Deep Manifold Computing and Visualization Using Elastic Locally Isometric Smoothness

The ability to preserve local geometry of highly nonlinear manifolds in high dimensional spaces and properly unfold them into lower dimensional hyperplanes is the key to the success of manifold computing, nonlinear dimensionality reduction (NLDR) and visualization. This paper proposes a novel method, called elastic locally isometric smoothness (ELIS), to empower deep neural networks with such an ability. ELIS requires that a desired metric between points should be preserved across layers in order to preserve local geometry; such a smoothness constraint effectively regularizes vector-based transformations to become well-behaved local metric-preserving homeomorphisms. Moreover, ELIS requires that the smoothness should be imposed in a way to render sufficient flexibility for tackling complicated nonlinearity and non-Euclideanity; this is achieved layer-wisely via nonlinearity in both the similarity and activation functions. The ELIS method incorporates a class of suitable nonlinear similarity functions into a two-way divergence loss and uses hyperparameter continuation in finding optimal solutions. Extensive experiments, comparisons, and ablation study demonstrate that ELIS can deliver results not only superior to UMAP and t-SNE for and visualization but also better than other leading counterparts of manifold and autoencoder learning for NLDR and manifold data generation.