In the forward diffusion process the probability distribution for the next step is [1]:

$$q(\mathbf{x}_t|\mathbf{x}_{t-1}) = N(\mathbf{x}_t;\sqrt{1-\beta_t}\mathbf{x}_{t-1},\beta_t\mathbf{I})$$

What is the purpose of scaling the mean of the distribution by $\sqrt{1 - \beta_t}$? I think this has something to do with maintaining the same variance across steps, but don't understand why keeping variance fixed would be helpful.

[1] https://arxiv.org/pdf/2006.11239.pdf

In the forward diffusion process described by Ho, et al. the probability distribution for the next step is:

$$q(\mathbf{x}_t|\mathbf{x}_{t-1}) = N(\mathbf{x}_t;\sqrt{1-\beta_t}\mathbf{x}_{t-1},\beta_t\mathbf{I})$$

What is the purpose of scaling the mean of the distribution by $\sqrt{1 - \beta_t}$? I think this has something to do with maintaining the same variance across steps, but don't understand why keeping variance fixed would be helpful.

Ho, J., Jain, A., Abbeel, P. (2020). Denoising Diffusion Probabilistic Models. Proceedings of the 34th Conference on Neural Information Processing Systems. NeurIPS 2020, Vancouver, CA.