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Added reference on how to generate noisy image at any timestep t during the forward diffusion process Tag: Reverted |
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The entire diffusion process then satisfies<math display="block">q(x_{0:T}) = q(x_0)q(x_1|x_0) \cdots q(x_T|x_{T-1}) = q(x_0) N(x_1 | \sqrt{\alpha_1} x_0, \beta_1 I) \cdots N(x_T | \sqrt{\alpha_T} x_{T-1}, \beta_T I)</math>or<math display="block">\ln q(x_{0:T}) = \ln q(x_0) - \sum_{t=1}^T \frac{1}{2\beta_t} \| x_t - \sqrt{1-\beta_t}x_{t-1}\|^2 + C</math>where <math>C</math> is a normalization constant and often omitted. In particular, we note that <math>x_{1:T}|x_0</math> is a [[gaussian process]], which affords us considerable freedom in [[Reparameterization trick|reparameterization]]. For example, by standard manipulation with gaussian process, <math display="block">x_{t}|x_0 \sim N\left(\sqrt{\bar\alpha_t} x_{0}, \sigma_{t}^2 I \right)</math><math display="block">x_{t-1} | x_t, x_0 \sim N(\tilde\mu_t(x_t, x_0), \tilde\sigma_t^2 I)</math>In particular, notice that for large <math>t</math>, the variable <math>x_{t}|x_0 \sim N\left(\sqrt{\bar\alpha_t} x_{0}, \sigma_{t}^2 I \right)</math> converges to <math>N(0, I)</math>. That is, after a long enough diffusion process, we end up with some <math>x_T</math> that is very close to <math>N(0, I)</math>, with all traces of the original <math>x_0 \sim q</math> gone.
For example, since<math display="block">x_{t}|x_0 \sim N\left(\sqrt{\bar\alpha_t} x_{0}, \sigma_{t}^2 I \right)</math>we can sample <math>x_{t}|x_0</math> directly "in one step", instead of going through all the intermediate steps <math>x_1, x_2, ..., x_{t-1}</math
{{Math proof|title=Derivation by reparameterization|proof=
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** {{Cite arXiv |last=Luo |first=Calvin |date=2022-08-25 |title=Understanding Diffusion Models: A Unified Perspective |class=cs.LG |eprint=2208.11970}}
** {{Cite web |last=Weng |first=Lilian |date=2021-07-11 |title=What are Diffusion Models? |url=https://lilianweng.github.io/posts/2021-07-11-diffusion-models/ |access-date=2023-09-25 |website=lilianweng.github.io |language=en}}
* Tutorials
** {{Cite arXiv |eprint=2406.08929 |first1=Preetum |last1=Nakkiran |first2=Arwen |last2=Bradley |title=Step-by-Step Diffusion: An Elementary Tutorial |date=2024 |last3=Zhou |first3=Hattie |last4=Advani |first4=Madhu|class=cs.LG }}
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