[Paper Review] SoftFlow: Probabilistic Framework for Normalizing Flow on Manifolds

Hyeong‐Ju Kim, H. S. Lee+3 more|arXiv (Cornell University)|Jun 8, 2020

3D Shape Modeling and AnalysisEngineering28 references46 citations

TL;DR

SoftFlow trains normalizing flows on manifolds by learning conditional distributions of perturbed data, enabling high-quality samples and extending to 3D point clouds with SoftPointFlow.

ABSTRACT

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.

Motivation & Objective

• Motivate and address the dimension-mismatch problem when data lie on low-dimensional manifolds.
• Propose SoftFlow to train flows by modeling perturbed data conditioned on noise, rather than the exact data distribution.
• Demonstrate that SoftFlow captures manifold structure and yields high-quality samples.
• Extend SoftFlow to 3D point clouds via SoftPointFlow to improve thin structure generation.

Proposed method

• Perturb each data point with noise drawn from a random distribution and condition the flow on the noise parameter.
• Train a normalizing flow to model the conditional distribution of perturbed data, allowing sampling by reducing the perturbation.
• Apply the framework to continuous normalizing flows by conditioning on noise in the dynamics (CNF context).
• Adapt the approach to discrete normalizing flows in SoftPointFlow to model 3D point clouds and latent shape variables.
• Use a two-level architecture (PriorFlow and DecoderFlow) with a latent S and an autoregressive decoding step for points.
• Evaluate using synthetic manifold data and point cloud benchmarks, comparing against Glow, FFJORD, and GAN-based baselines.

Experimental results

Research questions

• RQ1 Can normalizing flows accurately model data lying on low-dimensional manifolds by learning perturbation-conditioned distributions?
• RQ2 Does SoftFlow enable generation of samples that faithfully reflect manifold structure and avoid mode collapse or distorted geometries?
• RQ3 Can the SoftFlow idea improve 3D point cloud generation, particularly for thin structures, compared to existing flow-based methods?

Key findings

• SoftFlow generates high-quality samples that closely follow manifold data distributions, outperforming Glow and FFJORD on several synthetic manifolds.
• SoftPointFlow achieves state-of-the-art or competitive results for 3D point cloud generation, especially in preserving thin structures.
• SoftPointFlow shows improved 1-NNA (CD/EMD) metrics across airplane, chair, and car categories compared to PointFlow and GAN baselines.
• Preference tests indicate SoftPointFlow samples are generally preferred over PointFlow reconstructions across categories.
• SoftFlow provides a flexible and broadly applicable approach that can be integrated with both discrete and continuous normalizing flows.