DiffUCO: A Diffusion Model Framework for Unsupervised Neural Combinatorial Optimization

Sampling from complex, high-dimensional target distributions, such as the Boltzmann distribution, is of great significance in scientific research across various fields. These distributions often represent the probability of different states in a system, and sampling from them allows researchers to understand and analyze the behavior of the system under different conditions.

For example, in molecular simulations, the Boltzmann distribution represents the probability of a molecular system being in a particular state at a given temperature. Sampling from this distribution helps researchers predict molecular configurations and study their thermodynamic properties, such as free energy and entropy. This information is crucial in drug discovery, material science, and understanding biological processes at the molecular level.

In combinatorial optimization, sampling from high-dimensional target distributions can help find optimal solutions to complex problems. These problems often involve discrete variables and are computationally challenging to solve. By sampling from the distribution of possible solutions, researchers can explore the solution space more efficiently and find near-optimal solutions.

Moreover, high-dimensional sampling is also essential in machine learning, particularly in generative models and Bayesian inference. It allows for the generation of realistic data samples and the estimation of model parameters with uncertainty quantification.

In summary, sampling from complex, high-dimensional target distributions is crucial for understanding complex systems, solving optimization problems, and advancing scientific research in various fields5.

Diffusion Models are a type of latent variable model used in generative modeling tasks. They define a Markov chain of diffusion steps to slowly add random noise to data and then learn to reverse the diffusion process to construct desired data samples from the noise5. Unlike other latent variable models like Variational Autoencoders (VAEs) or flow models, diffusion models are learned with a fixed procedure and the latent variable has high dimensionality (same as the original data).

The training process of diffusion models involves two key processes: the forward process and the reverse process. The forward process gradually adds noise to the data, transforming it into a simple distribution (typically Gaussian). The reverse process then learns to undo this noise addition, generating new samples from the original data distribution5.

In the context of image generation, diffusion models have shown remarkable results. They can generate high-quality samples that are visually appealing and diverse. However, the training and sampling process can be computationally expensive, especially for high-resolution images.

To overcome this computational challenge, Latent Diffusion Models (LDMs) have been introduced. LDMs perform the diffusion process in a lower-dimensional latent space instead of the high-dimensional pixel space. By doing so, LDMs reduce the computational cost while preserving the generative capabilities of diffusion models.

The training process of LDMs involves encoding the input images into a latent space using an encoder network. The diffusion process is then applied to the latent representations, and the reverse process learns to generate new latent representations. Finally, a decoder network is used to reconstruct the images from the generated latent representations.

Overall, diffusion models play a crucial role in generative modeling tasks by gradually adding noise to the data and learning to reverse the diffusion process. LDMs further enhance the efficiency of diffusion models by performing the diffusion process in a lower-dimensional latent space.