非结构化数据流形的本征结构学习

摘要

我们提出了一种新颖框架，该框架能够直接从非结构化数据中学习用于形状和流形分析的谱基，无需依赖传统的算子选择、离散化和特征求解器。基于最优逼近理论的数学基础，我们通过最小化选定探测函数分布在所学基上的重构误差，训练神经网络分解隐式逼近算子。对于合适的分布，该方法可视为拉普拉斯算子及其特征分解的近似，这些在几何处理中具有基础性地位。此外，我们的方法以统一方式不仅恢复谱基，还能恢复隐式度量的采样密度和底层算子的特征值。值得注意的是，这种无监督方法不对数据流形做任何假设（如网格化或流形维度），使其能够扩展到任意维度的数据集。在三维曲面点云和高维图像流形上，我们的方法无需显式构建算子即可生成有意义的谱基，这些谱基与拉普拉斯算子的谱基具有相似性。通过用基于学习的方法取代传统的算子选择、构建和特征分解流程，本框架为传统处理管线提供了原则性的数据驱动替代方案。这为处理非结构化数据（尤其是高维空间数据）的几何处理开辟了新途径。

English

We introduce a novel framework that directly learns a spectral basis for shape and manifold analysis from unstructured data, eliminating the need for traditional operator selection, discretization, and eigensolvers. Grounded in optimal-approximation theory, we train a network to decompose an implicit approximation operator by minimizing the reconstruction error in the learned basis over a chosen distribution of probe functions. For suitable distributions, they can be seen as an approximation of the Laplacian operator and its eigendecomposition, which are fundamental in geometry processing. Furthermore, our method recovers in a unified manner not only the spectral basis, but also the implicit metric's sampling density and the eigenvalues of the underlying operator. Notably, our unsupervised method makes no assumption on the data manifold, such as meshing or manifold dimensionality, allowing it to scale to arbitrary datasets of any dimension. On point clouds lying on surfaces in 3D and high-dimensional image manifolds, our approach yields meaningful spectral bases, that can resemble those of the Laplacian, without explicit construction of an operator. By replacing the traditional operator selection, construction, and eigendecomposition with a learning-based approach, our framework offers a principled, data-driven alternative to conventional pipelines. This opens new possibilities in geometry processing for unstructured data, particularly in high-dimensional spaces.

非结构化数据流形的本征结构学习

Learning Eigenstructures of Unstructured Data Manifolds

摘要

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