[Paper Review] Deep Generative Learning via Variational Gradient Flow

Yuan Gao, Yuling Jiao+4 more|arXiv (Cornell University)|Jan 24, 2019

Generative Adversarial Networks and Image Synthesis59 references4 citations

TL;DR

This paper proposes Variational Gradient Flow (VGrow), a novel deep generative modeling framework that learns to transform a simple prior distribution into a target data distribution via gradient flow on probability space. By minimizing the first variation of f-divergence through a vector field derived from density ratio estimation, VGrow achieves stable, high-fidelity image generation competitive with state-of-the-art GANs, including a new 'logD' divergence that underpins the logD-trick GAN.

ABSTRACT

We propose a general framework to learn deep generative models via extbf{V}ariational extbf{Gr}adient Fl extbf{ow} (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.

Motivation & Objective

• To develop a unified framework for deep generative modeling that combines variational inference, gradient flow on probability space, and deep neural networks.
• To stabilize training of deep generative models by formulating the learning process as a continuous flow governed by the negative gradient of the first variation of f-divergence.
• To establish theoretical and practical connections between VGrow and existing methods such as VAE, GAN, and normalizing flows.
• To explore new divergences, including the newly discovered 'logD' divergence, as effective objectives for generative modeling.
• To achieve high-fidelity, stable image generation on benchmark datasets like MNIST, FashionMNIST, CIFAR10, and CelebA.

Proposed method

• The method evolves a distribution via a vector field that is the negative gradient of the first variation of the f-divergence between the evolving and target distributions.
• The evolving distribution is shown to coincide with the pushforward distribution through infinitesimal residual maps perturbing the identity along the vector field.
• The vector field depends on the density ratio between the pushforward and target distributions, which is estimated via a binary classification task.
• The framework generalizes and unifies VAE (via KL divergence), GAN (via JS divergence), and flow-based models (via exact KL minimization) under a single variational gradient flow perspective.
• A new divergence, 'logD', is derived and shown to underlie the logD-trick GAN, providing a principled alternative to standard GAN objectives.
• The method is trained using an outer-loop IS-Loop optimization strategy, with evaluation via inception score and FID for stability and performance assessment.

Experimental results

Research questions

• RQ1 How can a continuous gradient flow on probability space be used to learn deep generative models in a stable and efficient manner?
• RQ2 What is the theoretical connection between variational gradient flow and existing deep generative models such as VAE, GAN, and normalizing flows?
• RQ3 Can a new divergence, 'logD', be derived and used effectively as a training objective for GANs?
• RQ4 How does the proposed VGrow framework compare in performance and stability to state-of-the-art GANs on standard benchmarks?
• RQ5 To what extent can density ratio estimation via binary classification enable stable training of deep generative models?

Key findings

• VGrow achieves stable training with monotonically increasing inception scores on MNIST, FashionMNIST, and CIFAR10, indicating consistent learning progress.
• On MNIST and FashionMNIST, VGrow-JS achieves FID scores of 3.32 and 8.75, respectively, very close to the real data FID of 2.12 and 4.16.
• On CIFAR10 with 50k generated samples, VGrow-logD attains an FID of 28.8, outperforming WGAN-GP (31.1) and MMDGAN-GP-L2 (31.4), and approaching the best reported score of 28.5.
• The VGrow-KL model generates visually realistic samples on all datasets, with generated images easily identifiable by class and indistinguishable from real images in many cases.
• The method demonstrates robustness across multiple divergences, including KL, JS, JF, and the newly introduced logD, with consistent performance gains.
• Theoretical analysis confirms that the evolving distribution under VGrow is equivalent to the pushforward of the prior through infinitesimal residual maps, providing a solid foundation for the method.