学习无结构数据流形的特征结构
Learning Eigenstructures of Unstructured Data Manifolds
对做点云/流形几何处理的人，它提供了一条绕开网格化与显式拉普拉斯算子构造、直接从无结构数据学谱基（近似特征函数）的路子，可复用于形状对应、谱聚类、图信号处理等下游环节。
原文依据 we train a network to decompose an implicit approximation operator by minimizing the reconstruction error in the learned basis
摘要译文
由模型翻译 · 术语统一 · 请对照原文核对关键结论我们提出一个新框架，可直接从无结构数据中学习用于形状与流形分析的谱基，从而无需传统的算子选择、离散化与特征求解器。基于最优逼近理论，我们训练一个网络来分解隐式逼近算子，方法是在给定的探针函数分布上最小化所学基中的重建误差。对于合适的分布，这些基可视为对拉普拉斯算子及其特征分解的逼近，而后者是几何处理中的基础工具。此外，我们的方法以统一的方式不仅恢复谱基，还恢复隐式度量的采样密度以及底层算子的特征值。值得注意的是，我们的无监督方法不对数据流形作任何假设，例如网格化或流形维数，因而可扩展到任意维度的任意数据集。在位于三维曲面上的点云与高维图像流形上，我们的方法得到了有意义的谱基，它们可以类似于拉普拉斯算子的谱基，而无需显式构造算子。通过用基于学习的方法取代传统的算子选择、构造与特征分解，我们的框架为传统流程提供了一种有原则的、数据驱动的替代方案。这为无结构数据的几何处理开辟了新的可能，尤其是在高维空间中。
英文原文摘要 1487 字符
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.