# ARAPReg: An As-Rigid-As Possible Regularization Loss for Learning Deformable Shape Generators

arXiv: https://arxiv.org/abs/2108.09432 · HTML: https://mlsanity.com/paper/2108.09432/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 69 references that are in the map.
- First posted on arXiv 21 Aug 2021.

## Builds on (most cited first, 30 of 46)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [End-to-end Recovery of Human Shape and Pose](https://mlsanity.com/paper/1712.06584/) (2017)
- [Pixel2Mesh: Generating 3D Mesh Models from Single RGB Images](https://mlsanity.com/paper/1804.01654/) (2018)
- [Learning a Probabilistic Latent Space of Object Shapes via 3D Generative-Adversarial Modeling](https://mlsanity.com/paper/1610.07584/) (2016)
- [AtlasNet: A Papier-Mach'e Approach to Learning 3D Surface Generation](https://mlsanity.com/paper/1802.05384/) (2018)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Neural 3D Mesh Renderer](https://mlsanity.com/paper/1711.07566/) (2017)
- [3D Menagerie: Modeling the 3D shape and pose of animals](https://mlsanity.com/paper/1611.07700/) (2016)
- [Generating 3D faces using Convolutional Mesh Autoencoders](https://mlsanity.com/paper/1807.10267/) (2018)
- [Shape Completion using 3D-Encoder-Predictor CNNs and Shape Synthesis](https://mlsanity.com/paper/1612.00101/) (2016)
- [Octree Generating Networks: Efficient Convolutional Architectures for High-resolution 3D Outputs](https://mlsanity.com/paper/1703.09438/) (2017)
- [Learning Shape Abstractions by Assembling Volumetric Primitives](https://mlsanity.com/paper/1612.00404/) (2016)
- [DeepCap: Monocular Human Performance Capture Using Weak Supervision](https://mlsanity.com/paper/2003.08325/) (2020)
- [3D-CODED : 3D Correspondences by Deep Deformation](https://mlsanity.com/paper/1806.05228/) (2018)
- [SDM-NET: Deep Generative Network for Structured Deformable Mesh](https://mlsanity.com/paper/1908.04520/) (2019)
- [CSGNet: Neural Shape Parser for Constructive Solid Geometry](https://mlsanity.com/paper/1712.08290/) (2017)
- [Neural Cages for Detail-Preserving 3D Deformations](https://mlsanity.com/paper/1912.06395/) (2019)
- [The Hessian Penalty: A Weak Prior for Unsupervised Disentanglement](https://mlsanity.com/paper/2008.10599/) (2020)
- [Neural 3D Morphable Models: Spiral Convolutional Networks for 3D Shape Representation Learning and Generation](https://mlsanity.com/paper/1905.02876/) (2019)
- [Point Cloud GAN](https://mlsanity.com/paper/1810.05795/) (2018)
- [3D-PRNN: Generating Shape Primitives with Recurrent Neural Networks](https://mlsanity.com/paper/1708.01648/) (2017)
- [Variational Autoencoders for Deforming 3D Mesh Models](https://mlsanity.com/paper/1709.04307/) (2017)
- [Deep Generative Modeling for Scene Synthesis via Hybrid Representations](https://mlsanity.com/paper/1808.02084/) (2018)
- [Unsupervised Shape and Pose Disentanglement for 3D Meshes](https://mlsanity.com/paper/2007.11341/) (2020)
- [Transformation-Grounded Image Generation Network for Novel 3D View Synthesis](https://mlsanity.com/paper/1703.02921/) (2017)

## Built on by (most cited first, 30 of 40)

- [TextDeformer: Geometry Manipulation using Text Guidance](https://mlsanity.com/paper/2304.13348/) (2023)
- [Neural Jacobian Fields: Learning Intrinsic Mappings of Arbitrary Meshes](https://mlsanity.com/paper/2205.02904/) (2022)
- [Recent Trends in 3D Reconstruction of General Non-Rigid Scenes](https://mlsanity.com/paper/2403.15064/) (2024)
- [Frame Averaging for Equivariant Shape Space Learning](https://mlsanity.com/paper/2112.01741/) (2021)
- [Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework](https://mlsanity.com/paper/2204.04238/) (2022)
- [SPARF: Large-Scale Learning of 3D Sparse Radiance Fields from Few Input Images](https://mlsanity.com/paper/2212.09100/) (2022)
- [Geometry in Style: 3D Stylization via Surface Normal Deformation](https://mlsanity.com/paper/2503.23241/) (2025)
- [SRIF: Semantic Shape Registration Empowered by Diffusion-based Image Morphing and Flow Estimation](https://mlsanity.com/paper/2409.11682/) (2024)
- [GenCorres: Consistent Shape Matching via Coupled Implicit-Explicit Shape Generative Models](https://mlsanity.com/paper/2304.10523/) (2023)
- [DiffuMatch: Category-Agnostic Spectral Diffusion Priors for Robust Non-rigid Shape Matching](https://mlsanity.com/paper/2507.23715/) (2025)
- [DreamMesh: Jointly Manipulating and Texturing Triangle Meshes for Text-to-3D Generation](https://mlsanity.com/paper/2409.07454/) (2024)
- [A Survey on Computational Solutions for Reconstructing Complete Objects by Reassembling Their Fractured Parts](https://mlsanity.com/paper/2410.14770/) (2024)
- [Category-Agnostic Neural Object Rigging](https://mlsanity.com/paper/2505.20283/) (2025)
- [BaRe-ESA: A Riemannian Framework for Unregistered Human Body Shapes](https://mlsanity.com/paper/2211.13185/) (2022)
- [3D Generative Model Latent Disentanglement via Local Eigenprojection](https://mlsanity.com/paper/2302.12798/) (2023)
- [Explorable Mesh Deformation Subspaces from Unstructured Generative Models](https://mlsanity.com/paper/2310.07814/) (2023)
- [Basis restricted elastic shape analysis on the space of unregistered surfaces](https://mlsanity.com/paper/2311.04382/) (2023)
- [Implicit-ARAP: Efficient Handle-Guided Neural Field Deformation via Local Patch Meshing](https://mlsanity.com/paper/2405.12895/) (2024)
- [4DRecons: 4D Neural Implicit Deformable Objects Reconstruction from a single RGB-D Camera with Geometrical and Topological Regularizations](https://mlsanity.com/paper/2406.10167/) (2024)
- [ARAP-GS: Drag-driven As-Rigid-As-Possible 3D Gaussian Splatting Editing with Diffusion Prior](https://mlsanity.com/paper/2504.12788/) (2025)
- [Approximately Piecewise E(3) Equivariant Point Networks](https://mlsanity.com/paper/2402.08529/) (2024)
- [BLiSS: Bootstrapped Linear Shape Space](https://mlsanity.com/paper/2309.01765/) (2023)
- [TutteNet: Injective 3D Deformations by Composition of 2D Mesh Deformations](https://mlsanity.com/paper/2406.12121/) (2024)
- [Self Supervised Networks for Learning Latent Space Representations of Human Body Scans and Motions](https://mlsanity.com/paper/2411.03475/) (2024)
- [GenAnalysis: Joint Shape Analysis by Learning Man-Made Shape Generators with Deformation Regularizations](https://mlsanity.com/paper/2503.00807/) (2025)
- [NeuROK: Generative 4D Neural Object Kinematics](https://mlsanity.com/paper/2605.30347/) (2026)
- [MeshMamba: State Space Models for Articulated 3D Mesh Generation and Reconstruction](https://mlsanity.com/paper/2507.15212/) (2025)
- [PaNDaS: Learnable Deformation Modeling with Localized Control](https://mlsanity.com/paper/2412.02306/) (2024)
- [ARC-Flow : Articulated, Resolution-Agnostic, Correspondence-Free Matching and Interpolation of 3D Shapes Under Flow Fields](https://mlsanity.com/paper/2503.02606/) (2025)
- [ScanMove: Motion Prediction and Transfer for Unregistered Body Meshes](https://mlsanity.com/paper/2508.21095/) (2025)

## Abstract (arXiv)

This paper introduces an unsupervised loss for training parametric deformation shape generators. The key idea is to enforce the preservation of local rigidity among the generated shapes. Our approach builds on an approximation of the as-rigid-as possible (or ARAP) deformation energy. We show how to develop the unsupervised loss via a spectral decomposition of the Hessian of the ARAP energy. Our loss nicely decouples pose and shape variations through a robust norm. The loss admits simple closed-form expressions. It is easy to train and can be plugged into any standard generation models, e.g., variational auto-encoder (VAE) and auto-decoder (AD). Experimental results show that our approach outperforms existing shape generation approaches considerably on public benchmark datasets of various shape categories such as human, animal and bone.