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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf,len=1207
+page_content='ERA-Solver: Error-Robust Adams Solver  for Fast Sampling of Diffusion Probabilistic Models  Shengmeng Li 1 Luping Liu 2 Zenghao Chai 3 Runnan Li 1 Xu Tan 3  Abstract  Though denoising diffusion probabilistic models  (DDPMs) have achieved remarkable generation  results, the low sampling efficiency of DDPMs  still limits further applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Since DDPMs  can be formulated as diffusion ordinary differ-  ential equations (ODEs), various fast sampling  methods can be derived from solving diffusion  ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' However, we notice that previous sam-  pling methods with fixed analytical form are not  robust with the error in the noise estimated from  pretrained diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In this work, we  construct an error-robust Adams solver (ERA-  Solver), which utilizes the implicit Adams nu-  merical method that consists of a predictor and a  corrector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Different from the traditional predictor  based on explicit Adams methods, we leverage a  Lagrange interpolation function as the predictor,  which is further enhanced with an error-robust  strategy to adaptively select the Lagrange bases  with lower error in the estimated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Exper-  iments on Cifar10, LSUN-Church, and LSUN-  Bedroom datasets demonstrate that our proposed  ERA-Solver achieves 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='14, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='42, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='69 Fenchel  Inception Distance (FID) for image generation,  with only 10 network evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Introduction  In recent years, denoising diffusion probabilistic models  (DDPMs) (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020) have been proven to have poten-  tial in data generation tasks such as text-to-image genera-  tion(Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Kim & Ye, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022), speech synthesis(Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Lam  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Leng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022), and molecular conformation  formation(Hoogeboom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Jing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Wu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' They build a diffusion  process to add noise into the sample and a denoising process  to remove noise from the sample gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Compared with  1Microsoft Cloud+AI 2Zhejiang University 3Microsoft Re-  search Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Correspondence to: Xu Tan <xuta@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='com>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Implicit Adams  DPM-Solver  ERA-Solver (Ours)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We adopt the pretrained diffusion model from (Song  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a) and visualize the error between the estimated noise  and ground-truth noise in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We also provide the gener-  ated samples from the pretrained model based on implicit Adams  (traditional predictor-corrector method (Diethelm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2002)),  DPM-Solver(Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a), and our error-robust Adams solver  (ERA-Solver).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Our solver is robust for the error from pretrained  model so as to generate samples with better quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  generative adversarial networks (GANs)(Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=',  2014) and variational auto-encoders (VAEs)(Child, 2021),  DDPMs have achieved remarkable generation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' How-  ever, due to the properties of the Markov chain, the sampling  process requires hundreds or even thousands of denoising  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Such defects limit the wide applications of diffusion  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Thus, it is an urgent request for a fast sampling of  DDPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  There have already existed many works for accelerating  sampling speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Some works introduced an extra training  stage, such as knowledge distillation method (Salimans &  Ho, 2021), training sampler (Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021), or directly  combining with GANs (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022), to obtain a fast  sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' These methods require a cumbersome training  process for each task, and are black-box samplers due to  the lack of theoretical explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Denoising diffusion  implicit model (DDIM) (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a) and Score-  SDE(Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020b) revealed that the sampling can  be reformulated as a diffusion ordinary differential equa-  tion (ODE) solving process, which inspired many works to  design learning-free fast samplers based on numerical meth-  ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' PNDM (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021) introduced the analytic form  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='12935v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='LG]  30 Jan 2023  70  2  60  50  40  )03  30  20  3  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='1  time tError-Robust Adams Solver  of a 4-order linear multistep method, which is also called ex-  plicit Adams method (Małgorzata & Marciniak, 2002), and  utilized the estimated noises observed at previous sampling  steps to sample efficiently, with a warming initialization  based on Runge-Kutta methods (Butcher, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' DPM-  Solver(Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a) introduced exponential integrator  from ODE literature (Atkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2011) and required ex-  tra network evaluations to observe intermediate noise terms  for approximating Taylor expansion (Mohazzabi & Becker,  2017) in the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  The existing learning-free methods (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Lu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a) are based on the assump-  tion that the learned network has high accuracy in noise  estimations across all the sampling steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' However, we no-  tice that the noise estimated from the neural network is not  accurate enough and the error exists at almost every time  t, especially when time t approaches 0, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Existing sampling methods are not able to be robust for  noise estimation errors, since they consist of an analytic  sampling scheme with fixed coefficients to ensure sampling  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' For example, explicit Adams (Małgorzata &  Marciniak, 2002) consists of the analytic form with fixed  coefficients as formulated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  In this paper, we aim at designing an error-robust diffusion  ODE solver to speed up the sampling process of DDPMs  while achieving good sampling quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' To this end, we  focus on implicit Adams solver (Małgorzata & Marciniak,  2006), a kind of numerical ODE solver, which involves  unobserved terms to achieve high-order precision and con-  vergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In existing ODE literature(Atkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2011),  predictor-corrector has been introduced to perform implicit  Adams solver, which avoids solving the implicit equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Explicit Adams usually acts as the predictor to predict the  unobserved term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' However, traditional predictor-corrector  still suffers from the inaccurate estimation of diffusion noise  at each sampling step since it is composed of fixed coeffi-  cients, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Instead of utilizing explicit  Adams as the predictor, we adopt the Lagrange interpola-  tion function (Sauer & Xu, 1995) that interpolates several  Lagrange function bases as the predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We maintain a  buffer of estimated noises observed at previous sampling  steps during sampling and adaptively select those estimated  noises with low estimation error as the Lagrange function  bases, to ensure accurate interpolation result and thus an ac-  curate predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In this way, we can obtain a diffusion ODE  sampler with not only high convergence (thanks to the im-  plicit Adams method(Małgorzata & Marciniak, 2006)) but  also good error robustness (thanks to the adaptive strategy  for the Lagrange interpolation function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  However, it is not easy to select noises with low estima-  tion error as the Lagrange function bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' That is because,  unlike the training stage, there exist no reference noises  at the sampling stage to judge how accurate the estimated  noise is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Thus, we further propose an approach to roughly  measure the accuracy of the estimated noise by calculating  the difference between the noise obtained by the predictor  (as the prediction) and the noise observed at the previous  sampling step (as the reference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Based on this measure-  ment, we propose a selection strategy for the buffer which  adaptively chooses estimated noises that are more accurate  to construct the Lagrange function bases, so as to result in a  more accurate predictor, and thus a better ODE solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Our contributions can be summarized as follows:  We are the first to explore the potential of numerical im-  plicit Adams methods (Małgorzata & Marciniak, 2006)  to solve diffusion ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We propose an error-robust im-  plicit Adams Solver (ERA-Solver), which is training-free  and can be extended to pretrained DDPMs conveniently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We propose to adopt the Lagrange interpolation function  (Sauer & Xu, 1995) that selectively interpolates several  Lagrange function bases with low errors of estimated  noises to ensure the ERA-solver is robust to the error in  estimated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Experiment results show that we achieve better results  on LSUN-Bedroom, LSUN-Church, and Cifar10 datasets  when compared with previous methods in 10 function  evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' On LSUN-Church and LSUN-Bedroom, we  achieve state-of-the-art FID results of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='39 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='39 with  more function evaluations, compared with the previous  best results of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='74 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Preliminary  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Denoising Diffusion Probabilistic Models  Denoising diffusion probabilistic models (DDPMs)(Ho  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020) have demonstrated their great generation poten-  tial on various applications, such as text-to-image synthesis  (Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Kim & Ye, 2021), image  inpainting (Lugmayr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Kawar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022), speech synthesis (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Lam  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Leng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022), and molecular conformation  generation (Hoogeboom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Jing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Wu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It involves a diffusion  process to gradually add noise to data, and a parameterized  denoising process to reverse the diffusion process, sampling  through gradually removing the noise from random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  The diffusion process is modeled as a transition distribution:  q(xt|xt−1) := N(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' √αtxt−1, (1 − αt)I),  (1)  where α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', αT are fixed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' With the transition  distribution above, noisy distribution conditioned on clean  data x0 can be formulated as follows:  q(xt|x0) = N(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' √αtx0, (1 − αt)I),  (2)  Error-Robust Adams Solver  where ¯αt = �t  s=1 αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When t is large enough, the Markov  process will converge to a Gaussian steady-state distribution  N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  DDPMs also build the reverse process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', the sampling  process, to the diffusion process above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Similar to VAEs,  DDPMs introduce the variational bound to derive optimiza-  tion objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' After simplification (Luo, 2022), the main  objective of training optimization is formulated as a KL  divergence:  Lt−1 = DKL(q(xt−1|x0, xt)||pθ(xt−1|xt)),  (3)  where q(xt−1|x0, xt) has an analytical form derived from  the bayesian formula and can be expressed as a Gaussian  distribution parameterized with µt−1 =  1  √αt (xt −  1−αt  √1−¯αt ϵ)  and σt−1 = 1−¯αt−1  1−¯αt (1 − αt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  To simplify the optimization loss, the parameterized denois-  ing distribution is formulated as follows:  pθ(xt−1|xt) = N(xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' µθ(xt), σ2  t−1I)  µθ(xt) =  1  √αt  (xt − 1 − αt  √1 − ¯αt  ϵθ(xt, t)),  (4)  where ϵθ(xt, t) estimates the noise in noisy sample xt and  is called estimated noise in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The variance σ2  t−1 is  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In this way, the objective function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 3 can be  rewritten as following:  Lt−1 = Eq[  1  2σ2  t−1  ||µt−1 − µθ(xt)||2]  = Ex0,ϵ[  1  2σ2  t−1  ||  1  √αt (xt − 1 − αt  √1 − ¯αt  ϵ) − µθ(xt)||2]  = Ex0,ϵ[  1  2σ2  t−1  (1 − αt)2  αt(1 − ¯αt)||ϵ − ϵθ(xt, t)||2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  (5)  Though DDPMs have good theoretical properties, they suf-  fer from sampling efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It usually requires hundreds  or even thousands of network evaluations, which limits var-  ious downstream applications for DDPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' There already  existed methods (Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Lam  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Salimans & Ho, 2021) which depend on extra  training stage to derive fast sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The training-  based methods usually require tremendous training costs  for different data manifolds and tasks, which inspires many  works to explore a training-free sampler based on numerical  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Numerical Methods for Fast Sampling  Denoising diffusion implicit model (DDIM) (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=',  2020a) is a classic training-free approach for fast sampling,  which introduces a non-Markovian process that allows the  sampling with any number of evaluation steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We formu-  late the iteration time steps for solving diffusion ODEs as  {ti}N  i=0, where t0 is the beginning time and tN is the end  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The sequence of sampling steps maintains the prop-  erty that ti>ti+1 and tN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In particular, the last iterate  xtN represents the final generated sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The denoising  iteration process of DDIM can be formulated as follows:  xti+1 = �¯αti+1(xt − √1 − ¯αtiϵθ(xti, ti)  √¯αti)  +  �  1 − ¯αti+1 − σ2  tiϵθ(xti, ti) + σtiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  (6)  Furthermore, DDIM shares the same training objective with  DDPMs, which means the fast sampling scheme can be  applied to any pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When the variance σ2  ti is  set to 0, the sampling process becomes deterministic, which  can be regarded as the solving process of diffusion ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We use a term ϵti for the ease of description so that the  solving formula of diffusion ODE can be formulated (Song  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a) as follows:  ϵti = ϵθ(xti, ti),  (7)  xti+1 =  √¯αti+1  √¯αti  xti  + (  �  1 − ¯αti+1 −  �  ¯αti+1(1 − ¯αti)  √¯αti  )ϵti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  (8)  Many numerical solvers, such as Runge-Kutta (Butcher,  1996) method and linear multistep method (Wells, 1982),  can be applied to solve diffusion ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' PNDM(Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=',  2021) combined Runge-Kutta and linear multistep method  so as to solve the manifold problem of diffusion ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Essentially, the linear multistep method can be regarded  as explicit Adams (Małgorzata & Marciniak, 2002), which  utilized previously estimated noises to calculate the ϵti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The  ϵti in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 7 is reformulated as following:  ϵti = 1  24(55ϵθ(xti, ti) − 59ϵθ(xti−1, ti−1)  + 37ϵθ(xti−2, ti−2) − 9ϵθ(xti−3, ti−3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  (9)  DPM-Solver(Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a) introduced the method of  exponential integrator from ODE literature (Atkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=',  2011) to eliminate the discretization errors of the linear term  in the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Furthermore, DPM-Solver pro-  posed to utilize Taylor expansion (Mohazzabi & Becker,  2017) to approximate ϵti and contributed its analytical  forms for solving diffusion ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We notice that the estimated noises of pretrained diffusion  models are not precise across all sampling time, especially  when time ti is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Previous numerical methods  with analytical forms will suffer from the noise estimation  error of pretrained DDPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In this paper, we propose an  error-robust solver based on Adams numerical methods,  adaptively selecting noises with low estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Error-Robust Adams Solver  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' (8)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' (17)  Corrector  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' (11)  Predictor  Lagrange Buffer  Initial indexs  Selected Indexs  ERA-Solver  Push  Selected  Not Selected  Initial Indexs  Error-Robust Selection  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The pipeline of ERA-Solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The sampling scheme is based on the predictor-corrector method for implicit Adams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Our predictor  is robust to the errors of the estimated noises from pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The sampling starts from normal Gaussian noise xt0 and performs a  denoising scheme (from xti to xti+1) iteratively to get the final generated image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' ERA-Solver  In this section, we first point out that the error in the esti-  mated noise ϵθ(xti, ti) by the network θ limits the previ-  ous numerical fast samplers and introduce implicit Adams  numerical solver (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Then, we apply predictor-  corrector sampling and leverage a Lagrange interpolation  function as the predictor (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We design an error  distance to measure the accuracy of the estimated noise and  enhance the proposed predictor with an error-robust strategy  to adaptively select the Lagrange bases with lower noise  estimation error (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The whole sampling process is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Implicit Adams Methods  The sampling of DDPMs starts from a prior noise distribu-  tion xt0 ∼ N(0, I), and iteratively denoises xti to xti+1  until time t reaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In the sampling process, the most  time-consuming step is network evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Assuming we  have a pretrained noise estimation model ϵθ, we need to  achieve good generation quality with as few evaluation times  as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We notice that the noise estimation error exists across every  sampling time, especially when time ti is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It limits  previous numerical high-order solvers (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Lu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a) since they are based on  the assumption that the network has no estimation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Previous solvers usually involved observed noise terms to  achieve the analytic form of the sampling scheme that is  critical for sampling convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Thus, they are sensitive  to the errors of estimated noises from pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  In this paper, we explore the potential of implicit Adams  solver (Małgorzata & Marciniak, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Different from  explicit Adams (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 9), implicit Adams involves the unob-  served noise term and the ϵti in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 7 is reformulated as  follows:  ϵti = 1  24(9ϵθ(xti+1, ti+1) + 19ϵθ(xti, ti)  − 5ϵθ(xti−1, ti−1) + ϵθ(xti−2, ti−2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  (10)  It can be noticed that xti+1 can be observed only when  ϵti is achieved, while the ϵti contains unobserved term  ϵθ(xti+1, ti+1), which makes it challenging to solve implicit  equations and may need more time-consuming iteration  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' This greatly limits the implicit Adams method to be a  fast solver for diffusion ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Fortunately, in numerical ODE literature, the sampling effi-  ciency of implicit Adams can be improved with a predictor-  corrector sampling scheme (Diethelm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Specifi-  cally, the predictor makes a rough estimation of unobserved  term ¯ϵθ(xti+1, ti+1) and the corrector derives the precise  xti+1, which can reformulate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='10 as follows:  ϵti = 1  24(9¯ϵθ(xti+1, ti+1) + 19ϵθ(xti, ti)  − 5ϵθ(xti−1, ti−1) + ϵθ(xti−2, ti−2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  (11)  The traditional predictor-corrector utilizes explicit Adams  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 9) to perform predictor to make xti+1 observed so as  to derive ¯ϵθ(xti+1, ti+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' However, it still suffers from the  fixed form that is not robust to the noise estimation errors,  which can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Predictor with Lagrange Interpolation Function  In this paper, we propose to utilize noises observed at pre-  vious sampling steps and construct the Lagrange interpo-  lation function as the predictor to predict unobserved term  ϵθ(xti+1, ti+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In this way, we can design an adaptive  strategy to select Lagrange function bases to construct the  error-robust predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Specifically, we maintain a buffer about all previously esti-  mated noises, which have been observed and need no extra  Error-Robust Adams Solver  Selected  Not Selected  Sampling Process  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Error-robust selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' ∆ϵ is calculated based on  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 15 instead of the training loss in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The sampling NFE is  set to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  computations, and its corresponding time:  {(tn, ϵθ(xtn, tn)), n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='., i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  (12)  The maintained buffer is also called the Lagrange buffer  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Assume that the interpolation order is k, the  selected function bases to construct the Lagrange function  can be written as {(tτm, ϵθ(xtτm, tτm)), m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  The corresponding Lagrange interpolation function can be  formulated as:  lm(t) =  k−1  �  l=0,l̸=m  ( t − tτl  tτm − tτl  ),  Lϵ(t) =  k−1  �  m=0  lm(t) ∗ ϵθ(xtτm , tτm),  (13)  where τl belongs to the maintained Lagrange buffer and  has already been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' At time ti+1, we can derive an  estimation about ϵθ(xti+1, ti+1):  ¯ϵθ(xti+1, ti+1) = Lϵ(ti+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  (14)  With this prediction, we apply the corrector process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  11 to get the ϵti and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 8 to get the denoised sample xti+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  It can be noticed that the proposed predictor makes use of  the observed noise estimations and involves no network eval-  uations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Furthermore, the Lagrange bases in the predictor  can be adaptively selected from those noise estimations with  low errors, which is more error-robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We introduce this  error-robust selection strategy in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Error-Robust Selection Strategy  In this part, our goal is to design an error-robust selection  strategy for the maintained Lagrange buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When the  interpolation order is k, the intuitive selection approach is  to make a fixed selection of the last k estimated noises from  the maintained Lagrange buffer, which means τm = i − m  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' However, we notice that the noise estimation  Algorithm 1 ERA-Solver  1: Input: {ti}N  i=0, k, ϵθ  2: Instantiate: xt0 ∼ N(0, I), buffer Ω = ∅, ∆ϵ = λ  3: Ω = Ω ∪ {(t0, ϵθ(xt0, t0))}  4: for i in 0, 1, · · · , N − 1 do  5:  if i<k − 1 then  6:  Derive xti+1 based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 8 and ϵθ(xti, ti)  7:  Ω = Ω ∪ {(ti+1, ϵθ(xti+1, ti+1))}  8:  else  9:  Calculate {¯τm}k−1  m=0 via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 16  10:  Calculate {τm}k−1  m=0 via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 17 and ∆ϵ  11:  Derive Lagrange function Lϵ via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 13 and τm  12:  ¯ϵθ(xti+1, ti+1) ← Lϵ(ti+1)  13:  Calculate ϵti via Ω, ¯ϵθ(xti+1, ti+1), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 11  14:  Derive xti+1 based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 8 and ϵti  15:  Ω = Ω ∪ {(ti+1, ϵθ(xti+1, ti+1))}  16:  Update ∆ϵ via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 15 and ¯ϵθ(xti+1, ti+1)  17:  end if  18: end for  19: return xtN  error tends to increase as time ti approaches 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 1), in  which case the fixed selection strategy may aggregate the  noise estimation errors from Lagrange buffer and make the  constructed Lagrange function inaccurate for prediction at  time ti+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It motivates us to seek a reasonable measure of  the error in estimated noise so as to select those noises with  low estimation error for Lagrange interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Error Measure for Estimated Noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Since there exists  no ground-truth noise in sampling process, it is hard to mea-  sure the error in estimated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' To this end, we propose to  utilize the observed noise term ϵθ(xti, ti) as the target noise  and the predicted noise term ¯ϵθ(xti, ti) from last sampling  step as the estimated noise to calculate the approximation  error, which can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It can be formulated as  follows:  ∆ϵ = ||ϵθ(xti, ti) − ¯ϵθ(xti, ti)||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  (15)  ϵθ(xti, ti) is observed based on xti, which is achieved via  the ¯ϵθ(xti, ti) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When the error of estimated noise  from pretrained models increases, it tends to be hard for  ¯ϵθ(xti, ti) to approximate ϵθ(xti, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='3,  our error measure in the sampling process shares the same  trend as the error of estimated noises in the training process  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 1), which demonstrates the rationality of the proposed  error measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Selection Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  The high-level idea of our error-  robust selection strategy is that we tend to choose those  estimated noises from Lagrange buffer with low errors mea-  sured by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 15 as the Lagrange bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When sampling  14  12  10  8  3  6  4  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='1  time tiError-Robust Adams Solver  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Generation quality measured by FID ↓ on LSUN-Church, varying the number of function evaluation (NFE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Sampling method \\ NFE  5  10  12  15  20  40  50  100  DDIM (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a)  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='74  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='62  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='77  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='31  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='75  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='24  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='1  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='97  FON (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021)  \\  \\  \\  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='32  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='35  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='44  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='70  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='83  PNDM (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021)  \\  \\  \\  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='31  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='74  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='49  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='89  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='38  DPM-Solver-2 (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a)  310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='49  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='01  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='56  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='68  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='59  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='48  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='60  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='49  DPM-Solver-fast (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a)  346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='38  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='81  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='35  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='52  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='64  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='37  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='19  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='49  ERA-Solver  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='76  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='42  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='15  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='41  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='39  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='26  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='50  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='52  at time ti, the length of the Lagrange buffer is i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We  initialize k indexes uniformly to cover the whole buffer:  �τm = (i/k) ∗ m, m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='., k  (16)  Then we utilize the power function as an index translator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We parameterize the power function with the error measure  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 15) so that the translation of initial indexes can be  formulated as:  τm = ⌊(�τm/i)∆ϵ/λ ∗ i⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  (17)  where λ is a hyperparameter to adjust the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 3, our selection focus more on the be-  ginning of the Lagrange buffer, which tends to be more  accurate, when the error of estimated noises increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In  this way, our selection strategy of buffer makes our La-  grange interpolation function more accurate in an adaptive  way, which contributes to an error-robust Adams solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Overall Sampling Algorithm  In this part, we summarize our sampling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Given a  pretrained diffusion model ϵθ, we sample from a prior noise  xt0 ∼ N(0, I) and iteratively denoise from xti to xti+1  until time ti reaches 0 and xtN is our final generated sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  The sampling algorithm is based on the predict-corrector  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The k represents the Lagrange interpolation order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  For the initialization of Lagrange buffer, the first k sampling  steps are based on DDIM sampling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The details of  the sampling process can be found in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Experiment  In this section, we demonstrate that, as a training-free sam-  pler, ERA-Solver is able to accelerate the sampling process  and be robust for the error from existing pretrained diffusion  models on various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Experiment Setting  We conduct sampling experiments mainly based on three  datasets: Cifar10 (32 × 32) (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2009),  LSUN-Church (256 × 256), (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2015), LSUN-  Bedroom (256 × 256) (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2015), and Celeba (64  × 64) (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We adopt the pretrained models  from DDIM (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a) on three datasets above to  evaluate our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We adopt Fenchel Inception Distance (FID) (Heusel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=',  2017) as the evaluation metric to test the generation quality  of all sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' All evaluation results are based on  50k generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  When sampling on LSUN-Bedroom and LSUN-Church,  we set tN = 10−4 (withdraw the denoising trick (Song  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020b) at the final step) and adopt a uniform timestep  scheme (select ti uniformly) with discrete-time pretrained  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We set λ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 17 emprically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We set k = 4  for LSUN-Church and k = 3 for LSUN-Bedroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  When sampling on Cifar10, we follow DPM-Solver (Lu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a) and evaluate on both tN = 10−3 and tN =  10−4 settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We adopt logSNR timestep scheme applied  in DPM-Solver (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It has been demonstrated  (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a) that logSNR timestep scheme is more  suitable for Cifar10, which causes lower discretization er-  rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We set λ = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 17 and k = 4 emprically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The  experiment details and results on Celeba can be found in  Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Comparison with Previous Fast Sampling Methods  In this subsection, we show the comparison results of our  ERA-Solver and other training-free sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We  first compare ERA-Solver with previous fast sampling meth-  ods on LSUN-Church and LSUN-Bedroom datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We  directly use the code released in (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=',  2022a) to implement PNDM (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021), FON(Liu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021), and DPM-Solver(Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a) methods to  generate the samples for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We compare the sam-  pling quality of the training-free sampling methods based on  5, 10, 12, 15, 20, 40, 50, and 100 NFEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Note that PNDM  and FON (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021) use 4-order Runge-Kutta for  the first three sampling steps and each step costs 4 NFEs,  which means their methods need at least 13 NFEs to per-  form sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Thus, their FID results are missing on 5, 10,  and 12 NFEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' For DPM-Solver, we adopt a 2-order setting  (DPM-Solver-2) and its fast scheme (DPM-Solver-fast) for  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Error-Robust Adams Solver  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Generation quality measured by FID ↓ on LSUN-Bedroom, varying the number of function evaluation (NFE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Sampling method \\ NFE  5  10  12  15  20  40  50  100  DDIM (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a)  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='18  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='81  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='24  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='57  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='24  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='95  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='76  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='84  FON (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021)  \\  \\  \\  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='89  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='85  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='08  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='16  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='81  PNDM (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021)  \\  \\  \\  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='59  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='19  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='46  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='87  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='77  DPM-Solver-2 (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a)  354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='12  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='95  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='98  13  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='17  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='91  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='74  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='62  DPM-Solver-fast (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a)  376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='69  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='03  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='47  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='79  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='26  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='35  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='52  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='31  ERA-Solver  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='69  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='44  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='53  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='10  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='99  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='45  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='39  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='72  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Generation quality measured by FID ↓ on Cifar10, varying the number of function evaluation (NFE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Sampling method \\ NFE  5  10  12  15  20  40  50  100  DDPM(Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020)  \\  278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='67  246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='29  197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='63  137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='34  \\  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='63  \\  Analytic-DDPM (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022)  \\  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='03  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='69  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='82  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='35  \\  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='34  \\  Analytic-DDIM (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022)  \\  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='74  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='68  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='16  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='20  \\  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='28  \\  DDIM(Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a)  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='41  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='58  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='02  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='92  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='94  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='92  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='73  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='65  PNDM(Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021)  \\  \\  \\  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='47  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='31  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='35  DPM-Solver-fast (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a) (tN = 10−3)  269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='36  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='37  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='65  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='78  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='28  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='80  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='03  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='88  DPM-Solver-fast (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a) (tN = 10−4)  330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='08  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='32  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='31  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='75  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='80  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='51  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='54  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='44  ERA-Solver (tN = 10−3)  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='45  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='38  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='86  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='79  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='97  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='91  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  ERA-Solver (tN = 10−4)  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='31  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='84  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='84  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='45  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='42  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='51  LSUN-Church.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  The comparison results on LSUN-  Church are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When tested on extremely  few NFE like 5, our ERA-Solver obtains better FID 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='76,  achieving 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2% improvement compared with DDIM (Song  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When tested on a few NFE like 10 and 15, our  ERA-Solver obtains 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='42 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='41 FID results, achieving  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='9% and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='4% improvements compared with the previ-  ous best results of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='62 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When tested on larger  NFE, our ERA-Solver achieves state-of-the-art generation  quality with FID 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='39, compared with the previous best  result of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  LSUN-Bedroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  The comparison results on LSUN-  Bedroom are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When tested on extremely  small NFE like 5, our ERA-Solver obtains better FID 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='69,  achieving 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='3% improvement compared with 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='18 from  DDIM (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When tested on small NFEs like  10 and 12, our ERA-Solver obtains 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='44 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='53 FID re-  sults, achieving 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='1% and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5% improvements compared  with the previous best results of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='03 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When  tested on larger NFE, our ERA-Solver achieves state-of-the-  art generation quality with FID 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='39, compared with the  previous best result of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Cifar10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  The comparison results on Cifar10 are shown  in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Our ERA-Solver achieves better results at 5,  10, 12, 20, and 40 NFEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' When evaluated on 10 NFE, our  ERA-Solver obtains 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='14 FID result, achieving a 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='15%  improvement compared with the previous best result of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Fixed  NFE=10  NFE=20  NFE=30  ERS  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Generation quality comparison between error-robust se-  lection strategy (ERS) with λ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0 and fixed strategy (Fixed)  that select the last k estimated noises fixedly based on 5-order  Lagrange interpolation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Ablation Study  Effectiveness of error-robust selection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We  compare our error-robust selection strategy and fixed strat-  egy which selects the last k estimated noises (τm = i − m  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 13) based on various orders of Lagrange interpolation  function in the predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The results can be seen in Tab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We further visualize the samples generated from two strate-  gies based on a 5-order Lagrange interpolation function, the  results of which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It can be concluded  that our error-robust strategy is more effective than intuitive  strategies, especially when Lagrange function order is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Error-Robust Adams Solver  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' FID comparison between error-robust selection strategy  (ERS) and fixed strategy (fixed) that select the last k estimated  noises fixedly based on various Lagrange function orders (k =  3, 4, 5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' All experiments are conducted on LSUN-Church.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Method\\ NFE  10  15  20  40  50  ERA-Solver-3 fixed  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='83  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='52  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='72  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='58  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='69  ERS  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='03  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='63  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='11  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='37  ERA-Solver-4 fixed 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='56  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='72  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='99  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='89  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='92  ERS  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='42  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='41  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='39  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='26  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  ERA-Solver-5 fixed  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='7  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='36 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='58  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='09  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='12  ERS 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='85  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='48  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='28  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='26  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='64  ERA-Solver-6 fixed 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='91 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='69 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='6 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='22 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='44  ERS 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='79  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='41  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='41  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='43  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='7  Effectiveness of error measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We compare different  power scales based on the proposed error measure and var-  ious constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The results can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It can  be observed that the selection strategy based on our error  measure has better performance than those based on the  constant scale in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Discussion  Why FID gets worse when NFE >50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We can notice  that our solver performs worse when NFE increases to 50,  especially on LSUN dataset (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 2 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It can be  understood that FID results fluctuate around a value, which  is verified on PNDM (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Why ERA-Solver performs worse on Cifar10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  From  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 3, we can notice that ERA-Solver performs slightly  worse than DPM-Solver and PNDM when NFE increases,  which is different from the results on LSUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The reason  behind it is the resolution of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Cifar10 has a very low res-  olution (32 × 32) while LSUN has a higher resolution (256  × 256), which means the LSUN dataset is more informative  than Cifar10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Thus, the model tends to have lower train-  ing error (noise estimation error) when trained on Cifar10,  leading to the slightly worse performance of ERA-Solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Rationality of selection function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  In this paper, we de-  sign the selection function for the error robustness of the  sampling solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The changing trend of noise estimation  error with timestep (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 1) inspires us to choose the power  function as an error-robust selection function and experi-  ments demonstrate its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' There may exist a better  selection function or a learning-based approach to selecting  Lagrange functions and we leave it for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Why ERA-Solver has better error robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We no-  tice that the previous numeric solvers depend on the assump-  tion that the pretrained diffusion model achieves accurate  10  15  20  25  30  35  40  45  50  NFE  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  FID  ERS = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  ERS = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  ERS = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2  Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='8  Constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  Constant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Generation quality measured by FID ↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We compare our  error-aware scale (∆ϵ/λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 17) and constant scale (replace  ∆ϵ/λ with a constant) to demonstrate the effectiveness of the pro-  posed error measure based on the 3-order Lagrange interpolation  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' All FID results are evaluated on LSUN-Church with  5000 generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  noise estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In this paper, we demonstrate that there  exists an obvious estimation error (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 1) from pretrained  model, which limits the sampling efficiency of previous fast  solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Combined with the Lagrange interpolation as the predictor  and an error-robust selection strategy, we show that the Im-  plicit Adams solver has the potential to be error-robust so as  to achieve better sampling efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The predictor based on  Lagrange interpolation makes sure our solver has the conver-  gence property of the numerical solver (predictor-corrector  sampling scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The error-robust selection strategy adap-  tively introduces relatively accurate estimated noises and  mitigates the error accumulation in the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  More analysis can be found in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Conclusion  In this paper, we propose an error-robust Adams solver  (ERA-Solver) that consists of a predictor and a corrector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We leverage the Lagrange interpolation function to perform  the predictor, and further propose an error measure for the  sampling process and an error-robust strategy to enhance  the predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Experiments demonstrate that ERA-Solver  achieves better generation quality on LSUN-Church and  LSUN-Bedroom datasets at all NFEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Our ERA-Solver  might be misused with powerful diffusion models (Rombach  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Kim & Ye, 2021) to generate adverse fake  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We will restrict the usage of our solver and share it  with the fake detection community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  References  Atkinson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
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+page_content='  arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
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+page_content=' Very deep vaes generalize autoregressive models  and can outperform them on images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
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+page_content=' Nonlinear Dynamics, 29(1):3–22,  2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
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+page_content=', and Bengio, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2661, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Gu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Bao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Wen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Chen,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Yuan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', and Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Vector quantized diffusion  model for text-to-image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' In Proceedings of the  IEEE/CVF Conference on Computer Vision and Pattern  Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 10696–10706, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Heusel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
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+page_content='  Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Seff, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Funkhouser, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', and  Xiao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Lsun: Construction of a large-scale image  dataset using deep learning with humans in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='03365, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Ablation Study on Cifar10  In this part, we provide ablation experiments on Cifar10  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We replace our error-robust selection strategy with  the fixed selection strategy that fixedly selects the last k  estimated noises previously saved in the Lagrange buffer at  every sampling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' From  the table, we can see that our selection strategy achieves  better FID generation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We also conduct ablation studies for proposed error mea-  sures in the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We parameterize the power  function with various constants instead of our error mea-  sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The comparison results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Our  error measure can enhance the selection strategy with better  generation results in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Additional Results on Celeba  The comparison results on Celeba are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  When tested on a few NFEs like 10 and 12, our ERA-Solver  obtains 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='06 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='67 FID results, achieving 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='8% and  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='4% improvements compared with the previous best re-  sults of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='92 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It can also be observed that FID result  of ERA-Solver converges at about NFE 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Compared with  DPM-Solver which converges at NFE 36, ERA-Solver has  better convergence speed, while achieving comparable gen-  eration quality (FID 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='75 vs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Analysis of Error Robustness  Since data distributions like LSUN are high-dimensional  and complex, it is difficult to seek a ground truth for the pre-  dicted noise in the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Thus, we demonstrate  the error robustness of ERA-Solver from another perspec-  tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Assume that we achieve a generated sample xgen  0  from  a solver, we can utilize the diffusion process to add noise  to the sample, remapping the sample obtained from the de-  noising process back to the noise space to derive xgen  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We  measure the error robustness of the solver as the following:  ||ϵ − ϵθ(xgen  t  , t)||2,  (18)  The estimated noise from pretrained model can be seen  as the stepping direction of the noisy sample (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=',  2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' If a solver is not error-robust, its generated samples  will deviate from the original generation path in the sam-  pling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Since the generation process can be seen as  the reverse process of the diffusion process, the deviation of  generated samples will increase the error in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We select normal Implicit Adam solver (Małgorzata &  Marciniak, 2006), DPM-Solver (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a), and ERA-  Solver for comparisons and generate a batch of samples to  calculate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 15 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Three solvers share the same  sampling NFE, random seed, and the pretrained model ϵθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Error-Robust Adams Solver  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' FID comparison between error-robust selection strategy  (ERS) and fixed strategy (fixed) based on various Lagrange func-  tion orders (k = 3, 4, 5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Method\\ NFE  10  15  20  50  ERA-Solver-3  fixed  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='95  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='62  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='24  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  ERS  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='79  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='31  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='07  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='94  ERA-Solver-4  fixed  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='46  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='98  ERS  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='14  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='86  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='79  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='91  ERA-Solver-5  fixed  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='21  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='11  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='47  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='99  ERS  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='26  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='73  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='69  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='98  ERA-Solver-6  fixed  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='34  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='58  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='39  118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='38  ERS  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='26  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='16  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='73  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='04  10  15  20  25  30  35  40  45  50  NFE  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  FID  ERS = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  ERS = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  ERS = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2  Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  Constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='8  Constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0  Constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2  Constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Generation quality measured by FID ↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We compare our  error-aware scale (∆ϵ/λ) and constant scale based on the 4-order  Lagrange function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' All FID results are evaluated on Cifar10 with  50k generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 7, from which we can see that  ERA-Solver achieves lower error in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Qualitative Results  We sample and visualize the generated samples from LSUN-  Church and LSUN-Bedroom datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We evaluate DDIM,  DPM-Solver, and our method based on 5, 8, 10, 12, 15, and  20 NFEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Since PNDM (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021) requires at least 13  NFEs to perform sampling, we only compared our method  with DDIM (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a) and DPM-Solver-fast (Lu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We set λ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0 and apply a 4-order  Lagrange interpolation function (k = 4) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  The comparison results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Compared with DDIM, our ERA-Solver can obtain sharper  textures, benefiting from its high precision of the implicit  Adams scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Compared with DPM-Solver-fast, our ERA-  Solver can avoid excessive contrast and exposure, achieving  more natural texture details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We also provide generated samples based on the 4-order  ERA-Solver with λ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0 on Cifar10, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Results on Stable Diffusion  We conduct generation comparison based on large-scale  latent diffusion mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', Stable Diffusion (Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We apply the improved version (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022b) of  DPM-Solver for better conditional generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The PNDM  and improved DPM-Solver are all encapsulated in diffusers  (von Platen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' We directly apply them by diffusers  to make sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' For ERA-Solver, we set k = 4 and  λ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 11 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It  can be observed that ERA-Solver can generate promising  images when NFE is 15, which is faster than DPM-Solver  and PNDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  We also provide computation time for each sampling pro-  cess, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' It can be observed that at NFE  15, ERA-Solver consumes slightly more time (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='08s) than  DPM-Solver, which can be contributed to the cost of main-  taining the Lagrange buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The cost of the Lagrange buffer  will increase with NFE increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' However, ERA-Solver  has been able to generate exquisite images on NFE 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Thus,  the computation cost of the Lagrange buffer can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Error-Robust Adams Solver  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='9  time t  0  50  100  150  200  250  300  350  ||  (xgen  t  , t)||2  DPM-Solver  ERA-Solver (Ours)  Normal Implicit Adams  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The error comparison between three fast solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' For ERA-Solver, we set k = 4 and λ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The random seed and pretrained  model are shared with three solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Generation quality measured by FID ↓ on Celeba, varying the number of function evaluation (NFE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Sampling method \\ NFE  5  10  12  15  20  40  50  100  DDIM (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2020a)  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='26  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='4  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='23  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='63  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='62  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='87  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='08  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5  Analytic-DDPM (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022)  \\  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='99  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='27  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='80  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='14  \\  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='23  \\  Analytic-DDIM (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022)  \\  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='62  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='90  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='29  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='45  \\  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='13  \\  PNDM (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021)  \\  \\  \\  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='91  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='59  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='06  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='45  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='97  DPM-Solver-fast (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a)  \\  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='92  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='05  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='82  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='71 (NFE = 36)  ERA-Solver  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='32  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='06  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='67  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='99  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='69  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='73  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='72  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Generated samples from ERA-Solver-4 (20 NFE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Computation time of sampling on Stable Diffusion, vary-  ing different solvers and NFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Sampling Method \\ NFE  15  25  50  PNDM (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2021)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='69  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='74  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='13  DPM-Solver (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022a)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='86  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='76  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='30  ERA-Solver  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='94  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='05  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='01  Error-Robust Adams Solver  DDIM  DPM-  Solver-  fast\u200b  Ours  NFE=5  NFE=8  NFE=10  NFE=12  NFE=15  NFE=20  DDIM  DPM-  Solver-  fast\u200b  Ours  DDIM  DPM-  Solver-  fast\u200b  Ours  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Generation quality comparison with 5, 8, 10, 12, 15, and 20 NFEs on LSUN-Church dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Error-Robust Adams Solver  DDIM  DPM-  Solver-  fast\u200b  Ours  NFE=5  NFE=8  NFE=10  NFE=12  NFE=15  NFE=20  DDIM  DPM-  Solver-  fast\u200b  Ours  DDIM  DPM-  Solver-  fast\u200b  Ours  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Generation quality comparison with 5, 8, 10, 12, 15, and 20 NFEs on LSUN-Bedroom dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Error-Robust Adams Solver  PNDM  DPM-Solver  Ours  NFE=15  NFE=25  NFE=50  Prompt: Cute and adorable ferret wizard, wearing coat and suit  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Samples using the pretrained Stable-Diffusion (Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022) with a classifier-free guidance scale 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5 (the default  setting), varying different solvers and NFEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The main part of input prompt is: “Cute and adorable ferret wizard, wearing coat and suit”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='  Error-Robust Adams Solver  NFE=15  NFE=25  NFE=50  PNDM  DPM-Solver  Ours  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' Samples using the pretrained Stable-Diffusion (Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=', 2022) with a classifier-free guidance scale 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content='5 (the default  setting), varying different solvers and NFEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}
+page_content=' The main part of input prompt is: “A beautiful mansion beside a waterfall in the woods”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf'}