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.

翻译：在科学应用中，针对离散非欧几里得数据保持其内在几何与拓扑性质的几何表示学习至关重要。先前研究通常在表示学习过程中将非欧几里得离散数据映射至欧几里得空间，这可能导致部分关键几何信息的丢失。本文提出一种新颖的等距浸入核学习方法，从离散非欧几里得数据构建黎曼流形并等距诱导黎曼度量。我们证明等距浸入等价于流形切丛上的核函数，这显式保证了学习过程中任意切空间内向量间内积的不变性，从而保持了原始数据的几何结构。此外，本文引入了基于IIKL的参数化学习模型，并利用最大似然估计推导出该模型的交替训练方法，确保高效收敛。实验结果表明，利用学习得到的黎曼流形及其度量，我们的模型在三维及高维数据集中均成功保持了数据的内在几何表示，并显著提升了数据重建与分类等下游任务的准确率。研究表明，相较于最先进方法，本方法可将内积不变性损失降低90%以上，同时在下游重建准确率上平均提升40%，在涉及等距与共形的几何度量误差上减少90%。