Defining the Markov Chain

As introduced, the forward diffusion process gradually degrades an initial data point, such as a clear image $x_0$, into noise over a sequence of discrete timesteps, $t=1, 2, ..., T$. This structured degradation isn't arbitrary; it follows a specific probabilistic model known as a Markov chain.

A Markov chain is a sequence of random variables where the probability of transitioning to the next state depends only on the current state, not on the sequence of events that preceded it. In our context, the "states" are the increasingly noisy versions of our data: $x_0, x_1, x_2, ..., x_T$. The defining characteristic is that to get $x_t$, we only need to know $x_{t-1}$ and the rules for adding noise at step $t$. We don't need to know $x_{t-2}$ or any earlier states directly.

We can represent this sequence as:

$$x_0 \rightarrow x_1 \rightarrow x_2 \rightarrow \dots \rightarrow x_{t-1} \rightarrow x_t \rightarrow \dots \rightarrow x_T$$

Here:

• $x_0$ is the original data sample drawn from the true data distribution $q(x_0)$.
• $x_t$ is the data sample at timestep $t$, after $t$ steps of adding noise.
• $x_T$ is the final state after $T$ steps, which ideally resembles pure Gaussian noise.

The transition from one state $x_{t-1}$ to the next state $x_t$ is defined by a conditional probability distribution, denoted as $q(x_t | x_{t-1})$. This distribution specifies how to sample $x_t$ given $x_{t-1}$. Because this process only depends on the previous state, it satisfies the Markov property:

$$q(x_t | x_{t-1}, x_{t-2}, \dots, x_0) = q(x_t | x_{t-1})$$

This property significantly simplifies the analysis and implementation of the forward process. We don't need to track the entire history of noise additions to determine the next state; only the immediately preceding state is required.

Visually, we can picture this chain of dependencies:

The forward diffusion process modeled as a Markov chain. Each arrow represents a transition $q(x_t | x_{t-1})$, where noise is added based only on the previous state.

In the subsequent sections, we will get into the specific mathematical form of the transition $q(x_t | x_{t-1})$, which involves adding carefully scaled Gaussian noise at each step, controlled by a predefined schedule. Understanding this Markovian structure is the first step towards comprehending how diffusion models operate and, eventually, how they learn to reverse this process for generation.