Closed-Form Diffusion Models (2310.12395v3)

Abstract: Score-based generative models (SGMs) sample from a target distribution by iteratively transforming noise using the score function of the perturbed target. For any finite training set, this score function can be evaluated in closed form, but the resulting SGM memorizes its training data and does not generate novel samples. In practice, one approximates the score by training a neural network via score-matching. The error in this approximation promotes generalization, but neural SGMs are costly to train and sample, and the effective regularization this error provides is not well-understood theoretically. In this work, we instead explicitly smooth the closed-form score to obtain an SGM that generates novel samples without training. We analyze our model and propose an efficient nearest-neighbor-based estimator of its score function. Using this estimator, our method achieves competitive sampling times while running on consumer-grade CPUs.

• The paper introduces a smoothing technique for the exact score in diffusion models to enhance generalization without training neural networks.
• The paper develops a closed-form sampler that constructs samples as barycenters of training data using an efficient nearest-neighbor estimator.
• The paper demonstrates competitive image generation quality on high-dimensional tasks with reduced computational cost on consumer-grade CPUs.

Closed-Form Diffusion Models: Enhancements and Applications

The paper presents a novel approach to score-based generative models (SGMs) by proposing closed-form diffusion models that eschew the traditional reliance on neural network approximations. This research addresses the inherent limitations of closed-form SGMs, specifically the lack of generalization and scalability, without resorting to costly neural approximations.

At the core of this work lies the challenge of improving sample generalization in SGMs without the heavy computational burden of training neural networks. Traditional SGMs utilize approximations of the score function through neural networks, introducing both approximation and optimization errors. While these errors induce generalization, they also result in significant computational demands. This paper tackles this issue by introducing a method that explicitly smooths the closed-form score for SGMs, allowing the generation of novel samples without the need for training neural networks.

Methodology and Contributions

The authors present a comprehensive analysis of their model, known as smoothed closed-form diffusion models (smoothed CFDMs). These models achieve novel sample generation by smoothing the exact solution to the score-matching problem, effectively promoting generalization. Key contributions of this research include:

1. Smoothing the Exact Solution: The authors demonstrate that smoothing the exact score solution can promote generalization. This is achieved without incurring the costs associated with training a neural network.
2. Closed-form Sampler Construction: A closed-form sampler using the smoothed score is proposed, which generates samples that are barycenters of tuples of training data, circumventing the need for model training.
3. Acceleration through Nearest-Neighbor Estimation: The authors introduce an efficient nearest-neighbor-based estimator for the smoothed score, which streamlines sampling, achieving competitive performance relative to neural SGMs even on consumer-grade CPUs.
4. Evaluation on High-Dimensional Tasks: The method is tested on high-dimensional tasks such as image generation, where it shows the ability to generate high-quality samples and operate efficiently in the latent space of a pretrained autoencoder.

Practical Implications and Theoretical Insights

Practically, this research provides pathways to reduce the computational load traditionally associated with neural SGM training and sampling. By enabling efficient sample generation on standard hardware, the smoothed closed-form diffusion models represent a practical alternative for deploying SGMs in resource-constrained environments.

Theoretically, this work challenges existing paradigms that primarily rely on neural approximations for generative models by offering a viable alternative in the form of smoothed closed-form solutions. These findings may encourage further investigation into non-neural, classical approaches to generative modeling, potentially unveiling new insights into the fundamental principles of SGM generalization.

Future Directions

Future work could explore optimizing the choice of the smoothing parameter to balance generalization and fidelity closely. Additionally, extending this framework to other generative modeling tasks, such as text or video generation, would further validate the model's versatility and effectiveness. Lastly, investigating the interaction of smoothed CFDMs with manifold-hypothesis settings may yield deeper insights into the geometry of data distributions in generative models.

This research exemplifies an alternative perspective in SGM development, with potential impacts extending to both practical applications and theoretical advancements in artificial intelligence.