Research map / Probabilistic and scientific modeling / Molecules and proteins
Finsler-Laplace-Beltrami Operators with Application to Shape Analysis
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Derives a Finsler heat kernel from Randers-metric Finsler heat diffusion and a Finsler-Laplace-Beltrami operator (FLBO), then plugs it into ACSCNN-style spectral anisotropic convolutions for shape correspondence.
Problem
Anisotropic Laplace-Beltrami operators used in geometric deep learning for shape correspondence are heuristically designed and lack theoretical justification; the paper asks whether Finsler geometry offers a principled generalization.
Approach
Starting from the Finsler heat equation with a Randers metric and a short-time simplification with small asymmetry omega, the homogeneous part yields an anisotropic diffusion operator (FLBO) whose diffusivity is the dual Randers metric's Finsler diffusivity; this is discretized on triangle meshes and used as the basis of spectral anisotropic convolutions.
Key results
- On FAUST Remeshed, the FLBO variant with tau=0.3 reaches 62.67% accuracy at r=0 versus 61.95% for ACSCNN.
- On SCAPE Remeshed, tau=0.1 reaches 46.94% at r=0 versus 46.13% for ACSCNN.
- On SHREC'16 Partial Cuts the method is on par with or better than ACSCNN for default tau=0.1 (mixed per-category results, e.g. Michael 16.04 vs 12.82).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use FLBO (τ = 0.1) instead of ACSCNN | Shape correspondence · FAUST Remeshed | Accuracy (r = 0): 61.95 -> 62.61 | Helped |
| Use FLBO (τ = 0.1) instead of ACSCNN | Shape correspondence · SCAPE Remeshed | Accuracy (r = 0): 46.13 -> 46.94 | Helped |
| Vary τ from 0.1 to 0.2 (ablation) | FLBO shape correspondence · FAUST Remeshed | Accuracy (r = 0): 62.61 -> 62.58 | No clear effect |
| Vary τ from 0.1 to 0.3 (ablation) | FLBO shape correspondence · FAUST Remeshed | Accuracy (r = 0): 62.61 -> 62.67 | No clear effect |
| Vary τ from 0.1 to 0.4 (ablation) | FLBO shape correspondence · SCAPE Remeshed | Accuracy (r = 0): 46.94 -> 45.82 | Hurt |
| Vary τ from 0.1 to 0.5 (ablation) | FLBO shape correspondence · SCAPE Remeshed | Accuracy (r = 0): 46.94 -> 45.26 | Hurt |
| Vary τ from 0.1 to 0.2 (ablation) | FLBO shape correspondence · SHREC’16 Partial Cuts | Accuracy (r = 0): Cat: 40.13 -> 37.72; Dog: 59.31 -> 60.02 | Mixed |
| Vary τ from 0.1 to 0.3 (ablation) | FLBO shape correspondence · SHREC’16 Partial Cuts | Accuracy (r = 0): Cat: 40.13 -> 40.26; Michael: 16.04 -> 13.15 | Mixed |
| Vary τ from 0.1 to 0.4 (ablation) | FLBO shape correspondence · SHREC’16 Partial Cuts | Accuracy (r = 0): Cat: 40.13 -> 39.63; Horse: 42.72 -> 44.83 | Mixed |
| Vary τ from 0.1 to 0.5 (ablation) | FLBO shape correspondence · SHREC’16 Partial Cuts | Accuracy (r = 0): Cat: 40.13 -> 40.94; Victoria: 27.95 -> 28.46 | Mixed |
| Vary τ from 0.1 to 0.2 (ablation) | FLBO shape correspondence · SHREC’16 Partial Holes | Accuracy (r = 0): Cat: 19.95 -> 17.99; Horse: 47.87 -> 49.77 | Mixed |
| Vary τ from 0.1 to 0.3 (ablation) | FLBO shape correspondence · SHREC’16 Partial Holes | Accuracy (r = 0): Cat: 19.95 -> 21.27; Michael: 12.00 -> 11.31 | Mixed |
Surprising findings
- Performance is largely insensitive to the Finsler asymmetry hyperparameter tau, with results remaining consistent across tau from 0.1 to 0.5.
Limitations
- Derivation relies on strong assumptions, including small asymmetry omega and a short-time simplification with a specific initialization.
- Discretization is straightforward and only one choice of metric (M, omega) is explored.
- Gains over ACSCNN are often small and within noise across settings.
Experimental setup
Datasets: FAUST Remeshed SCAPE Remeshed SHREC'16 Partial Cuts SHREC'16 Partial Holes
Benchmarks: FAUST Remeshed SCAPE Remeshed SHREC'16 Partial Cuts SHREC'16 Partial Holes
Metrics: accuracy within r geodesic radius (Princeton benchmark proto
Baselines: FieldConv ACSCNN
Builds on: ACSCNN anisotropic Laplace-Beltrami operator FieldConv Finsler heat equation
Closest papers by meaning
- Riemannian Metric Matching for Scalable Geometric Modeling of Distributions (ICML 2026 · similarity 0.86)
- A New Finsler Minimal Path Model With Curvature Penalization for Image Segmentation and Closed Contour Detection (CVPR 2016 · similarity 0.86)
- Sinkhorn Normalization of Diffusion Kernels (ICML 2026 · similarity 0.85)
- Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding (CVPR 2025 · similarity 0.85)
- Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models (NeurIPS 2025 · similarity 0.83)
- RMLR: Extending Multinomial Logistic Regression into General Geometries (NeurIPS 2024 · similarity 0.83)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.