Unlocking the Power of Diffusion Models in Generative Modeling

Table of Contents

• What is Diffusion Modeling?
• The Basic Mechanism Behind Diffusion Models
• The Forward Diffusion Process
• The Reverse Process
• The Benefits of Small Step Size
• The Training Objective of Diffusion Models
• Diffusion Models as Latent Variable Generative Models
• Training Diffusion Models with VAE Objective
• The Variational Lower Bound in Diffusion Models
• Implementing Reverse Steps in Diffusion Models
• Conditional Generation with Diffusion Models
• Inpainting with Diffusion Models
• Diffusion Models vs. Other Generative Models
• Speeding up Sampling in Diffusion Models
• Continuous Time Formulation of Diffusion Models
• Connection with Score Matching Models
• The Relationship Between Score and Noise in Diffusion Models
• Conclusion
• Resources

What is Diffusion Modeling?

Diffusion modeling is an innovative approach in generative modeling, particularly in the domain of image generation. It involves gradually adding noise to an image and then learning to reverse this process, removing the noise and recovering a coherent image. Diffusion models have shown impressive performance in various tasks, surpassing other types of generative models like Generative Adversarial Networks (GANs), and have gained traction in the field. In this article, we will delve into the basic mechanism behind diffusion models, their training objective, and their applications in conditional generation and inpainting.

The Basic Mechanism Behind Diffusion Models

The basic idea behind diffusion models is to have a forward diffusion process that adds noise to an image over a series of time steps, and a reverse process that aims to undo this noise and reconstruct the original image. The forward process follows a Markov chain, with each time step's distribution depending on the sample from the previous step. The reverse process is also structured as a Markov chain, with each step aiming to reconstruct the image from the noisy sample. By gradually increasing the noise level over the time steps, the forward process pushes the image off the data manifold, while the reverse process brings it back to a coherent state.

The Forward Diffusion Process

The forward diffusion process, denoted as q, is designed to add noise to the image. It takes the form of a Markov chain, where each time step's distribution is conditioned on the sample from the previous step. In the case of continuous data like images, each transition is parameterized as a diagonal Gaussian. The variances at each time step, denoted as beta_t, are treated as hyperparameters and follow a fixed schedule. The variances generally increase with time, gradually pushing the samples towards a Gaussian distribution centered at zero with an identity covariance matrix.

Using a large but finite number of steps allows for setting the individual variances (beta_t) to be very small, maintaining proximity to the limiting distribution while making the process of undoing the steps less difficult. This small step size reduces ambiguity about the previous step's location, making it easier to model the posterior distribution of the forward step.

The Reverse Process

The reverse process aims to reconstruct the original image from the noisy sample obtained from the forward process. It is also structured as a Markov chain and follows a similar mechanism as the forward process but in reverse. Each reverse step is parameterized as a unimodal diagonal Gaussian. The means of these Gaussian distributions are learned during training, while the variances are set as time-specific constants. This choice of fixing the reverse process variances is due to instability issues and lower sample quality observed when attempting to learn them.

During inference, when generating samples, the reverse process is used to sample from the noisy image until producing the initial noisy image (x0). The reverse process gradually removes the noise, resulting in a coherent image.

The Benefits of Small Step Size

One of the benefits of using a small step size in diffusion models is the reduction in ambiguity about the previous step's location. This is particularly useful when inferring the posterior distribution over the previous step given the current step. With a small noise step, there is less uncertainty about where the sample came from, allowing for a more accurate estimation of the previous step's location. Modeling the posterior distribution with a unimodal Gaussian becomes justified, further enhancing the reconstruction process.

The Training Objective of Diffusion Models

The training objective of diffusion models is to maximize a lower bound on the marginal log-likelihood of the data. This lower bound is known as the variational lower bound or evidence lower bound (ELBO). The objective consists of a reconstruction term and a KL-divergence term.

The reconstruction term encourages the model to maximize the expected density assigned to the data, while the KL-divergence term encourages the approximate posterior to be similar to the prior on the latent variable. The reconstruction term is subtracted by the KL-divergence term, resulting in the lower bound.

By viewing the diffusion model as a latent variable generative model, similar to Variational Autoencoders (VAEs), we can borrow the basic training objective used in VAEs to train diffusion models. This allows for the optimization of both the forward and reverse processes simultaneously.

Diffusion Models as Latent Variable Generative Models

Diffusion models can be seen as latent variable generative models. The forward process is analogous to the encoder in VAEs, producing a latent representation from the data. The reverse process is analogous to the decoder, reconstructing the data from the latent representation.

However, unlike VAEs, where both the encoder and decoder are trained jointly, in diffusion models, the forward process is typically fixed and the focus is solely on training the reverse process. This means that only a single network needs to be trained, simplifying the training process.

Training Diffusion Models with VAE Objective

Since diffusion models can be viewed as a type of VAE, the training objective used in VAEs can be adapted for training diffusion models. The variational lower bound (ELBO) serves as the training objective, allowing the model to maximize the likelihood of the observed data.

By substituting the variables in the ELBO derivation with the corresponding elements in the diffusion model framework, we can obtain a Simplified version of the ELBO specifically tailored for diffusion models. This simplified objective discards certain term weights Present in the original ELBO, leading to better sample quality.

The Variational Lower Bound in Diffusion Models

The variational lower bound (ELBO) in diffusion models can be simplified, resulting in a more efficient training process. By discarding certain term weights, we can emphasize more challenging steps with greater noise during training.

The forward process (q) and the fixed reverse process are treated as fixed terms in the objective. The KL-divergence term between each reverse step and the posterior of the forward process conditioned on x0 can be evaluated in closed form, reducing the variance in the training process.

The objective is rearranged to put more emphasis on steps with greater noise, allowing the model to focus on challenging reconstructions. This modification helps improve the efficiency of the training process and leads to higher quality samples.

Implementing Reverse Steps in Diffusion Models

The implementation of reverse steps in diffusion models can be done in different ways. In the paper "Denoising Diffusion Probabilistic Models" (DDPM), the authors suggest setting the reverse process variances as time-specific constants. Learning these variances was found to result in unstable training and lower sample quality. By focusing solely on learning the means, the network can be trained more effectively.

Another important aspect of implementing reverse steps is the reparameterization technique used to predict the noise added in each step. By introducing an auxiliary noise variable called epsilon, which follows a constant distribution independent of the forward time step, the reverse step network can be designed to predict this epsilon. This reparameterization simplifies the training process and leads to better sample quality.

Conditional Generation with Diffusion Models

Diffusion models can be extended to conditional generation tasks, where samples are generated based on some condition, such as class labels or text descriptions. One common approach is to feed the conditioning variable during training. The model learns to utilize this additional information to guide the generation process.

Another approach involves guiding the diffusion process with a separate classifier. By utilizing the gradient of the target label probability with respect to the current noise image, the reverse diffusion process is pushed in the direction of the desired condition. This technique can be applied not only with single-WORD labels but also with higher-dimensional text descriptions.

An alternative approach, known as "classifier-free diffusion guidance," eliminates the need for a separate classifier. During training, the conditioning label is set to a null label with a certain probability. At inference time, the reconstructed samples are artificially pushed further towards the conditional direction, leading to higher quality samples. Despite not receiving new information, this technique has shown superior sample quality compared to classifier-guided methods.

Inpainting with Diffusion Models

Inpainting is the process of filling in missing or corrupted regions of an image. Diffusion models can be used for inpainting tasks by fine-tuning a pre-trained model specifically for this purpose.

The naive approach to inpainting with diffusion models involves replacing known regions of an image with samples from the forward process after each reverse step. However, this method may result in edge artifacts as the model is not aware of the full surrounding context, only a hazy version of it.

To overcome these limitations, a more effective approach involves randomly removing sections of training images and conditioning the model to fill in the missing regions based on the clear context. This fine-tuning process leads to better inpainting results, reducing edge artifacts and producing more coherent images.

Diffusion Models vs. Other Generative Models

Diffusion models offer several advantages and differences compared to other prominent generative models like Generative Adversarial Networks (GANs). One key difference is the sampling procedure. Diffusion models generate samples by following a slow Markov chain, which can be time-consuming. In contrast, GANs can generate images in a single forward pass.

Despite the slower sampling process, diffusion models have shown promising performance. Recent diffusion models have outperformed GANs in perceptual quality metrics and demonstrated impressive results in various conditional settings, such as converting text descriptions to images and image manipulation tasks. Ongoing research aims to improve the sampling efficiency of diffusion models.

Speeding up Sampling in Diffusion Models

The slow Markov chain sampling procedure in diffusion models can be a limitation in terms of sampling efficiency. Generating an image using diffusion models typically requires simulating the entire chain, which can be time-consuming.

Efforts are being made to speed up the sampling process in diffusion models. By leveraging acceleration techniques or approximating the limiting distribution using numerical integration, the sampling time can be significantly reduced. Ongoing research aims to explore these avenues and improve the efficiency of diffusion models.

Continuous Time Formulation of Diffusion Models

Diffusion models can be formulated in continuous time, leading to a framework known as probability flow ODE (Ordinary Differential Equation). This formulation allows for approximating log-likelihood via numerical integration.

Continuous time diffusion models offer an alternative perspective and provide flexibility in modeling complex distributions. These models have shown promising results in various tasks and can contribute to advancing the field of generative modeling. Further research and exploration in continuous time diffusion models are expected in the future.

Connection with Score Matching Models

There is a close connection between denoising diffusion models and score matching models. Score matching models aim to estimate the gradient of the log of the target probability density with respect to the data. A score network is trained to estimate this gradient, and a Markov chain is set up to produce samples guided by this gradient.

In denoising diffusion models, the noise predicted in each reverse step can be shown to be equivalent, up to a scaling factor, to the score in score matching models. This connection highlights the relationship between diffusion models and score-based models, providing insight into the underlying mechanisms and further expanding the possibilities in generative modeling.

The Relationship Between Score and Noise in Diffusion Models

The noise predicted in diffusion models plays a crucial role in the reconstruction process. It can be seen as an approximation of the gradient of the data log density, also known as the score. By attempting to undo the noise, the diffusion model is effectively following the gradient of the data log density.

Diffusion models leverage this connection to approximate the true score and perform high-quality reconstructions. The relationship between score and noise provides a deeper understanding of the inner workings of diffusion models and their probabilistic nature.

Conclusion

Diffusion models have gained significant Momentum in the field of generative modeling, particularly in tasks like image generation, conditional generation, and inpainting. The basic mechanism behind diffusion models involves a forward diffusion process that adds noise to an image and a reverse process that aims to undo this noise and reconstruct the original image.

Diffusion models offer advantages in terms of sample quality, perceptual metrics, and flexibility in modeling complex distributions. Despite the slower sampling procedure, ongoing research aims to improve sampling efficiency. Continuous time formulation and the connection with score matching models contribute to advancing the field of generative modeling.

Diffusion models continue to evolve and show promise, offering new possibilities in generative modeling and pushing the boundaries of what can be achieved in image generation and related tasks.

Resources

• Denoising Diffusion Probabilistic Models (DDPM)
• Variational Autoencoders (VAEs)
• Generative Adversarial Networks (GANs)