Dimensionality reduction (DR) of high-dimensional data is of theoretical and practical interest in machine learning. However, there exist intriguing, non-intuitive discrepancies between the geometry of high- and low-dimensional space. We look into such discrepancies and propose a novel visualization method called Space-based Manifold Approximation and Projection (SpaceMAP). Our method establishes an analytical transformation on distance metrics between spaces to address the “crowding problem" in DR. With the proposed equivalent extended distance (EED), we are able to match the capacity of high- and low-dimensional space in a principled manner. To handle complex data with different manifold properties, we propose hierarchical manifold approximation to model the similarity function in a data-specific manner. We evaluated SpaceMAP on a range of synthetic and real datasets with varying manifold properties, and demonstrated its excellent performance in comparison with classical and state-of-the-art DR methods. In particular, the concept of space expansion provides a generic framework for understanding nonlinear DR methods including the popular t-distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection

Embedding high-dimensional data onto a low-dimensional manifold is of both theoretical and practical value. In this article, we propose to combine deep neural networks (DNN) with mathematics-guided embedding rules for high-dimensional data embedding. We introduce a generic deep embedding network (DEN) framework, which is able to learn a parametric mapping from high-dimensional space to low-dimensional space, guided by well-established objectives such as Kullback-Leibler (KL) divergence minimization. We further propose a recursive strategy, called deep recursive embedding (DRE), to make use of the latent data representations for boosted embedding performance. We exemplify the flexibility of DRE by different architectures and loss functions, and benchmarked our method against the two most popular embedding methods, namely, t-distributed stochastic neighbor embedding (t-SNE) and uniform manifold approximation and projection (UMAP). The proposed DRE method can map out-of-sample data and scale to extremely large datasets. Experiments on a range of public datasets demonstrated improved embedding performance in terms of local and global structure preservation, compared with other state-of-the-art embedding methods. Code is available at https://github.com/tao-aimi/DeepRecursiveEmbedding.