Learning Eigenstructures of Unstructured Data Manifolds

TL;DR

The paper addresses the challenge of obtaining spectral decompositions for geometry and manifold data without explicit operator discretization. It introduces a neural framework that predicts an orthonormal spectral basis from unstructured data and uses optimal-approximation theory to recover both eigenvectors and implicit eigenvalues, bypassing traditional operator construction and eigensolvers. Empirically, the method yields Laplacian-like spectra on 3D shapes and meaningful embeddings for high-dimensional image manifolds, often matching or surpassing conventional baselines. This data-driven approach scales to high dimensions and suggests a foundation-model-style paradigm for spectral analysis on 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.

Figures (28)

• Figure 1: Overview of our neural framework to compute spectral bases directly from unstructured point clouds of any dimensionality, based on optimal-approximation theory, without first explicitly choosing, discretely approximating, and eigendecomposing an operator.
• Figure 2: Learned eigenfunctions on [0,1] recover frequency-ordered harmonics resembling the Laplacian's spectrum.
• Figure 3: Unnormalized spectral basis (top) and $xyz$ reconstruction from $k$ basis vectors (bottom), using either the oracle cotangent Laplacian or our method (overfitting setting). Scalars are cosine similarities between basis vectors. We get similar if not more detailed reconstructions. More in \ref{['sec: additional experimental details sup mat']}.
• Figure 4: Eigenvalues of the oracle cotangent Laplacian and our estimated ones (overfitting setting). More in \ref{['sec: additional experimental details sup mat']}.
• Figure 5: Estimated mass metric $M$ from $\mathbf{q}_1$ (overfitting setting).

Theorems & Definitions (2)

• Theorem 3.1: Min-Max Optimality aflalo2016best
• Theorem 3.2: Operator-Bounded PCA aflalo2016best