October 29, 2023

Last updated: November 29, 2023

Table of Contents

Diffusion Models are a class of generative models that can produce realistic and diverse data, such as images, text, audio, and video. They are based on the idea of transforming the data distribution into a simple noise distribution through a series of random diffusion steps. By reversing this process, we can sample new data from the noise distribution using a learned score function that guides the diffusion towards the data distribution.

Diffusion Models have several advantages over other generative models, such as Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs). Some of these advantages are:

- They do not suffer from mode collapse, where the stable diffusion models only generate a few modes of data distribution and ignore the rest.
- They do not require adversarial training, which can be unstable and hard to tune.
- They can handle discrete and continuous data without any special tricks or modifications.
- They can generate high-resolution and high-fidelity data with fewer parameters and less computation.

The forward and reverse diffusion processes are the core components of the Diffusion Model. They define how the data is transformed into noise and how the noise is transformed back into data.

The forward diffusion process is a Markov chain that starts from the original data x and ends at a noise sample Îµ. At each step t, the data is corrupted by adding Gaussian noise to it. The noise level increases as t increases until it reaches 1 at the final step T. At this point, x_T is completely random and independent of x.

**x_t = âˆš(1 – Î²t) * x(t-1) + âˆšÎ²_t * Î·_t**

where Î²_t is the noise level at step t, and Î·_t is a standard Gaussian random variable. The noise level Î²_t increases as t increases until it reaches 1 at the final step T. At this point, x_T is completely random and independent of x.

The reverse diffusion process is the inverse of the forward diffusion process. It starts from a noise sample Îµ and ends at a data sample x. At each step t, the noise is reduced by subtracting Gaussian noise from it. The noise level decreases as t decreases until it reaches 0 at the initial step 0. At this point, Îµ*0 is equal to x. *

*Îµ_t = âˆš(1 – Î²_t) * Îµ*(t+1) – âˆšÎ²_t * Î·_t

where Î²_t is the same noise level as in the forward diffusion process, and Î·_t is a standard Gaussian random variable. The noise level Î²_t decreases as t decreases until it reaches 0 at the initial step 0. At this point, Îµ_0 is equal to x.

In practice, we do not know the exact value of Î·*t at each step. Therefore, we need a score function s_t(x_t) that estimates the conditional distribution of x*(t-1) given x_t. The score function s_t(x_t) tells us how likely x_(t-1) is for a given x_t, and how to adjust x_t to make it closer to x_(t-1). We can use the score function s_t(x_t) to sample from the reverse diffusion process using Langevin dynamics:

*x_(t-1) = x_t + Î±_t * s_t(x_t) + âˆš(2 * Î±_t) * Î¶*

where Î±_t is the step size at step t, and Î¶ is a standard Gaussian random variable. By repeating this process from t = T to t = 0, we can generate a data sample x from a noise sample Îµ.

The noise schedule and the number of steps are two important hyperparameters that affect the performance of the Diffusion Model. They determine how fast and how smoothly the data is transformed into noise and vice versa.

The noise schedule is a sequence of noise levels Î²_t that control the amount of Gaussian noise added or subtracted at each step t. A common choice for the noise schedule is to use a geometric progression:

*Î²_t = Î² * (1 – Î²)^(T – 1 – t) *

where Î² is a constant between 0 and 1, and T is the total number of steps. This noise schedule ensures that the variance of x_t is constant for all t, which simplifies the score function estimation.

The number of steps T is the length of the forward and reverse diffusion processes. It affects the quality and diversity of the generated data. A larger T means that the data is more corrupted by noise, which makes it harder to recover from the noise, but also allows for more variation in the data. A smaller T means that the data is less corrupted by noise, which makes it easier to recover from the noise, but also limits the variation in the data.

There is a trade-off between the noise schedule and the number of steps. A more aggressive noise schedule (larger Î²) requires more steps to achieve better quality, while a less aggressive noise schedule (smaller Î²) requires fewer steps to achieve good diversity. The optimal choice of these hyperparameters depends on the data domain, the score function architecture, and the computational budget.

Î² is a constant between 0 and 1 that controls the noise level in the Diffusion Model. A larger Î² means that more noise is added or subtracted at each step, while a smaller Î² means that less noise is added or subtracted at each step. A larger Î² makes the data more corrupted by noise, while a smaller Î² makes the data less corrupted by noise. 0.5 is the middle value of Î² and is neither considered as a small nor a large Î². It is a middle value of Î² that balances the trade-off between quality and diversity in the Diffusion Model. It means that the noise level is 50% at the final step of the forward diffusion process and 50% at the initial step of the reverse diffusion process. It is a balanced choice that preserves some information and some variation in the data. However, it may not be the optimal choice for every data domain or score function architecture. You may need to experiment with different values of Î² to find the best one for your task.

To sample from the trained Diffusion Model, we need to follow the reverse diffusion process using the score function and Langevin dynamics.

Here are the steps to do that:

- Start from a random noise sample Îµ ~ N(0, I), where I is the identity matrix.

- Set t = T, where T is the total number of steps in the forward and reverse diffusion processes.

- While t > 0, do the following:

- Compute the score function output s_t(x_t) by feeding x_t to the neural network.

- Update x_(t-1) by using the Langevin dynamics formula:

*x_(t-1) = x_t + Î±_t * s_t(x_t) + âˆš(2 * Î±_t) * Î¶*

where Î±_t is the step size at step t, and Î¶ is a standard Gaussian random variable.

- Decrease t by 1.
- Return x_0 as the sampled data.

Diffusion Model in AI is a promising research direction in the field of generative AI modeling. They have shown impressive results in various data domains, such as images, text, audio, and video. Applications of diffusion models can be found in areas such as data augmentation, super-resolution, inpainting, style transfer, and more.

However, there are still some challenges and limitations that need to be addressed in the future. Experts are working on solutions to overcome the challenges and improve its results but until then Happy Diffusing readers.

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