# How is the noise in the forward process in Denoising Diffusion Probabilistic Models computed?

The inputs are decayed towards the origin using this formula within Denoising Diffusion Probabilistic Models (DDPMs):

$$q\left(\mathbf{x}_{1: T} \mid \mathbf{x}_0\right):=\prod_{t=1}^T q\left(\mathbf{x}_t \mid \mathbf{x}_{t-1}\right), \quad q\left(\mathbf{x}_t \mid \mathbf{x}_{t-1}\right):=\mathcal{N}\left(\mathbf{x}_t ; \sqrt{1-\beta_t} \mathbf{x}_{t-1}, \beta_t \mathbf{I}\right)$$

I however do not understand how the origin is determined. How is the noise in the forward process or diffusion process computed? In the original DDPM paper it is only stated that:

the forward process variances $$β_t$$ can be learned by reparameterization [33] or held constant as hyperparameters.

You can generate a noise $$\epsilon_t$$ by sampling a distribution (which you already explicitly stated as $$\mathcal{N}(\mathbf{x}_t; \sqrt{1-\beta_t}\mathbf{x}_{t-1}, \beta_t\mathbf{I})$$) during the sampling phase. The "origin" is the standard normal distribution $$\mathcal{N}(\mathbf{0}, \mathbf{I})$$, which is a constant instead of a variable ground truth to learn.
The variable ground truth to learn (using a U-Net) during the training phase is rather the noise $$\epsilon_t$$ added during the sampling phase. Reverse to the diffusion (sampling) process, the noise $$\epsilon_t$$, when subtracted from the noisier $$\mathbf{x}_{t}$$, can produce a less noisy $$\mathbf{x}_{t+1}$$. Therefore, given $$\mathbf{x}_{t}$$, the noise $$\epsilon_t$$ encodes certain information regarding the input data $$\mathbf{x}_{T}$$.