# SoftFlow: Probabilistic Framework for Normalizing Flow on Manifolds

arXiv: https://arxiv.org/abs/2006.04604 · HTML: https://mlsanity.com/paper/2006.04604/

## Key facts

- Cited by 98 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2020.

## Built on by (most cited first, 30 of 98)

- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [3D Shape Generation and Completion through Point-Voxel Diffusion](https://mlsanity.com/paper/2104.03670/) (2021)
- [Neural Wavelet-domain Diffusion for 3D Shape Generation](https://mlsanity.com/paper/2209.08725/) (2022)
- [DiT-3D: Exploring Plain Diffusion Transformers for 3D Shape Generation](https://mlsanity.com/paper/2307.01831/) (2023)
- [SDF-StyleGAN: Implicit SDF-Based StyleGAN for 3D Shape Generation](https://mlsanity.com/paper/2206.12055/) (2022)
- [SetVAE: Learning Hierarchical Composition for Generative Modeling of Set-Structured Data](https://mlsanity.com/paper/2103.15619/) (2021)
- [Towards Implicit Text-Guided 3D Shape Generation](https://mlsanity.com/paper/2203.14622/) (2022)
- [3D-LDM: Neural Implicit 3D Shape Generation with Latent Diffusion Models](https://mlsanity.com/paper/2212.00842/) (2022)
- [Advances in 3D Generation: A Survey](https://mlsanity.com/paper/2401.17807/) (2024)
- [Fast Point Cloud Generation with Straight Flows](https://mlsanity.com/paper/2212.01747/) (2022)
- [RangeLDM: Fast Realistic LiDAR Point Cloud Generation](https://mlsanity.com/paper/2403.10094/) (2024)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [JANA: Jointly Amortized Neural Approximation of Complex Bayesian Models](https://mlsanity.com/paper/2302.09125/) (2023)
- [Controllable Mesh Generation Through Sparse Latent Point Diffusion Models](https://mlsanity.com/paper/2303.07938/) (2023)
- [Learning to design drug-like molecules in three-dimensional space using deep generative models](https://mlsanity.com/paper/2104.08474/) (2021)
- [Sketch and Text Guided Diffusion Model for Colored Point Cloud Generation](https://mlsanity.com/paper/2308.02874/) (2023)
- [DiffFacto: Controllable Part-Based 3D Point Cloud Generation with Cross Diffusion](https://mlsanity.com/paper/2305.01921/) (2023)
- [Universal approximation property of invertible neural networks](https://mlsanity.com/paper/2204.07415/) (2022)
- [Wasserstein Flow Matching: Generative modeling over families of distributions](https://mlsanity.com/paper/2411.00698/) (2024)
- [Neural Wavelet-domain Diffusion for 3D Shape Generation, Inversion, and Manipulation](https://mlsanity.com/paper/2302.00190/) (2023)
- [iFlow: Numerically Invertible Flows for Efficient Lossless Compression via a Uniform Coder](https://mlsanity.com/paper/2111.00965/) (2021)
- [3DQD: Generalized Deep 3D Shape Prior via Part-Discretized Diffusion Process](https://mlsanity.com/paper/2303.10406/) (2023)
- [AI-Generated Content (AIGC) for Various Data Modalities: A Survey](https://mlsanity.com/paper/2308.14177/) (2023)
- [GECCO: Geometrically-Conditioned Point Diffusion Models](https://mlsanity.com/paper/2303.05916/) (2023)
- [Quasi-Monte Carlo for 3D Sliced Wasserstein](https://mlsanity.com/paper/2309.11713/) (2023)
- [FloMo: Tractable Motion Prediction with Normalizing Flows](https://mlsanity.com/paper/2103.03614/) (2021)
- [SP-GAN: Sphere-Guided 3D Shape Generation and Manipulation](https://mlsanity.com/paper/2108.04476/) (2021)
- [Noise Conditional Flow Model for Learning the Super-Resolution Space](https://mlsanity.com/paper/2106.04428/) (2021)
- [Not-So-Optimal Transport Flows for 3D Point Cloud Generation](https://mlsanity.com/paper/2502.12456/) (2025)
- [Go with the Flows: Mixtures of Normalizing Flows for Point Cloud Generation and Reconstruction](https://mlsanity.com/paper/2106.03135/) (2021)

## Abstract (arXiv)

Flow-based generative models are composed of invertible transformations between two random variables of the same dimension. Therefore, flow-based models cannot be adequately trained if the dimension of the data distribution does not match that of the underlying target distribution. In this paper, we propose SoftFlow, a probabilistic framework for training normalizing flows on manifolds. To sidestep the dimension mismatch problem, SoftFlow estimates a conditional distribution of the perturbed input data instead of learning the data distribution directly. We experimentally show that SoftFlow can capture the innate structure of the manifold data and generate high-quality samples unlike the conventional flow-based models. Furthermore, we apply the proposed framework to 3D point clouds to alleviate the difficulty of forming thin structures for flow-based models. The proposed model for 3D point clouds, namely SoftPointFlow, can estimate the distribution of various shapes more accurately and achieves state-of-the-art performance in point cloud generation.