Deep MMD Gradient Flow without adversarial training (2405.06780v1)

Abstract: We propose a gradient flow procedure for generative modeling by transporting particles from an initial source distribution to a target distribution, where the gradient field on the particles is given by a noise-adaptive Wasserstein Gradient of the Maximum Mean Discrepancy (MMD). The noise-adaptive MMD is trained on data distributions corrupted by increasing levels of noise, obtained via a forward diffusion process, as commonly used in denoising diffusion probabilistic models. The result is a generalization of MMD Gradient Flow, which we call Diffusion-MMD-Gradient Flow or DMMD. The divergence training procedure is related to discriminator training in Generative Adversarial Networks (GAN), but does not require adversarial training. We obtain competitive empirical performance in unconditional image generation on CIFAR10, MNIST, CELEB-A (64 x64) and LSUN Church (64 x 64). Furthermore, we demonstrate the validity of the approach when MMD is replaced by a lower bound on the KL divergence.

• The paper proposes DMMD, a novel method that replaces adversarial training with noise-adaptive MMD gradient flow for generative modeling.
• It utilizes a noise-conditional discriminator trained via a progressive forward diffusion process to refine data representations at multiple noise levels.
• Experimental results on CIFAR-10 reveal competitive FID and inception scores, demonstrating improved stability and scalability in image generation.

Understanding Deep MMD Gradient Flow Without Adversarial Training

Introduction

Generative modeling has seen significant strides, powering applications from image and audio generation to protein modeling and 3D creation. Typically, effective generative models fall into two categories: Generative Adversarial Networks (GANs) and Diffusion Models. Each comes with its strengths and challenges. This article focuses on a novel approach that combines insights from both, particularly streamlining the training of a discriminator without adversarial dynamics, leveraging something called Deep MMD Gradient Flow.

The Problem with Traditional Methods

GANs feature a generator and a discriminator that are trained in a min-max game, which often faces issues like instability and mode collapse. Even though GANs can create high-quality samples, fine-tuning the training procedure to avoid pitfalls is arduous.

Diffusion Models rely on a forward noising process followed by a learned backward denoising process. These models can handle multi-step processes well but often require many steps and can suffer from inefficiencies particularly as the gradient becomes unstable near the data distribution.

Enter DMMD: Diffusion-MMD-Gradient Flow

Combining the strengths of GANs' discriminators and Diffusion Models' noise processes, this new method introduces DMMD (Diffusion-MMD-Gradient Flow), offering a technique for generative modeling without requiring adversarial training and instead leveraging insights from both diffusion models and Maximum Mean Discrepancy (MMD).

How DMMD Works

Key Concepts:

1. Maximum Mean Discrepancy (MMD): This is a measure of the distance between two distributions by comparing their mean embeddings in a Reproducing Kernel Hilbert Space (RKHS). In GANs, it's used as a loss to train discriminators.
2. Noise-Adaptive Gradient Flow: Instead of only measuring MMD between two distributions, DMMD trains a noise-adaptive MMD that changes as samples move from a noisy initial distribution to the target distribution.

Training the Noise-Conditional Discriminator

The process begins by learning a noise-conditional MMD discriminator:

1. Forward Diffusion Process: Gradually add noise to the data to create multiple noisy versions of the dataset.
2. Training Process: A neural network learns to discriminate between noisy and actual data at various noise levels, improving its understanding incrementally, much like the progressive refinement in GANs but without creating adversarial conditions.

Sampling New Data

Using the trained noise-conditional MMD discriminator, DMMD allows for the creation of new samples by following these steps:

1. Initialization: Start with random samples from a Gaussian distribution.
2. Gradient Flow: Move these samples in the direction of the target distribution using MMD Gradient Flow, adjusting for the corresponding noise level adapted by the discriminator.

The Why: Benefits and Results

Numerical Insights

DMMD shows competitive results in unconditional image generation tasks such as CIFAR-10, achieving FID scores as low as 7.74 and inception scores above 9 in some configurations, rivaling traditional GANs and Diffusion models.

Adaptive Progression

One of the strong claims made by this paper is that an adaptive discriminator, which changes its kernel width based on the noise level, leads to faster convergence. This is crucial in high-dimensional settings like image generation where fixed methods may falter.

Broader Implications

Practical Impact

1. Scalability: Since DMMD eliminates the need for adversarial training, it offers a more stable and scalable paradigm for training discriminators.
2. Generative Flexibility: By leveraging gradient flows and adaptive measures, it opens up new avenues for generative modeling in complex settings such as high-dimensional data.

Theoretical Considerations

The forward diffusion process combined with MMD gradient flows offers an interesting mix of theory from optimal transport and kernel methods. It could pave the way for more robust theoretical frameworks that address issues with both GANs and diffusion models.

Future Prospects

1. Expanding Scope: It would be interesting to explore DMMD in advanced settings like 3D object generation or in contexts where traditional diffusion models struggle, such as with highly irregular data distributions.
2. Theoretical Optimizations: Further research might focus on optimizing the training procedure or understanding the convergence properties of the gradient flows better.

Conclusion

The proposed DMMD method exemplifies how combining established concepts from GANs and Diffusion models can lead to more stable, efficient generative modeling techniques. By training discriminators without adversarial dynamics and leveraging noise-adaptive mechanisms, DMMD provides a promising new direction in the landscape of generative modeling.