    \begin{algorithm}[H]
      \caption{Our gradient estimation sampler}
      \label{alg:second-order}
      \begin{algorithmic}
        \Require $(\sigma_N, \ldots, \sigma_0)$, $x_N \sim \mathcal{N}(0, I)$, $\epsilon_\theta$
        \Ensure Compute $x_0$ with $N$ evaluations of $\epsilon_\theta$
        \State $x_{N-1} \gets x_N + (\sigma_{N-1} - \sigma_N)\epsilon_\theta(x_N, \sigma_N)$
        \For{$t = N-1, \ldots, 1$}
        \State $\bar{\epsilon}_t \gets 2\epsilon_\theta(x_t, \sigma_t) - \epsilon_\theta(x_{t+1}, \sigma_{t+1})$
        \State $x_{t-1} \gets x_t + (\sigma_{t-1} - \sigma_t) \bar{\epsilon}_t$
        \EndFor
        \State \Return $x_0$
      \end{algorithmic}
  \end{algorithm}

\begin{algorithm}
[H]
\caption{Our gradient estimation sampler}
\begin{algorithmic}
\Require $(\sigma_N, \ldots, \sigma_0)$, $x_N \sim \mathcal{N}(0, I)$, $\epsilon_\theta$
\Ensure Compute $x_0$ with $N$ evaluations of $\epsilon_\theta$
\State $x_{N-1} \gets x_N + (\sigma_{N-1} - \sigma_N)\epsilon_\theta(x_N, \sigma_N)$
\For{$t = N-1, \ldots, 1$}
\State $\bar{\epsilon}_t \gets 2\epsilon_\theta(x_t, \sigma_t) - \epsilon_\theta(x_{t+1}, \sigma_{t+1})$
\State $x_{t-1} \gets x_t + (\sigma_{t-1} - \sigma_t) \bar{\epsilon}_t$
\EndFor
\State \Return $x_0$
\end{algorithmic}
\end{algorithm}
{
    "title": "interpreting and improving diffusion models using the euclidean distance   function",
    "id": "2306.04848",
    "abstract": "denoising is intuitively related to projection. indeed, under the manifold hypothesis, adding random noise is approximately equivalent to orthogonal perturbation. hence, learning to denoise is approximately learning to project. in this paper, we use this observation to reinterpret denoising diffusion models as approximate gradient descent applied to the euclidean distance function. we then provide straight-forward convergence analysis of the ddim sampler under simple assumptions on the projection-error of the denoiser. finally, we propose a new sampler based on two simple modifications to ddim using insights from our theoretical results. in as few as 5-10 function evaluations, our sampler achieves state-of-the-art fid scores on pretrained cifar-10 and celeba models and can generate high quality samples on latent diffusion models.",
    "categories": "cs.lg cs.cv math.oc stat.ml",
    "doi": "",
    "created": "2023-06-07",
    "updated": "2024-02-13",
    "authors": [
        "frank permenter",
        "chenyang yuan"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2306.04848"
}
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