Isometric Immersion Learning with Riemannian Geometry

Manifold learning has been proven to be an effective method for capturing the implicitly intrinsic structure of non-Euclidean data, in which one of the primary challenges is how to maintain the distortion-free (isometry) of the data representations. Actually, there is still no manifold learning method that provides a theoretical guarantee of isometry. Inspired by Nash's isometric theorem, we introduce a new concept called isometric immersion learning based on Riemannian geometry principles. Following this concept, an unsupervised neural network-based model that simultaneously achieves metric and manifold learning is proposed by integrating Riemannian geometry priors. What's more, we theoretically derive and algorithmically implement a maximum likelihood estimation-based training method for the new model. In the simulation experiments, we compared the new model with the state-of-the-art baselines on various 3-D geometry datasets, demonstrating that the new model exhibited significantly superior performance in multiple evaluation metrics. Moreover, we applied the Riemannian metric learned from the new model to downstream prediction tasks in real-world scenarios, and the accuracy was improved by an average of 8.8%.