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Learning Eigenstructures of Unstructured Data Manifolds

Roy Velich, Arkadi Piven, David Bensaïd, Daniel Cremers, Thomas Dagès, Ron Kimmel

arXiv:2512.01103v1cs.CV

TL;DR

Conventional operator-based eigendecomposition pipelines do not scale gracefully from surfaces to high-dimensional manifolds. This paper learns spectral bases directly through optimal approximation, reporting competitive agreement with oracle LBO baselines on surfaces and scalability to higher-dimensional data.

  • Problem

    Traditional operator eigendecomposition pipelines designed for surface data do not scale gracefully to high-dimensional manifolds.

    Source: §1 · p. 1

  • Method

    The method learns a basis that optimally reconstructs probe functions, whose distribution determines the implicit operator, and derives metric and eigenvalue information from the basis and reconstruction error.

    Source: §1 · p. 1, §3.2 · p. 4, §3.3 · p. 5

  • Results

    The learned eigenstructures perform competitively with oracle discrete LBO baselines on 3D surfaces, and the method scales to higher-dimensional manifolds.

    Source: §1 · p. 2, §4.2 · p. 7

  • Takeaways & Limitations

    The framework offers a data-driven route to spectral bases without explicit operator selection, construction, or classical numerical eigensolvers.

    Source: §1 · p. 2

    Training is computationally expensive, and approximating the Laplacian well requires manually tuning probe-distribution hyperparameters per shape.

    Source: §5 · p. 8

Abstract

from arXiv · show

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.

1. Introduction

Spectral decompositions provide core computational tools for geometry processing, but conventional operator-based pipelines designed for surfaces do not scale gracefully to high-dimensional manifolds. The paper proposes learning spectral decompositions directly from data, with probe-function choices determining the implicit operator.

1. Introduction

  • Operator spectral decompositions underpin geometry processing: LBO eigenvalues serve as intrinsic shape signatures, while eigenfunctions support kernels, descriptors, and functional-map matching.The passage cites applications including heat- and wavekernel methods, feature descriptors, and functional maps.Source: §1 · p. 1
  • The standard pipeline chooses a discrete approximation, constructs an operator matrix, and solves a generalized eigenproblem; this surface-oriented pipeline does not scale gracefully to high-dimensional manifolds.Discrete approximations aim to preserve properties such as positive semi-definiteness, symmetry, consistency, or weak formulations.Source: §1 · p. 1
  • Optimal-approximation theory yields a learned eigenbasis that minimizes reconstruction error over probe functions, with probe classes inducing different implicit metrics and operators.The basis and associated eigenvalues make the implicit operator reconstructible a posteriori; suitable cases approximate the LBO.Source: §1 · p. 1
  • The proposed approach learns spectral decompositions from data rather than selecting, explicitly discretizing, and eigendecomposing a specific operator.The learned basis avoids explicit operator construction.Source: §1 · pp. 1–2
  • The learned bases and estimated metrics perform competitively with oracle discrete LBO baselines on 3D surfaces, while the method scales to higher-dimensional manifolds.The authors frame this as enabling manifold learning where mesh-based pipelines are inapplicable and common graph-based ones are unreliable.Source: §1 · p. 2

2. Related Works

Related work establishes the importance of operator-defined spectral bases while highlighting the challenges of obtaining reliable operators on unstructured and high-dimensional data. Existing learned-operator and neural eigenproblem approaches retain constraints that motivate learning spectral bases directly.

2. Related Works

  • Operator eigenbases provide the spectral representations used for filtering, diffusion, and convolution on meshes and graphs.Keeping only the first eigenvectors projects signals onto the operator's lowest-energy subspace, retaining smooth low-frequency components.Source: §2 · p. 2
  • The cotangent Laplacian is a reference LBO discretization, but its mesh requirements and surface focus complicate use on point clouds and higher-dimensional manifolds.Graph Laplacians are used for higher-dimensional manifolds but depend strongly on connectivity, including local density, unlike the targeted smooth LBO.Source: §2 · p. 2
  • Most learned Laplacians still assemble mass and stiffness matrices and use eigensolvers sensitive to those approximations, while triangle-based methods target surfaces.Other approaches learn operator action without eigendecompositions or eigenvalues without eigenvectors; Rayleigh-quotient construction for implicit neural representations assumes a 2D surface and Euclidean ambient metric.Source: §2 · p. 2
  • Changing the metric through scaling, anisotropy, or asymmetry changes the LBO and its approximation; intrinsic and extrinsic alternatives remain understudied.Source: §2 · p. 2
  • Neural eigenproblem methods use explicit domain parameterizations and Euclidean gradients, and require training from scratch for each domain, limiting direct use on curved point-cloud manifolds.The cited approaches include Rayleigh-quotient training of eigenfunctions and a semi-group-flow formulation for high-dimensional problems.Source: §2 · p. 2

3. Method

Optimal-approximation theory links reconstruction-optimal bases to operator eigenvectors, providing a route to learn spectral bases directly from point clouds. The framework predicts an orthonormal basis, learns metric-related sampling weights, and estimates eigenvalues through probe-function reconstruction.

3.1. Foundations

  • For signal classes bounded by an SPD operator, the optimal orthonormal basis consists of that operator’s eigenvectors ordered by increasing eigenvalues.The worst reconstruction error using the first k eigenvectors is inversely related to the (k + 1)-th eigenvalue.Source: §3.1 · p. 3
  • For smooth signals with bounded Dirichlet energy, the paper uses the Laplacian to motivate a framework intended to generalize beyond Laplacians and to high-dimensional manifolds.The normalized Laplacian’s first eigenvector encodes sampling weights, motivating the framework’s choice to encode the metric in its first basis vector.Source: §3.1 · pp. 3–4

3.2. Neural Framework for Direct Spectral Bases

  • A network predicts per-point features that are QR-decomposed into an orthonormal basis interpreted as the first eigenvectors of a normalized implicit operator.The approach targets arbitrary-dimensional point clouds without first choosing and discretizing an operator or applying an eigensolver.Source: §3.2 · p. 4
  • The first predicted eigenvector encodes the mass-matrix diagonal, while maximum probe-reconstruction errors provide estimates of subsequent eigenvalues.For each resolution, the method estimates λ_(k+1) from the reciprocal of the maximum reconstruction error, without additional network parameters.Source: §3.2 · p. 4

3.3. Learning by Optimal Approximations

  • Training minimizes average reconstruction error over probe functions progressively projected onto truncated predicted bases.The loss switches from the worst-case min-max objective to its PCA counterpart, averaging probe contributions for stabler optimization; training does not compute the implicit operator or its eigenvalues.Source: §3.3 · pp. 4–5
  • The default probes are random functions smoothed with Gaussian kernels on the data’s k-nearest-neighbor graph, loosely approximating smooth Laplacian-constrained signals.This choice does not aim to reproduce the Laplacian exactly, and the method is not restricted to that operator or probe distribution.Source: §3.3 · p. 5
  • At inference, worst-case reconstruction errors estimate eigenvalues, and the basis and eigenvalues can reconstruct the implicit operator for downstream tasks.The normalized basis can also be converted to an unnormalized basis, which may be preferable downstream because it is more sampling-invariant.Source: §3.3 · p. 5

4. Experiments

Experiments span a toy interval, 3D surface and volume point clouds, and high-dimensional image manifolds. The learned bases recover Laplacian-like structure and produce competitive manifold embeddings.

4.1. Toy 1D Segment Manifold

  • On the unit interval, the learned basis forms frequency-increasing harmonics resembling Fourier basis functions and Laplacian eigenfunctions.The first vector is constant, while subsequent vectors exhibit increasing numbers of half-waves.Source: §4.1 · p. 6, Figure 2 · p. 6

4.2. 2D Surface and 3D Volume Manifolds

  • In the overfitting setting, unsupervised bases closely match cotangent-Laplacian eigenvectors, approximate its eigenvalues, and can retain more reconstruction detail.On some shapes, higher-frequency vectors diverge slightly; the estimated area weights from q1 are reported as well-behaved.Source: Figure 3 · p. 6, Figure 4 · p. 6, Figure 5 · p. 6
  • A model trained on surface point clouds generalizes to unseen surfaces and volumes, though with less precision than in the overfitting setting.The volume shapes test a manifold dimensionality not seen during training.Source: §4.2 · p. 7, Figure 6 · p. 7

4.3. High-Dimensional Manifolds

  • Learned spectral bases provide competitive or superior embeddings to manifold-learning baselines on high-dimensional image data.The experiments use image embeddings and compare against PCA, Isomap, Laplacian Eigenmaps, t-SNE, and UMAP; Figure 7 reports average clustering performance over 50 runs, and Figure 8 visualizes a 2D embedding.Source: §4.3 · pp. 7–8, Figure 7 · p. 8, Figure 8 · p. 8

5. Conclusion

The framework learns orthonormal spectral bases from unstructured data using probe distributions that encode implicit operators, and recovers the associated metric and eigenvalues. It handles data of varying dimensionality, while training cost and per-shape probe-distribution tuning remain limitations.

5. Conclusion

  • The neural framework learns spectral bases directly from unstructured data, recovers the associated metric and eigenvalues, and scales in practice to high-dimensional datasets.Different probe distributions encode different implicit operators; the method recovers the Laplace-Beltrami operator in experiments on one- to three-dimensional manifolds.Source: §5 · p. 8
  • Training is computationally expensive, and best approximating the Laplacian requires manually tuning probe-distribution hyperparameters per shape.The authors identify efficient compilation and CUDA implementations as possible ways to improve training efficiency.Source: §5 · p. 8
  • The authors plan to explore further high-dimensional applications and learn probe distributions to approximate operators beyond traditional Laplacians.Proposed directions include using the spectral basis in graph neural networks and developing a foundation model for data of any dimensionality.Source: §5 · p. 8

Supplementary Material

The supplementary material relates PCA's error-minimization and eigenvector formulations, describes conditions under which normalized-operator eigenstructures reveal the metric, and details an efficient M-weighted projection procedure.

A.1. PCA Alternative Formulations

  • Minimizing expected squared approximation error is equivalent to maximizing cumulative Rayleigh quotients, yielding covariance eigenvectors in decreasing eigenvalue order.The PCA formulation works iteratively over the basis vectors.Source: §A.1 · p. 14

A.2. Normalized Operators and Eigenstructures

  • For operators with a one-dimensional nullspace whose vector is a diagonally scaled all-ones vector, learning the normalized eigenbasis can recover the scaling matrix and thus the metric.The method is suited to operators with Euclidean-orthogonal eigenvectors; the normalized Laplacian is one example.Source: §A.2 · p. 14

B.1. Projecting onto Optimal Reconstruction Bases

  • The method projects probe functions M-orthogonally onto basis spans by solving small, batched linear systems for each projection size.The systems are symmetric positive definite and their size depends on the number of basis vectors, not the input size.Source: §B.1 · p. 15

B.2. Replacing the M-norm with the Euclidean norm for Approximation Errors

The method combines M-weighted projection with Euclidean-norm reconstruction error, which supports unsupervised learning of density-related mass elements; the appendix also specifies a toy 1D experiment.

B.2. Replacing the M-norm with the Euclidean norm for Approximation Errors

  • Using Euclidean-norm error with M-weighted projection implicitly encourages mass elements tied to the Euclidean-induced sample volume or surface area.Using the theoretically prescribed M-norm for error led to unstable training, NaNs, and metric collapse to zero.Source: §B.2 · p. 15

C.1. Toy 1D Segment Manifold

  • The toy 1D experiment uses 100 uniformly sampled points, smoothed noise probes on a 16-neighbor kNN graph, and learns five basis vectors.A three-hidden-layer MLP is trained on batches of 500,000 probe functions.Source: §C.1 · p. 15
  • The toy experiment trains with Adam at an initial learning rate of 10^-2 and schedules reductions at 30% and 70% of training.The model has three hidden layers of width 64 with ReLU activations.Source: §C.1 · p. 15

C.2.1. Datasets

The evaluation covers overfitting on benchmark shapes and generalization from diverse surface point-cloud training datasets to multiple established benchmarks, including volumetric point clouds.

C.2.1. Datasets

  • The overfitting setup uses shapes from a geometry-processing benchmark and removes mesh connectivity, retaining point clouds.Source: §C.2.1 · p. 15
  • For generalization, training spans synthetic human bodies, protein domains, CAD models, real human scans, and dynamic non-rigid surfaces.The named datasets are SUR-REAL, SHREC’21 Protein Domains, ABC, FAUST, and DeformingThings4D.Source: §C.2.1 · p. 15
  • Evaluation uses ten established benchmarks, and also tests—but does not train—on volumetric point clouds generated from SHREC’20 Non-Isometric surface meshes.Volumetric samples are drawn within padded mesh bounding boxes, classified as inside or outside, and retained interior points are subsampled.Source: §C.2.1 · pp. 15–16

C.2.2. Implementation – Generalization Setting

The generalization model predicts spectral bases across diverse 3D shapes, while the overfitting configuration specializes a smaller model to one geometry. Both use point-cloud inputs and generated probe functions, with overfitting also using multiple smoothing scales.

C.2.2. Implementation – Generalization Setting

  • Generalization training uses AdamW over 100 epochs, with cosine annealing, gradient clipping, and gradients accumulated across two batches.Training uses batches of 30 shapes; the experiments run on eight A100 GPUs.Source: §C.2.2 · p. 16

C.2.3. Implementation – Overfitting setting

  • For a single target geometry, the overfitting model uses a smaller 8-layer, 8-head Transformer and trains on one mesh for 12,000 epochs.The mesh is sampled to 4,000 points, and training uses batch size 1.Source: §C.2.3 · p. 16
  • Overfitting probe fields use 40 smoothing iterations with σ sampled from 0.01–0.2, a multiscale choice associated with improved results across shapes.At inference, eigenvalue alignment depends on probe-generation settings; the default is k = 70, 48 iterations, and σ = 0.101, while reported results generally use per-shape tuned settings.Source: §C.2.3 · p. 17

C.2.4. Additional Results

Additional experiments extend the overfitting and generalization results, examining recovered bases, metrics, and eigenvalues across shapes. The section also situates manifold learning as low-dimensional representation that preserves local, global, or combined structure.

C.2.4. Additional Results

  • Additional overfitting results include spectral-basis and filtering visualizations, estimated metrics, and eigenvalue curves under tuned and fixed probe-generation settings.Table 3 further reports average cosine similarity at truncation levels and mean relative eigenvalue discrepancy.Source: §C.2.4 · p. 17, Table 2 · p. 17, Table 3 · p. 18
  • Across additional shapes, the direct neural method recovers oracle Laplacian spectral decompositions without access to mesh structure, with low-frequency bases most similar and higher frequencies differing in detail.The method still preserves the shape manifold’s overall low-pass information.Source: §C.2.4 · pp. 17–18
  • Generalization experiments accurately estimate the first Fourier eigenfunctions, which encode low-pass and globally important manifold information, though performance is below single-shape overfitting.The authors describe the results as potential for a foundation model and note that their training setting is limited.Source: §C.2.4 · p. 18

C.3.1. Short Manifold Learning Overview

  • Manifold learning seeks low-dimensional representations of high-dimensional data by preserving pairwise dissimilarities, targeting local, global, or combined manifold structure.The overview cites representative methods for each category.Source: §C.3.1 · p. 18

C.3.2. Implementation

Image-manifold experiments learn spectral decompositions of pretrained vision-feature manifolds across four datasets. The setup evaluates clustering against classical methods and tunes probe-function smoothing parameters per dataset.

C.3.2. Implementation

  • Experiments use CLIP and DINOv2 embeddings as coordinates for kNN construction, across Caltech256, CIFAR100, Imagenette, and STL-10.CLIP embeddings have 512 dimensions and DINOv2 embeddings have 384 dimensions; kNN uses cosine similarity.Source: §C.3.2 · p. 18
  • Evaluation compares PCA, UMAP, t-SNE, Isomap, and Laplacian Eigenmaps using 32 runs and class-weighted samples with imbalanced class distributions.Probe smoothing parameters are tuned per dataset, and the authors identify this sensitivity as a limitation of the current results.Source: §C.3.2 · p. 19
  • Across image-manifold experiments, the model predicts 11 eigenvectors using 5,000 sampled images per dataset and generates 300 probe functions per manifold; the kNN graph uses cosine similarity with self-loops.Source: §C.3.2 · p. 18, Table 4 · p. 19

C.3.3. Results

Across image datasets and feature extractors, Optimal Approximation Eigenmaps are competitive on clustering and generally outperform PCA, Isomap, and Laplacian Eigenmaps. Visualizations also indicate semantic clustering with smoother manifold structure than methods that overcluster.

C.3.3. Results

  • Optimal Approximation Eigenmaps perform competitively across most settings, generally outperforming PCA, Isomap, and Laplacian Eigenmaps, while UMAP and t-SNE often achieve higher clustering scores through overclustering.Results average 32 runs across datasets and feature extractors and report seven clustering metrics.Source: §C.3.3 · p. 19
  • For Imagenette and STL-10 visualizations, learned bases form semantically meaningful clusters with smooth transitions, unlike the artificial overclustering reported for t-SNE and UMAP.The Imagenette CLIP figure compares the learned basis with PCA, Isomap, t-SNE, UMAP, and Laplacian Eigenmaps.Source: §C.3.3 · p. 19, Figure 21 · p. 35
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