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

arXiv: https://arxiv.org/abs/2505.06288 · HTML: https://mlsanity.com/paper/2505.06288/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 40 references that are in the map.
- First posted on arXiv 7 May 2025.

## Builds on (most cited first, 22 of 22)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [TriMap: Large-scale Dimensionality Reduction Using Triplets](https://mlsanity.com/paper/1910.00204/) (2019)
- [How rotational invariance of common kernels prevents generalization in high dimensions](https://mlsanity.com/paper/2104.04244/) (2021)
- [Learning Topology-Preserving Data Representations](https://mlsanity.com/paper/2302.00136/) (2023)
- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach](https://mlsanity.com/paper/2203.00126/) (2022)
- [Learning the Geodesic Embedding with Graph Neural Networks](https://mlsanity.com/paper/2309.05613/) (2023)
- [Depth-discriminative Metric Learning for Monocular 3D Object Detection](https://mlsanity.com/paper/2401.01075/) (2024)
- [Learning Manifold Implicitly via Explicit Heat-Kernel Learning](https://mlsanity.com/paper/2010.01761/) (2020)
- [Inferring manifolds using Gaussian processes](https://mlsanity.com/paper/2110.07478/) (2021)
- [Neural Operators for Nonlinear Functionals on RKHS](https://mlsanity.com/paper/2403.12187/) (2024)
- [A Manifold-based Airfoil Geometric-feature Extraction and Discrepant Data Fusion Learning Method](https://mlsanity.com/paper/2206.12254/) (2022)
- [Curvature Augmented Manifold Embedding and Learning](https://mlsanity.com/paper/2403.14813/) (2024)
- [Transient anisotropic kernel for probabilistic learning on manifolds](https://mlsanity.com/paper/2407.21435/) (2024)
- [Uniform Interpolation Constrained Geodesic Learning on Data Manifold](https://mlsanity.com/paper/2002.04829/) (2020)
- [Kernel function impact on convolutional neural networks](https://mlsanity.com/paper/2302.10266/) (2023)
- [A Perceptron-based Fine Approximation Technique for Linear Separation](https://mlsanity.com/paper/2309.06049/) (2023)
- [Dimensionality Reduction using Elastic Measures](https://mlsanity.com/paper/2209.04933/) (2022)
- [Isometric Immersion Learning with Riemannian Geometry](https://mlsanity.com/paper/2409.14760/) (2024)

## Abstract (arXiv)

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.