limit their applicability for handling large-scale
data. Here, we propose a scalable manifold learning (scML) method that can
manipulate large-scale and high-dimensional data in an efficient manner. It
starts by seeking a set of landmarks to construct the low-dimensional skeleton
of the entire data, and then incorporates the non-landmarks into the learned
space based on the constrained locally linear embedding (CLLE). We empirically
validated the effectiveness of scML on synthetic datasets and real-world
benchmarks of different types, and applied it to analyze the single-cell
transcriptomics and detect anomalies in electrocardiogram (ECG) signals. scML
scales well with increasing data sizes and embedding dimensions, and exhibits
promising performance in preserving the global structure. The experiments
demonstrate notable robustness in embedding quality as the sample rate
decreases.
