It’s interesting to see that the two versions of the explanation are quite different, with ChatGPT’s version being more centered on intuition and Mistral Large-2’s version being more technical and mathematical.

Prompt: “Can you help me write about the connection between the Central limit theorem and the reverse process of diffusion models being able to initialize from normal distributions”

ChatGPT’s version

The Central Limit Theorem and Diffusion Models

Overview of the Central Limit Theorem (CLT)

The Central Limit Theorem (CLT) is a fundamental principle in probability theory and statistics. It states that the sum (or average) of a large number of independent, identically distributed (i.i.d.) random variables, regardless of their original distribution, will tend to follow a normal (Gaussian) distribution, provided that certain conditions are met. This convergence to a normal distribution occurs as the number of variables increases.

Mathematically, if $X_1, X_2, \dots, X_n$ are i.i.d. random variables with mean $\mu$ and variance $\sigma^2$, the normalized sum:

$$ Z_n = \frac{1}{\sqrt{n}} \sum_{i=1}^{n} \left( \frac{X_i - \mu}{\sigma} \right) $$

approaches a standard normal distribution $N(0,1)$ as $n$ becomes large.

Diffusion Models and Their Reverse Process

Diffusion models are a type of generative model used in machine learning and statistics. These models simulate a diffusion process where data points are progressively noised until they resemble a simple distribution, typically a normal distribution. The reverse process of this diffusion aims to recover the original data by starting from a sample in this simple distribution and reversing the noise.

In the forward process, a data point $x_0$ is incrementally noised through a series of steps:

$$ x_t = x_{t-1} + \text{noise} $$

where $t$ indicates the time step, and the noise is often Gaussian. After many such steps, the distribution of the data converges to a normal distribution due to the CLT, because the accumulated effect of adding small, independent noise at each step approximates a normal distribution.

In the reverse process, the model aims to reverse this diffusion to generate data samples by denoising a sample from the normal distribution back to the original data distribution. This reverse process is crucially dependent on the fact that the forward process converges to a normal distribution.

Connection Between CLT and the Reverse Process

The connection between the CLT and the reverse process in diffusion models lies in the justification for initializing the reverse process from a normal distribution. The CLT explains why the forward process in a diffusion model, after many iterations, leads to a normal distribution, regardless of the initial data distribution. This normal distribution, reached through the forward diffusion, serves as the starting point for the reverse process.

When we initialize the reverse process from a normal distribution, we’re leveraging the fact that the forward process transforms any complex data distribution into this normal form. Therefore, by starting the reverse process from a normal distribution, we can effectively “trace back” through the noise to recover a sample from the original complex distribution.

In summary, the CLT provides the theoretical underpinning for why the diffusion process leads to a normal distribution in the first place, and this understanding allows us to confidently start the reverse process of diffusion models from a normal distribution to generate realistic data samples.

Mistral Large-2’s version

Title: The Central Limit Theorem and the Reverse Process of Diffusion Models

Introduction

The Central Limit Theorem (CLT) is a fundamental concept in probability theory and statistics, asserting that the distribution of the sum (or average) of a large number of independent, identically distributed (i.i.d.) random variables approaches a normal distribution, regardless of the original distribution. Diffusion models, on the other hand, are a class of generative models that have shown promise in various applications, including image and audio synthesis. Interestingly, the reverse process of diffusion models is intrinsically connected to the CLT, particularly when initialized from normal distributions.

The Central Limit Theorem

The CLT states that for a sequence of i.i.d. random variables $X_1, X_2, \ldots, X_n$ with finite mean $\mu$ and variance $\sigma^2$, the distribution of the standardized sum

$$ Z_n = \frac{\sum_{i=1}^{n} X_i - n\mu}{\sigma \sqrt{n}} $$

converges to a standard normal distribution $N(0, 1)$ as $n \to \infty$. This theorem explains why the normal distribution appears so frequently in nature and forms the basis for many statistical tests and confidence intervals.

Diffusion Models and the Reverse Process

Diffusion models define a Markov chain that gradually adds Gaussian noise to the data, transforming it into a simple, tractable distribution, such as a standard normal distribution. The reverse process then learns to denoise the data, generating new samples from the original data distribution.

Given a data distribution $q(x_0)$, the forward diffusion process is defined as:

$$ q(x_t | x_{t-1}) = \mathcal{N}(x_t; \sqrt{1 - \beta_t} x_{t-1}, \beta_t I) $$

where $\beta_t$ is a variance schedule. After a large number of time steps $T$, the data distribution $q(x_T)$ approaches a standard normal distribution due to the cumulative effect of the added Gaussian noise.

The reverse process is also a Markov chain with Gaussian transitions:

$$ p_\theta(x_{t-1} | x_t) = \mathcal{N}(x_{t-1}; \mu_\theta(x_t, t), \Sigma_\theta(x_t, t)) $$

where $\mu_\theta$ and $\Sigma_\theta$ are parameterized neural networks that predict the mean and covariance of the Gaussian distribution, respectively.

Connection to the Central Limit Theorem

The connection between the CLT and the reverse process of diffusion models lies in the initialization from a normal distribution. As the forward diffusion process involves summing independent Gaussian noise increments, the CLT guarantees that the resulting distribution after a large number of time steps will be approximately normal.

Moreover, the reverse process can be viewed as a series of denoising steps, where each step refines the current estimate by removing some of the added noise. Starting from a normal distribution, the reverse process gradually transforms the data back to the original distribution, essentially inverting the summation of noise increments described by the CLT.

Conclusion

The Central Limit Theorem provides a theoretical foundation for the forward diffusion process, explaining why the data distribution approaches a normal distribution after a large number of noise additions. The reverse process of diffusion models leverages this property, initializing from a normal distribution and learning to denoise the data, ultimately generating new samples from the original data distribution. This intrinsic connection highlights the importance of the CLT in the context of diffusion models and contributes to our understanding of their underlying mechanisms.