IIKL: Isometric Immersion Kernel Learning with Riemannian Manifold for Geometric Preservation

TL;DR

This paper introduces IIKL, a novel method for learning Riemannian manifolds that preserves the intrinsic geometric properties of non-Euclidean data, leading to improved accuracy in data reconstruction and classification tasks.

Contribution

The paper proposes a new isometric immersion kernel learning approach that explicitly maintains geometric invariance, with a novel parameterized model and efficient training method.

Findings

Reduced inner product invariant loss by over 90% compared to SOTA methods.

Achieved 40% improvement in data reconstruction accuracy.

Significantly improved geometric metric errors in experiments.

Abstract

Geometric representation learning in preserving the intrinsic geometric and topological properties for discrete non-Euclidean data is crucial in scientific applications. Previous research generally mapped non-Euclidean discrete data into Euclidean space during representation learning, which may lead to the loss of some critical geometric information. In this paper, we propose a novel Isometric Immersion Kernel Learning (IIKL) method to build Riemannian manifold and isometrically induce Riemannian metric from discrete non-Euclidean data. We prove that Isometric immersion is equivalent to the kernel function in the tangent bundle on the manifold, which explicitly guarantees the invariance of the inner product between vectors in the arbitrary tangent space throughout the learning process, thus maintaining the geometric structure of the original data. Moreover, a novel parameterized learning model based on IIKL is introduced, and an alternating training method for this model is derived using Maximum Likelihood Estimation (MLE), ensuring efficient convergence. Experimental results proved that using the learned Riemannian manifold and its metric, our model preserved the intrinsic geometric representation of data in both 3D and high-dimensional datasets successfully, and significantly improved the accuracy of downstream tasks, such as data reconstruction and classification. It is showed that our method could reduce the inner product invariant loss by more than 90% compared to state-of-the-art (SOTA) methods, also achieved an average 40% improvement in downstream reconstruction accuracy and a 90% reduction in error for geometric metrics involving isometric and conformal.