Learning Eigenstructures of Unstructured Data Manifolds

2025-11-30

AI Summary

Learning the spectral decomposition of the Laplacian operator from unstructured high-dimensional data—such as 3D point clouds or image manifolds—typically requires explicit discretization, mesh construction, or solving eigenvalue problems, limiting scalability and applicability to unknown geometries. Method: We propose an end-to-end framework that jointly learns the implicit spectral basis, eigenvalues, and density-induced metric of the Laplacian directly from raw data—without constructing differential operators, discretizing domains, or solving eigenproblems. Leveraging optimal approximation theory, we parameterize the operator and its spectrum via neural networks, minimizing reconstruction error under a probe function distribution. Contribution/Results: The method is fully unsupervised, mesh-free, dimension-agnostic, and geometry-agnostic. Experiments demonstrate that the learned spectral basis exhibits Laplacian-like properties—yielding interpretable, generalizable, and scalable representations across diverse unstructured datasets, including high-dimensional settings.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

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.

Problem

• Learns spectral basis from unstructured data without operator construction
• Recovers implicit metric sampling density and operator eigenvalues
• Scales to arbitrary datasets without manifold assumptions

Innovation

• Learns spectral basis directly from unstructured data
• Trains network to decompose implicit operator via reconstruction error
• Recovers spectral basis, sampling density, and eigenvalues unsupervised