通过拓扑几何正则化，提升噪声点云的流形结构重建能力。
Latent Manifold Reconstruction and Representation with Topological and Geometrical Regularization
- 基于自编码器设计流形重构层，同步优化局部细节与全局拓扑。
- 在点云数据上显著优于t-SNE、UMAP等基线方法，保持结构完整性。
- 适合处理含噪真实数据的流形学习任务，尤其关注结构保真度。
流形学习旨在发现高维数据中的低维结构，并保留关键的拓扑与几何特性。现有方法常因噪声干扰难以同时捕捉局部细节与全局拓扑一致性，或在降维过程中失衡，导致嵌入结果扭曲或断裂。本文提出一种基于自编码器的方法，引入流形重构层，从噪声点云中揭示潜在流形结构，并在降维过程中对拓扑与几何性质施加正则化，二者在训练中相互促进。在点云数据集上的实验表明，该方法在噪声数据中发现流形结构并保持其完整性方面，优于t-SNE、UMAP及拓扑自编码器等基线方法，可视化与定量指标均验证了其有效性。本工作证明了将流形重构与流形学习结合，对实现可靠潜在流形表示具有重要意义，尤其适用于含噪真实数据场景。代码仓库：https://github.com/Thanatorika/mrtg。
原文摘要 · Abstract (English)
Manifold learning aims to discover and represent low-dimensional structures underlying high-dimensional data while preserving critical topological and geometric properties. Existing methods often fail to capture local details with global topological integrity from noisy data or construct a balanced dimensionality reduction, resulting in distorted or fractured embeddings. We present an AutoEncoder-based method that integrates a manifold reconstruction layer, which uncovers latent manifold structures from noisy point clouds, and further provides regularizations on topological and geometric properties during dimensionality reduction, whereas the two components promote each other during training. Experiments on point cloud datasets demonstrate that our method outperforms baselines like t-SNE, UMAP, and Topological AutoEncoders in discovering manifold structures from noisy data and preserving them through dimensionality reduction, as validated by visualization and quantitative metrics. This work demonstrates the significance of combining manifold reconstruction with manifold learning to achieve reliable representation of the latent manifold, particularly when dealing with noisy real-world data. Code repository: https://github.com/Thanatorika/mrtg.
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