Image generation with shortest path diffusion (2306.00501v1)

Abstract: The field of image generation has made significant progress thanks to the introduction of Diffusion Models, which learn to progressively reverse a given image corruption. Recently, a few studies introduced alternative ways of corrupting images in Diffusion Models, with an emphasis on blurring. However, these studies are purely empirical and it remains unclear what is the optimal procedure for corrupting an image. In this work, we hypothesize that the optimal procedure minimizes the length of the path taken when corrupting an image towards a given final state. We propose the Fisher metric for the path length, measured in the space of probability distributions. We compute the shortest path according to this metric, and we show that it corresponds to a combination of image sharpening, rather than blurring, and noise deblurring. While the corruption was chosen arbitrarily in previous work, our Shortest Path Diffusion (SPD) determines uniquely the entire spatiotemporal structure of the corruption. We show that SPD improves on strong baselines without any hyperparameter tuning, and outperforms all previous Diffusion Models based on image blurring. Furthermore, any small deviation from the shortest path leads to worse performance, suggesting that SPD provides the optimal procedure to corrupt images. Our work sheds new light on observations made in recent works and provides a new approach to improve diffusion models on images and other types of data.

• The paper demonstrates that minimizing the Fisher metric-based path length yields an optimal image corruption strategy for diffusion models.
• It introduces Shortest Path Diffusion (SPD), combining image sharpening and noise deblurring to outperform traditional blurring methods.
• Empirical results on CIFAR10 and ImageNet 64x64 reveal improved FID scores, underscoring SPD's effectiveness in high-quality synthesis.

Image Generation with Shortest Path Diffusion: An Analytical Perspective

The paper "Image Generation with Shortest Path Diffusion" by Ayan Das et al. presents a novel approach to optimizing diffusion models for image generation through the computation of the shortest path in the space of probability distributions. This work contributes to the existing literature by addressing the optimality of image corruption methodologies preceding the denoising process within diffusion models.

Summary and Contributions

Diffusion models have established a strong foothold in image generation tasks due to their iterative noise-to-image synthesis capabilities. Typically, these models employ a stochastic corruption of images via a noise process, subsequently reversed during image generation. However, previous approaches, including those leveraging blurring for corruption, have been largely empirical. This paper sets forth a hypothesis that the "optimal" corruption procedure minimizes the path length from an original to a corrupted image, using the Fisher metric in probability distribution space.

The key contributions of the paper can be summarized as follows:

• Optimal Path Calculation: The authors derive the shortest path using the Fisher Information Matrix as the metric, assuming its suitability for guiding likelihood-based optimization often seen in diffusion model training.
• Shortest Path Diffusion (SPD): The analytical derivation reveals that a combination of image sharpening and noise deblurring constitutes the optimal corruption strategy, contrary to the previously favored image blurring techniques.
• Empirical Validation: Experimental results demonstrate that the proposed SPD method outperforms previous noise- and blur-based diffusion models on standard datasets (e.g., CIFAR10 and ImageNet 64x64) in terms of FID scores, emphasizing the efficacy of the shortest path strategy.

The paper further highlights that any deviation from this optimal path formulation degrades performance, supporting the hypothesis that SPD provides a robust corruption mechanism.

Theoretical Insights and Implications

Theoretically, the work is anchored in minimizing the path length in a Riemannian manifold described by image distribution covariances. This geometric interpretation offers significant advancements over arbitrary noise and blur schedulers. While the Fisher Information provides a logical bridge due to its reparameterization invariance and ties to likelihood (Cramer-Rao bounds), the adaptation to non-Gaussian distributions (such as natural images) is handled by approximating these paths via Gaussian analogs using the spectral properties of images.

The implications extend beyond purely visual metrics. SPD potentially alleviates computational burdens in hyperparameter tuning and adjusts naturally to dataset-specific statistics, given the data-driven nature of the corruption filter based on power-spectral density.

Future Directions

The paper opens multiple avenues for future research:

• Generalization to Other Domains: Extending the SPD framework to other modalities like audio and beyond require contextual adaptation of the covariance estimation process.
• Alternative Metrics: Investigating the utility of other metrics like the Wasserstein distance in calculating the shortest paths could yield different optimization landscapes.
• Integration with Advanced Techniques: SDP could be synergistically combined with advances in variational distributions or latent space diffusion processes, potentially leveraging neural ODEs or stochastic calculus.

In conclusion, the shortest path diffusion method redefines the corruption process as a problem of geometric optimization, presenting a structured solution rooted in theoretical underpinnings. This work suggests that meticulous analytical groundwork paired with empirical validation can chart pathways to more efficient and high-fidelity generative models. The implications for AI are substantial, emphasizing the need for principled approaches in the design of synthetically creative systems.