# Structured Regularization of Functional Map Computations

arXiv: https://arxiv.org/abs/2009.14624 · HTML: https://mlsanity.com/paper/2009.14624/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Sep 2020.

## Built on by (most cited first, 26 of 26)

- [DPFM: Deep Partial Functional Maps](https://mlsanity.com/paper/2110.09994/) (2021)
- [Unsupervised Learning of Robust Spectral Shape Matching](https://mlsanity.com/paper/2304.14419/) (2023)
- [Unsupervised Deep Multi-Shape Matching](https://mlsanity.com/paper/2207.09610/) (2022)
- [Spectral Meets Spatial: Harmonising 3D Shape Matching and Interpolation](https://mlsanity.com/paper/2402.18920/) (2024)
- [Self-Supervised Learning for Multimodal Non-Rigid 3D Shape Matching](https://mlsanity.com/paper/2303.10971/) (2023)
- [A Scalable Combinatorial Solver for Elastic Geometrically Consistent 3D Shape Matching](https://mlsanity.com/paper/2204.12805/) (2022)
- [Hybrid Functional Maps for Crease-Aware Non-Isometric Shape Matching](https://mlsanity.com/paper/2312.03678/) (2023)
- [Memory-Scalable and Simplified Functional Map Learning](https://mlsanity.com/paper/2404.00330/) (2024)
- [Hierarchical Neural Semantic Representation for 3D Semantic Correspondence](https://mlsanity.com/paper/2509.17431/) (2025)
- [Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning](https://mlsanity.com/paper/2403.01781/) (2024)
- [SRIF: Semantic Shape Registration Empowered by Diffusion-based Image Morphing and Flow Estimation](https://mlsanity.com/paper/2409.11682/) (2024)
- [On Unsupervised Partial Shape Correspondence](https://mlsanity.com/paper/2310.14692/) (2023)
- [DiffuMatch: Category-Agnostic Spectral Diffusion Priors for Robust Non-rigid Shape Matching](https://mlsanity.com/paper/2507.23715/) (2025)
- [Revisiting Map Relations for Unsupervised Non-Rigid Shape Matching](https://mlsanity.com/paper/2310.11420/) (2023)
- [Synchronous Diffusion for Unsupervised Smooth Non-Rigid 3D Shape Matching](https://mlsanity.com/paper/2407.08244/) (2024)
- [FRIDU: Functional Map Refinement with Guided Image Diffusion](https://mlsanity.com/paper/2506.14322/) (2025)
- [MorphAny3D: Unleashing the Power of Structured Latent in 3D Morphing](https://mlsanity.com/paper/2601.00204/) (2026)
- [Symmetry Understanding of 3D Shapes via Chirality Disentanglement](https://mlsanity.com/paper/2508.05505/) (2025)
- [SGMatch: Semantic-Guided Non-Rigid Shape Matching with Flow Regularization](https://mlsanity.com/paper/2603.12937/) (2026)
- [RINO: Rotation-Invariant Non-Rigid Correspondences](https://mlsanity.com/paper/2603.27773/) (2026)
- [ARC-Flow : Articulated, Resolution-Agnostic, Correspondence-Free Matching and Interpolation of 3D Shapes Under Flow Fields](https://mlsanity.com/paper/2503.02606/) (2025)
- [DcMatch: Unsupervised Multi-Shape Matching with Dual-Level Consistency](https://mlsanity.com/paper/2509.01204/) (2025)
- [DeepShapeMatchingKit: Accelerated Functional Map Solver and Shape Matching Pipelines Revisited](https://mlsanity.com/paper/2604.10377/) (2026)
- [Extracting a functional representation from a dictionary for non-rigid shape matching](https://mlsanity.com/paper/2305.10332/) (2023)
- [Deep Frequency-Aware Functional Maps for Robust Shape Matching](https://mlsanity.com/paper/2402.03904/) (2024)
- [TokenMatch: 3D Mesh Correspondence Transformer with Curvature-Guided Tokenisation](https://mlsanity.com/paper/2609.04202/) (2026)

## Abstract (arXiv)

We consider the problem of non-rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to commute with the Laplace-Beltrami operators on the source and target shapes. We show that this approach has certain undesirable fundamental theoretical limitations, and can be undefined even for trivial maps in the smooth setting. Instead we propose a novel, theoretically well-justified approach for regularizing functional maps, by using the notion of the resolvent of the Laplacian operator. In addition, we provide a natural one-parameter family of regularizers, that can be easily tuned depending on the expected approximate isometry of the input shape pair. We show on a wide range of shape correspondence scenarios that our novel regularization leads to an improvement in the quality of the estimated functional, and ultimately pointwise correspondences before and after commonly-used refinement techniques.