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<h1 class="title is-2 publication-title">E-TSDM: <span class="is-size-3"><span class="is-size-2">E</span>arly - <span class="is-size-2">T</span>ime <span class="is-size-2">S</span>hared <span class="is-size-2">D</span>iffusion <span class="is-size-2">M</span>odels</span></h1>
<h3 class="title is-3 publication-title">Lipschitz Singularities in Diffusion Models</h3>
<h5 class="subtitle is-5 publication-awards" style="font-weight: bold; color: red;">ICLR 2024 (Oral)</h5>
<div class="is-size-5 publication-authors">
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Zhantao Yang</a>,
</span>
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Ruili Feng</a>,
</span>
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Han Zhang</a>,
</span>
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Yujun Shen</a>,
</span>
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Kai Zhu</a>,
</span>
<br> <!-- 插入换行 -->
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Lianghua Huang</a>,
</span>
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Yifei Zhang</a>,
</span>
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Yu Liu</a>,
</span>
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Deli Zhao</a>,
</span>
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Jingren Zhou</a>,
</span>
<span class="author-block">
<a style="color:#f68946;font-weight:normal;">Fan Cheng<sup>†</sup></a>
</span>
</div>
<div class="is-size-5 publication-authors">
<span class="author-block"><b style="color:#f68946; font-weight:normal">▶ </b> Shanghai Jiao Tong University</b></span>
<br> <!-- 插入换行 -->
<span class="author-block"><b style="color:#F2A900; font-weight:normal">▶ </b> University of Science and Technology of China</span>
<span class="author-block"><b style="color:#008AD7; font-weight:normal">▶ </b> Ant Group</span>
<span class="author-block"><b style="color:#008AD7; font-weight:normal">▶ </b> Alibaba Group</span>
</div>
<div class="is-size-6 publication-authors">
<span class="author-block"><sup>†</sup>Corresponding Author</span>
</div>
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<span>Code (Coming Soon)</span>
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<h2 class="title is-3">Abstract</h2>
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<p>
Diffusion models, which employ stochastic differential equations to sample images through integrals, have emerged as a dominant class of generative models. However, the rationality of the diffusion process itself receives limited attention, leaving the question of whether the problem is well-posed and well-conditioned. In this paper, we explore a perplexing tendency of diffusion models: they often display the infinite Lipschitz property of the network with respect to time variable near the zero point. We provide theoretical proofs to illustrate the presence of infinite Lipschitz constants and empirical results to confirm it. The Lipschitz singularities pose a threat to the stability and accuracy during both the training and inference processes of diffusion models. Therefore, the mitigation of Lipschitz singularities holds great potential for enhancing the performance of diffusion models. To address this challenge, we propose a novel approach, dubbed \methodabbr, which alleviates the Lipschitz singularities of the diffusion model near the zero point of timesteps. Remarkably, our technique yields a substantial improvement in performance. Moreover, as a byproduct of our method, we achieve a dramatic reduction in the Fréchet Inception Distance of acceleration methods relying on network Lipschitz, including DDIM and DPM-Solver, by over 33%. Extensive experiments on diverse datasets validate our theory and method. Our work may advance the understanding of the general diffusion process, and also provide insights for the design of diffusion models.
</p>
</div>
</div>
</div>
</div>
</section>
<section class="section">
<!-- Results. -->
<div class="columns is-centered has-text-centered">
<div class="column is-six-fifths">
<h2 class="title is-3"> Lipschitz Singularities</h2>
</div>
</div>
<!-- </div> -->
<!--/ Results. -->
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<p>
Diffusion models often exhibit an alarming propensity to possess infinite Lipschitz near the zero point of timesteps
<br>
It may causes
<ul type="1">
<li><b>Inaccuracy with skipped timesteps.</span></li>
<li><b>Numerical instability.</span></li>
<li><b>Limiting performance of ode-solvers.</span></li>
</ul>
<!-- Please check out our
<a href="https://github.com/haotian-liu/LLaVA/blob/main/docs/MODEL_ZOO.md">[Model Zoo]</a>. -->
</p>
</div>
<centering>
<div style="text-align: center;">
<img id="teaser" width="50%" src="images/Lipschitz Constants.png">
</div>
</centering>
</div>
</div>
</section>
<section class="section">
<!-- Results. -->
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<div class="column is-six-fifths">
<h2 class="title is-3"> Analysis </h2>
</div>
</div>
<!-- </div> -->
<!--/ Results. -->
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<p>
For noise-predicting diffusion models
<p>$$ \boldsymbol{\epsilon_\theta}(\mathbf{x}_t, t) = - \sigma_t \nabla_{\mathbf{x}_t} \log q_t (\mathbf{x}_t) $$</p>
Derivative
<p>$$ \frac{\partial \boldsymbol{\epsilon_\theta}(\mathbf{x}_t, t)}{\partial t} = \frac{\alpha_t}{\sqrt{1 - \alpha_t^2}} \frac{d \alpha_t}{dt}\nabla_\mathbf{x} \log q_t(\mathbf{x}) - \frac{\partial \nabla_{\mathbf{x}} \log q_t (\mathbf{x})}{\partial t} \sigma_t $$</p>
For all noise schedules
<p>$$ \sigma_t = \sqrt{1 - \alpha_t^2} $$</p>
<p>$$ t\rightarrow 0, \alpha_t \rightarrow 1, \frac{\alpha_t}{\sqrt{1 - \alpha_t^2}} \rightarrow \infty $$</p>
For commonly used noise schedules, we can prove
<p>$$ \alpha_t \rightarrow 1, \frac{d \alpha_t}{dt} \neq 0 $$</p>
Smooth assumption
<p>$$ \Vert \frac{\partial \nabla_{\mathbf{x}} \log q_t (\mathbf{x})}{\partial t} \Vert_2 < \infty, \quad \Vert \nabla_{\mathbf{x}} \log q_t (\mathbf{x}) \Vert_2 < \infty $$</p>
Conclusion
<p>$$ \text{When} \quad t \rightarrow 0 , \quad \Vert \frac{\partial \nabla_{\mathbf{x}} \log q_t (\mathbf{x})}{\partial t} \Vert_2 \rightarrow \infty $$</p>
<!-- Please check out our
<a href="https://github.com/haotian-liu/LLaVA/blob/main/docs/MODEL_ZOO.md">[Model Zoo]</a>. -->
</p>
</div>
</centering>
</div>
</div>
</section>
<section class="section">
<!-- Results. -->
<div class="columns is-centered has-text-centered">
<div class="column is-six-fifths">
<h2 class="title is-3"> E-TSDM: Early-Time Shared Diffusion Models</h2>
</div>
</div>
<!-- </div> -->
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<p>
<!-- Please check out our
<a href="https://github.com/haotian-liu/LLaVA/blob/main/docs/MODEL_ZOO.md">[Model Zoo]</a>. -->
</p>
</div>
<centering>
<div style="text-align: center;">
<img id="teaser" width="50%" src="images/method.png">
</div>
</centering>
</div>
</div>
<!-- Reduce Lipschitz Constants. -->
<div class="columns is-centered">
<div class="column is-full-width">
<h2 class="title is-4"> <span style="font-size: 100%;">Reduce Lipschitz Constants</span> </h2>
<div>
<a href="https://plotly.com/~lichunyuan24/5/?share_key=d78QObaCAYCIy8PJpe3gd1" target="_blank" title="llava_gpt4_pie" style="display: block; text-align: center;"> <img id="painting_icon" width="50%" src="images/reduce Lipschitz constants.png"> </a>
</div>
</div>
</div>
<!-- Improve stability. -->
<div class="columns is-centered">
<div class="column is-full-width">
<h2 class="title is-4"> <span style="font-size: 100%;">More stable network</span> </h2>
<div>
<a href="https://plotly.com/~lichunyuan24/5/?share_key=d78QObaCAYCIy8PJpe3gd1" target="_blank" title="llava_gpt4_pie" style="display: block; text-align: center;"> <img id="painting_icon" width="50%" src="images/stability.png"> </a>
</div>
</div>
</div>
</section>
<section class="section">
<!-- Results. -->
<div class="columns is-centered has-text-centered">
<div class="column is-six-fifths">
<h2 class="title is-3"> Performance</h2>
</div>
</div>
<!-- </div> -->
<!--/ Results. -->
<div class="container is-max-desktop">
<!-- Unconditional generation. -->
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<h2 class="title is-4"> <span style="font-size: 100%;">Unconditional Generation</span> </h2>
<p style="font-family:Times New Roman"><b>E-TSDM exceeds baseline on multiple datasets.</b>
<div>
<a href="https://plotly.com/~lichunyuan24/5/?share_key=d78QObaCAYCIy8PJpe3gd1" target="_blank" title="llava_gpt4_pie" style="display: block; text-align: center;"> <img id="painting_icon" width="50%" src="images/fid_results.png"> </a>
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<!-- Fast samplers. -->
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<h2 class="title is-4"> <span style="font-size: 100%;"> Fast Samplers</span> </h2>
<p style="font-family:Times New Roman"><b>E-TSDM well supports the popular fast samplers.</b>
<div>
<a href="https://plotly.com/~lichunyuan24/1/?share_key=v4opE3TJpxqQ08RYsDD4iv" target="_blank" title="Plot 1" style="display: block; text-align: center;"><img id="painting_icon" width="65%" src="images/fast_sampler.png"></a>
<script data-plotly="lichunyuan24:1" sharekey-plotly="v4opE3TJpxqQ08RYsDD4iv" src="https://plotly.com/embed.js" async></script>
</div>
</div>
</div>
<!-- Generalizability. -->
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<h2 class="title is-4"> <span style="font-size: 100%;"> Generalizability</span> </h2>
<p style="font-family:Times New Roman"><b>E-TSDM is still effective for other noise schedules and the continuous-time diffusion models.</b>
<div>
<a href="https://plotly.com/~lichunyuan24/1/?share_key=v4opE3TJpxqQ08RYsDD4iv" target="_blank" title="Plot 1" style="display: block; text-align: center;"><img id="painting_icon" width="65%" src="images/generalizability.png"></a>
<script data-plotly="lichunyuan24:1" sharekey-plotly="v4opE3TJpxqQ08RYsDD4iv" src="https://plotly.com/embed.js" async></script>
</div>
</div>
</div>
<!-- Conditional generation. -->
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<h2 class="title is-4"> <span style="font-size: 100%;">Conditional Generation</span> </h2>
<p style="font-family:Times New Roman"><b>E-TSDM well supports the super-resolution task.</b>
<div>
<a href="https://plotly.com/~lichunyuan24/1/?share_key=v4opE3TJpxqQ08RYsDD4iv" target="_blank" title="Plot 1" style="display: block; text-align: center;"><img id="painting_icon" width="65%" src="images/super resolution.png"></a>
<script data-plotly="lichunyuan24:1" sharekey-plotly="v4opE3TJpxqQ08RYsDD4iv" src="https://plotly.com/embed.js" async></script>
</div>
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</section>
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