# Deep Generative Learning via Variational Gradient Flow

arXiv: https://arxiv.org/abs/1901.08469 · HTML: https://mlsanity.com/paper/1901.08469/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jan 2019.

## Built on by (most cited first, 30 of 32)

- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Variational Wasserstein gradient flow](https://mlsanity.com/paper/2112.02424/) (2021)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Scalable Wasserstein Gradient Flow for Generative Modeling through Unbalanced Optimal Transport](https://mlsanity.com/paper/2402.05443/) (2024)
- [Unifying GANs and Score-Based Diffusion as Generative Particle Models](https://mlsanity.com/paper/2305.16150/) (2023)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Gaussian Interpolation Flows](https://mlsanity.com/paper/2311.11475/) (2023)
- [SAN: Inducing Metrizability of GAN with Discriminative Normalized Linear Layer](https://mlsanity.com/paper/2301.12811/) (2023)
- [MonoFlow: Rethinking Divergence GANs via the Perspective of Wasserstein Gradient Flows](https://mlsanity.com/paper/2302.01075/) (2023)
- [Neural Wasserstein Gradient Flows for Maximum Mean Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2301.11624/) (2023)
- [The Score-Difference Flow for Implicit Generative Modeling](https://mlsanity.com/paper/2304.12906/) (2023)
- [Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regression](https://mlsanity.com/paper/2305.00608/) (2023)
- [Neural Sinkhorn Gradient Flow](https://mlsanity.com/paper/2401.14069/) (2024)
- [Minimizing f-Divergences by Interpolating Velocity Fields](https://mlsanity.com/paper/2305.15577/) (2023)
- [Relative Entropy Gradient Sampler for Unnormalized Distributions](https://mlsanity.com/paper/2110.02787/) (2021)
- [GANs as Gradient Flows that Converge](https://mlsanity.com/paper/2205.02910/) (2022)
- [PINS: Proximal Iterations with Sparse Newton and Sinkhorn for Optimal Transport](https://mlsanity.com/paper/2502.03749/) (2025)
- [Module-wise Training of Neural Networks via the Minimizing Movement Scheme](https://mlsanity.com/paper/2309.17357/) (2023)
- [Efficient Multimodal Sampling via Tempered Distribution Flow](https://mlsanity.com/paper/2304.03933/) (2023)
- [DiffFlow: A Unified SDE Framework for Score-Based Diffusion Models and Generative Adversarial Networks](https://mlsanity.com/paper/2307.02159/) (2023)
- [Statistical and Geometrical properties of regularized Kernel Kullback-Leibler divergence](https://mlsanity.com/paper/2408.16543/) (2024)
- [Generative Modeling with Flow-Guided Density Ratio Learning](https://mlsanity.com/paper/2303.03714/) (2023)
- [Trust the Critics: Generatorless and Multipurpose WGANs with Initial Convergence Guarantees](https://mlsanity.com/paper/2111.15099/) (2021)
- [Block-wise Training of Residual Networks via the Minimizing Movement Scheme](https://mlsanity.com/paper/2210.00949/) (2022)
- [A New Paradigm for Generative Adversarial Networks based on Randomized Decision Rules](https://mlsanity.com/paper/2306.13641/) (2023)
- [Nested Annealed Training Scheme for Generative Adversarial Networks](https://mlsanity.com/paper/2501.11318/) (2025)
- [Guiding Time-Varying Generative Models with Natural Gradients on Exponential Family Manifold](https://mlsanity.com/paper/2502.07650/) (2025)
- [Boosting Statistic Learning with Synthetic Data from Pretrained Large Models](https://mlsanity.com/paper/2505.04992/) (2025)
- [SONA: Learning Conditional, Unconditional, and Mismatching-Aware Discriminator](https://mlsanity.com/paper/2510.04576/) (2025)

## Abstract (arXiv)

We propose a general framework to learn deep generative models via Variational Gradient Flow (VGrow) on probability spaces. The evolving distribution that asymptotically converges to the target distribution is governed by a vector field, which is the negative gradient of the first variation of the f-divergence between them. We prove that the evolving distribution coincides with the pushforward distribution through the infinitesimal time composition of residual maps that are perturbations of the identity map along the vector field. The vector field depends on the density ratio of the pushforward distribution and the target distribution, which can be consistently learned from a binary classification problem. Connections of our proposed VGrow method with other popular methods, such as VAE, GAN and flow-based methods, have been established in this framework, gaining new insights of deep generative learning. We also evaluated several commonly used divergences, including Kullback-Leibler, Jensen-Shannon, Jeffrey divergences as well as our newly discovered `logD' divergence which serves as the objective function of the logD-trick GAN. Experimental results on benchmark datasets demonstrate that VGrow can generate high-fidelity images in a stable and efficient manner, achieving competitive performance with state-of-the-art GANs.