Add files using upload-large-folder tool

parse/dev/42zs3qa2kpy/42zs3qa2kpy.md

@@ -0,0 +1,550 @@
+# OFFLINE REINFORCEMENT LEARNING VIA HIGHFIDELITY GENERATIVE BEHAVIOR MODELING
+
+Huayu Chen1, Cheng ${ { \bf L } } { \bf u } ^ { 1 }$ , Chengyang $\mathbf { Y i n g ^ { 1 } }$ , Hang $\mathbf { S u } ^ { 1 , 2 }$ ∗, $ { \mathbf { J u n } }  { \mathbf { Z } }  { \mathbf { h u } } ^ { 1 , 2 * }$   
+1Department of Computer Science & Technology, Institute for AI, BNRist Center,   
+Tsinghua-Bosch Joint ML Center, THBI Lab, Tsinghua University   
+2Pazhou Lab, Guangzhou, 510330, China   
+chenhuay21@mails.tsinghua.edu.cn   
+{lucheng.lc15,yingcy17}@gmail.com   
+{suhangss,dcszj}@tsinghua.edu.cn
+
+# ABSTRACT
+
+In offline reinforcement learning, weighted regression is a common method to ensure the learned policy stays close to the behavior policy and to prevent selecting out-of-sample actions. In this work, we show that due to the limited distributional expressivity of policy models, previous methods might still select unseen actions during training, which deviates from their initial motivation. To address this problem, we adopt a generative approach by decoupling the learned policy into two parts: an expressive generative behavior model and an action evaluation model. The key insight is that such decoupling avoids learning an explicitly parameterized policy model with a closed-form expression. Directly learning the behavior policy allows us to leverage existing advances in generative modeling, such as diffusionbased methods, to model diverse behaviors. As for action evaluation, we combine our method with an in-sample planning technique to further avoid selecting outof-sample actions and increase computational efficiency. Experimental results on D4RL datasets show that our proposed method achieves competitive or superior performance compared with state-of-the-art offline RL methods, especially in complex tasks such as AntMaze. We also empirically demonstrate that our method can successfully learn from a heterogeneous dataset containing multiple distinctive but similarly successful strategies, whereas previous unimodal policies fail. The source code is provided at https://github.com/ChenDRAG/SfBC.
+
+# 1 INTRODUCTION
+
+Offline reinforcement learning seeks to solve decision-making problems without interacting with the environment. This is compelling because online data collection can be dangerous or expensive in many realistic tasks. However, relying entirely on a static dataset imposes new challenges. One is that policy evaluation is hard because the mismatch between the behavior and the learned policy usually introduces extrapolation error (Fujimoto et al., 2019). In most offline tasks, it is difficult or even impossible for the collected transitions to cover the whole state-action space. When evaluating the current policy via dynamic programming, leveraging actions that are not presented in the dataset (out-of-sample) may lead to highly unreliable results, and thus performance degrade. Consequently, in offline RL it is critical to stay close to the behavior policy during training.
+
+Recent advances in model-free offline methods mainly include two lines of work. The first is the adaptation of existing off-policy algorithms. These methods usually include value pessimism about unseen actions or regulations of feasible action space (Fujimoto et al., 2019; Kumar et al., 2019; 2020). The other line of work (Peng et al., 2019; Wang et al., 2020; Nair et al., 2020) is derived from constrained policy search and mainly trains a parameterized policy via weighted regression. Evaluations of every state-action pair in the dataset are used as regression weights.
+
+The main motivation behind weighted policy regression is that it helps prevent querying out-of-sample actions (Nair et al., 2020; Kostrikov et al., 2022). However, we find that this argument is untenable in certain settings. Our key observation is that policy models in existing weighted policy regression methods are usually unimodal Gaussian models and thus lack distributional expressivity, while in the real world collected behaviors can be highly diverse. This distributional discrepancy might eventually lead to selecting unseen actions. For instance, given a bimodal target distribution, fitting it with a unimodal distribution unavoidably results in covering the low-density area between two peaks. In Section 3.1, we empirically show that lack of policy expressivity may lead to performance degrade.
+
+Ideally, this problem could be solved by switching to a more expressive distribution class. However, it is nontrivial in practice since weighted regression requires exact and derivable density calculation, which places restrictions on distribution classes that we can choose from. Especially, we may not know what the behavior or optimal policy looks like in advance.
+
+To overcome the limited expressivity problem, we propose to decouple the learned policy into two parts: an expressive generative behavior model and an action evaluation model. Such decoupling avoids explicitly learning a policy model whose target distribution is difficult to sample from, whereas learning a behavior model is much easier because sampling from the behavior policy is straightforward given the offline dataset collected by itself. Access to data samples from the target distribution is critical because it allows us to leverage existing advances in generative methods to model diverse behaviors. To sample from the learned policy, we use importance sampling to select actions from candidates proposed by the behavior model with the importance weights computed by the action evaluation model, which we refer to as Selecting from Behavior Candidates (SfBC).
+
+However, the selecting-from-behavior-candidates approach introduces new challenges because it requires modeling behaviors with high fidelity, which directly determines the feasible action space. A prior work (Ghasemipour et al., 2021) finds that typically-used VAEs do not align well with the behavior dataset, and that introducing building-in good inductive biases in the behavior model improves the algorithm performance. Instead, we propose to learn from diverse behaviors using a much more expressive generative modeling method, namely diffusion probabilistic models (Ho et al., 2020), which have recently achieved great success in modeling diverse image distributions, outperforming other existing generative models (Dhariwal & Nichol, 2021). We also propose a planning-based operator for Q-learning, which performs implicit planning strictly within dataset trajectories based on the current policy, and is provably convergent. The planning scheme greatly reduces bootstrapping steps required for dynamic programming and thus can help to further reduce extrapolation error and increase computational efficiency.
+
+The main contributions of this paper are threefold: 1. We address the problem of limited policy expressivity in conventional methods by decoupling policy learning into behavior learning and action evaluation, which allows the policy to inherit distributional expressivity from a diffusion-based behavior model. 2. The learned policy is further combined with an implicit in-sample planning technique to suppress extrapolation error and assist dynamic programming over long horizons. 3. Extensive experiments demonstrate that our method achieves competitive or superior performance compared with state-of-the-art offline RL methods, especially in sparse-reward tasks such as AntMaze.
+
+# 2 BACKGROUND
+
+# 2.1 CONSTRAINED POLICY SEARCH IN OFFLINE RL
+
+Consider a Markov Decision Process (MDP), described by a tuple $\langle S , \mathcal { A } , P , r , \gamma \rangle$ . $s$ denotes the state space and $\mathcal { A }$ is the action space. $P ( \pmb { s } ^ { \prime } | \pmb { s } , \pmb { a } )$ and $r ( s , a )$ respectively represent the transition and reward functions, and $\gamma \in ( 0 , 1 ]$ is the discount factor. Our goal is to maximize the expected discounted return $J ( \pi ) = \mathbb { E } _ { s \sim \rho _ { \pi } ( s ) } \mathbb { E } _ { \pmb { a } \sim \pi ( \cdot | s ) } \left[ r ( s , \pmb { a } ) \right]$ of policy $\pi$ , where $\begin{array} { r } { \rho _ { \pi } ( \pmb { \mathscr { s } } ) = \sum _ { n = 0 } ^ { \infty } \gamma ^ { n } \bar { p } _ { \pi } ( \pmb { \mathscr { s } } _ { n } = \pmb { \mathscr { s } } ) } \end{array}$ is the discounted state visitation frequencies induced by the policy $\pi$ (Sutton $\&$ Barto, 1998).
+
+According to the policy gradient theorem (Sutton et al., 1999), given a parameterized policy $\pi _ { \theta }$ , and the policy’s state-action function $Q ^ { \pi }$ , the gradient of $J ( \pi _ { \theta } )$ can be derived as:
+
+$$
+\nabla _ { \theta } J ( \pi _ { \theta } ) = \int _ { S } \rho _ { \pi } ( s ) \int _ { A } \nabla _ { \theta } \pi _ { \theta } ( \boldsymbol { a } | s ) Q ^ { \pi } ( s , \boldsymbol { a } ) \mathrm { d } a \ \mathrm { d } s .
+$$
+
+When online data collection from policy $\pi$ is not possible, it is difficult to estimate $\rho _ { \pi } ( s )$ in Equation 1, and thus the expected value of the Q-function $\begin{array} { r } { \eta ( \bar { \boldsymbol { \pi } } _ { \boldsymbol { \theta } } ) : = \int _ { S } \rho _ { \pi } ( \pmb { \mathscr { s } } ) \int _ { \mathcal { A } } \pi _ { \boldsymbol { \theta } } ( \pmb { a } | \pmb { \mathscr { s } } ) Q ^ { \pi } ( \pmb { \mathscr { s } } , \dot { \pmb { a } } ) } \end{array}$ . Given a static dataset ${ \mathcal { D } } ^ { \mu }$ consisting of multiple trajectories $\{ ( s _ { n } , a _ { n } , r _ { n } ) \}$ collected by a behavior policy $\mu ( { \pmb a } | { \pmb s } )$ , previous off-policy methods (Silver et al., 2014; Lillicrap et al., 2016) estimate $\eta ( \pi _ { \theta } )$ with a surrogate objective $\hat { \eta } ( \pi _ { \boldsymbol { \theta } } )$ by replacing $\rho _ { \pi } ( s )$ with $\rho _ { \mu } ( s )$ . In offline settings, due to the importance of sticking with the behavior policy, prior works (Peng et al., 2019; Nair et al., 2020) explicitly constrain the learned policy $\pi$ to be similar to $\mu$ , while maximizing the expected value of the Q-functions:
+
+$$
+\arg \operatorname* { m a x } _ { \pi } \quad \int _ { S } \rho _ { \mu } ( s ) \int _ { \mathcal { A } } \pi ( a | s ) Q _ { \phi } ( s , a ) \mathrm { d } a \mathrm { d } s - \frac { 1 } { \alpha } \int _ { S } \rho _ { \mu } ( s ) D _ { \mathrm { K L } } \left( \pi ( \cdot | s ) \right| | \mu ( \cdot | s ) ) \mathrm { d } s .
+$$
+
+The first term in Equation 2 corresponds to the surrogate objective $\hat { \eta } ( \pi _ { \boldsymbol { \theta } } )$ , where $Q _ { \phi } ( s , \pmb { a } )$ is a learned Q-function of the current policy $\pi$ . The second term is a regularization term to constrain the learned policy within support of the dataset ${ \mathcal { D } } ^ { \mu }$ with $\alpha$ being the coefficient.
+
+# 2.2 POLICY IMPROVEMENT VIA WEIGHTED REGRESSION
+
+The optimal policy $\pi ^ { * }$ for Equation 2 can be derived (Peters et al., 2010; Peng et al., 2019; Nair et al., 2020) by use of Lagrange multiplier:
+
+$$
+\pi ^ { * } ( { \pmb a } | { \pmb s } ) = \frac { 1 } { Z ( { \pmb s } ) } \mu ( { \pmb a } | { \pmb s } ) \exp \left( \alpha Q _ { \phi } ( { \pmb s } , { \pmb a } ) \right) ,
+$$
+
+where $Z ( s )$ is the partition function. Equation 3 forms a policy improvement step.
+
+Directly sampling from $\pi ^ { * }$ requires explicitly modeling behavior $\mu$ , which itself is challenging in continuous action-space domains since $\mu$ can be very diverse. Prior methods (Peng et al., 2019; Wang et al., 2020; Chen et al., 2020) bypass this issue by projecting $\pi ^ { * }$ onto a parameterized policy $\pi _ { \theta }$ :
+
+$$
+\begin{array} { r l } { \underset { \theta } { \mathrm { a r g } \mathrm { m i n } } } & { \mathbb { E } _ { s \sim \mathcal { D } ^ { \mu } } \left[ D _ { \mathrm { K L } } \left( \pi ^ { * } ( \cdot | s ) | | \pi _ { \theta } ( \cdot | s ) \right) \right] } \\ { \underset { \theta } { = \mathrm { a r g } \mathrm { m a x } } } & { \mathbb { E } _ { ( s , a ) \sim \mathcal { D } ^ { \mu } } \left[ \cfrac { 1 } { Z ( s ) } \mathrm { l o g } \pi _ { \theta } ( a | s ) \exp \left( \alpha Q _ { \phi } ( s , a ) \right) \right] . } \end{array}
+$$
+
+Such a method is usually referred to as weighted regression, with $\exp { ( \alpha Q _ { \phi } ( s , pmb { a } ) ) }$ being the regression weights.
+
+Although weighted regression avoids the need to model the behavior policy explicitly, it requires calculating the exact density function $\pi _ { \boldsymbol { \theta } } ( \pmb { a } | \pmb { s } )$ as in Equation 4. This constrains the policy $\pi _ { \theta }$ to distribution classes that have a tractable expression for the density function. We find this in practice limits the model expressivity and could be suboptimal in some cases (Section 3.1).
+
+# 2.3 DIFFUSION PROBABILISTIC MODEL
+
+Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2021b) are generative models by first defining a forward process to gradually add noise to an unknown data distribution $p _ { 0 } ( { \pmb x } _ { 0 } )$ and then learning to reverse it. The forward process $\{ \pmb { x } ( t ) \} _ { t \in [ 0 , T ] }$ is defined by a stochastic differential equation (SDE) $\mathrm { d } \pmb { x } _ { t } = f ( \pmb { x } _ { t } , t ) \mathrm { d } t + g ( t ) \mathrm { d } \pmb { w } _ { t }$ , where ${ \pmb w } _ { t }$ is a standard Brownian motion and $f ( t ) , g ( t )$ are hand-crafted functions (Song et al., 2021b) such that the transition distribution $p _ { t 0 } ( \pmb { x } _ { t } | \pmb { x } _ { 0 } ) = \mathcal { N } ( \pmb { x } _ { t } | \alpha _ { t } \pmb { x } _ { 0 } , \sigma _ { t } ^ { 2 } \pmb { I } )$ for some $\alpha _ { t } , \sigma _ { t } > 0$ and $p _ { T } ( \pmb { x } _ { T } ) \approx \mathcal { N } ( \pmb { x } _ { T } | 0 , \pmb { I } )$ . To reverse the forward process, diffusion models define a scored-based model $\scriptstyle { \pmb { s } } \theta$ and optimize the parameter $\theta$ by:
+
+$$
+\underset { \theta } { \arg \operatorname* { m i n } } \quad \mathbb { E } _ { t , { \boldsymbol { x } } _ { 0 } , { \boldsymbol { \epsilon } } } [ \| { \boldsymbol { \sigma } } _ { t } { \bf s } _ { \theta } ( { \boldsymbol { x } } _ { t } , t ) + { \boldsymbol { \epsilon } } \| _ { 2 } ^ { 2 } ] ,
+$$
+
+where $t \sim \mathcal { U } ( 0 , T )$ $\begin{array} { r } { \hat { \mathbf { \Phi } } ) , \pmb { x } _ { 0 } \sim p _ { 0 } ( \pmb { x } _ { 0 } ) , \pmb { \epsilon } \sim \mathcal { N } ( 0 , \pmb { I } ) , \pmb { x } _ { t } = \alpha _ { t } \pmb { x } _ { 0 } + \sigma _ { t } \pmb { \epsilon } . } \end{array}$
+
+Sampling by diffusion models can be alternatively viewed as discretizing the diffusion ODEs (Song et al., 2021b), which are generally faster than discretizing the diffusion SDEs (Song et al., 2021a; Lu et al., 2022). Specifically, the sampling procedure needs to first sample a pure Gaussian $x _ { T } \sim$ $\mathcal { N } ( 0 , \pmb { I } )$ , and then solve the following ODE from time $T$ to time 0 by numerical ODE solvers:
+
+$$
+\mathrm { d } { \pmb x } _ { t } = \bigg [ f ( { \pmb x } _ { t } , t ) - \frac { 1 } { 2 } g ^ { 2 } ( t ) { \pmb s } _ { \theta } ( { \pmb x } _ { t } , t ) \bigg ] \mathrm { d } t .
+$$
+
+Then the final solution $\scriptstyle { \pmb x } _ { 0 }$ at time $0$ is the sample from the diffusion models.
+
+# 3 METHOD
+
+We propose a Selecting-from-Behavior-Candidates (SfBC) approach to address the limited expressivity problem in offline RL. Below we first motivate our method by highlighting the importance of a distributionally expressive policy in learning from diverse behaviors. Then we derive a high-level solution to this problem from a generative modeling perspective.
+
+# 3.1 LEARNING FROM DIVERSE BEHAVIORS
+
+In this section, we show that the weighted regression broadly used in previous works might limit the distributional expressivity of the policy and lead to performance degrade. As described in Section 2.2, conventional policy regression methods project the optimal policy $\pi ^ { * }$ in Equation 3 onto a parameterized policy set. In continuous action-space domains, the projected policy is usually limited to a narrow range of unimodal distributions (e.g., squashed Gaussian), whereas the behavior policy could be highly diverse (e.g., multimodal). Lack of expressivity directly prevents the RL agent from exactly mimicking a diverse behavior policy. This could eventually lead to sampling undesirable out-of-sample actions during policy evaluation and thus large extrapolation error. Even if Q-values can be accurately estimated, an inappropriate unimodal assumption about the optimal policy might still prevent extracting a policy that has multiple similarly rewarding but distinctive strategies.
+
+We design a simple task named Bidirectional Car to better explain this point. Consider an environment where a car placed in the middle of two endpoints can go either side to gain the final reward. If an RL agent finds turning left and right similarly rewarding, by incorrectly assuming a unimodal distribution of the behavior policy, it ends up staying put instead of taking either one of the optimal actions (Figure 1). As a result, unimodal policies fail to completely solve this task or loss diversity whereas a more distributionally expressive policy easily succeeds.
+
+![](images/591df92c0ef0d29924a2012250a43e016d1fe0ad6c22f9897283aff063517013.jpg)  
+Figure 1: Illustration of the Bidirectional-Car task and comparison between SfBC and unimodal policies. See Section 6.2 for experimental details.
+
+We therefore deduce that distributional expressivity is a necessity to enable diverse behavior learning. To better model the complex behavior policy, we need more powerful generative modeling for the policy distribution, instead of the simple and unimodal Gaussians.
+
+# 3.2 SELECTING FROM BEHAVIOR CANDIDATES
+
+In this section, we provide a generative view of how to model a potentially diverse policy. Specifically, in order to model $\pi ^ { * }$ with powerful generative models, essentially we need to perform maximum likelihood estimation for the model policy $\pi _ { \theta }$ , which is equivalent to minimizing KL divergence between the optimal and model policy:
+
+$$
+\begin{array} { r l } { \underset { \theta } { \arg \operatorname* { m a x } } } & { \mathbb { E } _ { s \sim \mathcal { D } ^ { \mu } } \mathbb { E } _ { a \sim \pi ^ { * } ( \cdot | s ) } \left[ \log \pi _ { \theta } ( a | s ) \right] \Leftrightarrow \underset { \theta } { \arg \operatorname* { m i n } } \quad \mathbb { E } _ { s \sim \mathcal { D } ^ { \mu } } \left[ D _ { \mathrm { K L } } \left( \pi ^ { * } ( \cdot | s ) | | \pi _ { \theta } ( \cdot | s ) \right) \right] . } \end{array}
+$$
+
+However, drawing samples directly from $\pi ^ { * }$ is difficult, so previous methods (Peng et al., 2019; Nair et al., 2020; Wang et al., 2020) rely on the weighted regression as described in Equation 4.
+
+The main reason that limits the expressivity of $\pi _ { \theta }$ is the need of calculating exact and derivable density function $\pi _ { \boldsymbol { \theta } } ( \pmb { a } | \pmb { s } )$ in policy regression, which places restrictions on distribution classes that we can choose from. Also, we might not know what the behavior or optimal policy looks like previously.
+
+Our solution is based on a key observation that directly parameterizing the policy $\pi$ is not necessary. To better model a diverse policy, we propose to decouple the learning of $\pi$ into two parts. Specifically, we leverage Equation 3 to form a policy improvement step:
+
+$$
+\pi ( { \pmb a } | { \pmb s } ) \propto \mu _ { \theta } ( { \pmb a } | { \pmb s } ) \exp \left( \alpha Q _ { \phi } ( { \pmb s } , { \pmb a } ) \right) .
+$$
+
+One insight of the equation above is that minimizing KL divergence between $\mu$ and $\mu _ { \theta }$ is much easier compared with directly learning $\pi _ { \theta }$ because sampling from $\mu$ is straightforward given $D ^ { \mu }$ . This allows to us to leverage most existing advances in generative modeling (Section 4.1). $Q _ { \phi } ( s , a )$ could be learned using the existing Q-learning framework (Section 4.2).
+
+The inverse temperature parameter $\alpha$ in Equation 8 serves as a trade-off between conservative and greedy improvement. We can see that when $\alpha  0$ , the learned policy falls back to the behavior policy, and when $\alpha  + \infty$ the learned policy becomes a greedy policy.
+
+To sample actions from $\pi$ , we use an importance sampling technique. Specifically, for any state $\pmb { s }$ , first we draw $M$ action samples from a learned behavior policy $\mu _ { \boldsymbol { \theta } } ( \cdot | \boldsymbol { s } )$ as candidates. Then we evaluate these action candidates with a learned critic $Q _ { \phi }$ . Finally, an action is resampled from $M$ candidates with $\exp { ( \alpha Q _ { \phi } ( s , \pmb { a } ) ) }$ being the sampling weights. We summarize this procedure as selecting from behavior candidates (SfBC), which could be understood as an analogue to rejection sampling.
+
+Although generative modeling of the behavior policy has been explored by several works (Fujimoto et al., 2019; Kumar et al., 2019), it was mostly used to form an explicit distributional constraint for the policy model $\pi _ { \theta }$ . In contrast, we show directly leveraging the learned behavior model to generate actions is not only feasible but beneficial on the premise that high-fidelity behavior modeling can be achieved. We give a practical implementation in the next section.
+
+# 4 PRACTICAL IMPLEMENTATION
+
+In this section, we derive a practical implementation of SfBC, which includes diffusion-based behavior modeling and planning-based Q-learning. An algorithm overview is given in Appendix A.
+
+# 4.1 DIFFUSION-BASED BEHAVIOR MODELING
+
+It is critical that the learned behavior model is of high fidelity because generating any out-of-sample actions would result in unwanted extrapolation error, while failing to cover all in-sample actions would restrict feasible action space for the policy. This requirement brings severe challenges to existing behavior modeling methods, which mainly include using Gaussians or VAEs. Gaussian models suffer from limited expressivity as we have discussed in Section 3.1. VAEs, on the other hand, need to introduce a variational posterior distribution to optimize the model distribution, which has a trade-off between the expressivity and the tractability (Kingma et al., 2016; Lucas et al., 2019). This still limits the expressivity of the model distribution. An empirical study is given in Section 6.3.
+
+To address this problem, we propose to learn from diverse behaviors using diffusion models (Ho et al., 2020), which have recently achieved great success in modeling diverse image distributions (Ramesh et al., 2022; Saharia et al., 2022), outperforming other generative models (Dhariwal & Nichol, 2021). Specifically, we follow Song et al. (2021b) and learn a state-conditioned diffusion model $s _ { \theta }$ to predict the time-dependent noise added to the action $^ { a }$ sampled from the behavior policy $\mu ( \cdot | s )$ :
+
+$$
+\theta = \underset { \theta } { \arg \operatorname* { m i n } } \quad \mathbb { E } _ { ( s , a ) \sim D ^ { \mu } , \epsilon , t } [ \| \sigma _ { t } \mathbf { s } _ { \theta } ( \alpha _ { t } \pmb { a } + \sigma _ { t } \epsilon , s , t ) + \epsilon \| _ { 2 } ^ { 2 } ] ,
+$$
+
+where $\epsilon \sim \mathcal { N } ( 0 , I )$ , $t \sim \mathcal { U } ( 0 , T )$ . $\alpha _ { t }$ and $\sigma _ { t }$ are determined by the forward diffusion process. Intuitively $s _ { \theta }$ is trained to denoise $\mathbf { \sigma } _ { \mathbf { } } \mathbf { a } _ { t } : = \alpha _ { t } \mathbf { \pmb { a } } + \sigma _ { t } \mathbf { \epsilon } \mathbf { \epsilon }$ into the unperturbed action $^ { a }$ such that $a _ { T } \sim$ $\mathcal { N } ( 0 , \pmb { I } )$ can be transformed into $\begin{array} { r } { \pmb { a } \sim \mu _ { \theta } ( \cdot | \pmb { s } ) } \end{array}$ by solving an inverse ODE defined by $s _ { \theta }$ (Equation 6).
+
+# 4.2 Q-LEARNING VIA IN-SAMPLE PLANNING
+
+Generally, Q-learning can be achieved via the Bellman expectation operator:
+
+$$
+\begin{array} { r } { \mathcal { T } ^ { \pi } Q ( s , a ) = r ( s , a ) + \gamma \mathbb { E } _ { s ^ { \prime } \sim P ( \cdot \mid s , a ) , a ^ { \prime } \sim \pi ( \cdot \mid s ^ { \prime } ) } Q ( s ^ { \prime } , a ^ { \prime } ) . } \end{array}
+$$
+
+However, $\tau ^ { \pi }$ is based on one-step bootstrapping, which has two drawbacks: First, this can be computationally inefficient due to its dependence on many steps of extrapolation. This drawback is exacerbated in diffusion settings since drawing actions from policy $\pi$ in Equation 10 is also time-consuming because of many iterations of Langevin-type sampling. Second, estimation errors may accumulate over long horizons. To address these problems, we take inspiration from episodic learning methods (Blundell et al., 2016; Ma et al., 2022) and propose a planning-based operator $\mathcal { T } _ { \mu } ^ { \pi }$
+
+$$
+\begin{array} { r } { \mathcal { T } _ { \mu } ^ { \pi } Q ( s , \pmb { a } ) : = \underset { n \geq 0 } { \operatorname* { m a x } } \{ ( \mathcal { T } ^ { \mu } ) ^ { n } \mathcal { T } ^ { \pi } Q ( s , \pmb { a } ) \} , } \end{array}
+$$
+
+![](images/4de8fba46e273afb595ba61890452a4a5121add659582f5fbaf2bd8e5ce12955.jpg)  
+Figure 2: Visualizations of the implicitly planned $\mathrm { Q }$ -targets $R _ { n } ^ { ( k ) }$ sampled from the dataset of an AntMaze task in four consecutive value iterations. The red pentagram stands for the reward signal. Implicit planning helps to iteratively stitch together successful subtrajectories.
+
+where $\mu$ is the behavior policy. $\mathcal { T } _ { \mu } ^ { \pi }$ combines the strengths of both the n-step operator $( { \mathcal { T } } ^ { \mu } ) ^ { n }$ , which enjoys a fast contraction property, and the operator $\mathcal { T } ^ { \pi }$ , which has a more desirable fixed point. We prove in Appendix $\textrm { C }$ that $\mathcal { T } _ { \mu } ^ { \pi }$ is also convergent, and its fixed point is bounded between $Q ^ { \pi }$ and $Q ^ { * }$ .
+
+Practically, given a dataset $\mathcal { D } ^ { \mu } = \{ ( s _ { n } , a _ { n } , r _ { n } ) \}$ collected by behavior $\mu$ , with $n$ being the timestep in a trajectory. We can rewrite Equation 11 in a recursive manner to calculate the Q-learning targets:
+
+$$
+\begin{array} { r l } & { R _ { n } ^ { ( k ) } = r _ { n } + \gamma \operatorname* { m a x } ( R _ { n + 1 } ^ { ( k ) } , V _ { n + 1 } ^ { ( k - 1 ) } ) , } \\ { \mathrm { w h e r e } \quad V _ { n } ^ { ( k - 1 ) } : = \mathbb { E } _ { a \sim \pi ( \cdot \vert s _ { n } ) } Q _ { \phi } ( s _ { n } , a ) , } \\ { \mathrm { a n d } \quad \phi = \underset { \phi } { \arg \operatorname* { m i n } } \quad \mathbb { E } _ { ( s _ { n } , a _ { n } ) \sim \mathcal { D } ^ { \mu } } \Vert Q _ { \phi } ( s _ { n } , a _ { n } ) - R _ { n } ^ { ( k - 1 ) } \Vert _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+Above $k \in \{ 1 , 2 , \dots \}$ is the iteration number. We define ${ \cal R } _ { n } ^ { ( 0 ) }$ as the vanilla return of trajectories. Equation 12 offers an implicit planning scheme within dataset trajectories that mainly helps to avoid bootstrapping over unseen actions and to accelerate convergence. Equation 13 enables the generalization of actions in similar states across different trajectories (stitching together subtrajectories). Note that we have omitted writing the iteration superscript of $\pi$ and $\mu$ for simplicity. During training, we alternate between calculating new Q-targets $R _ { n }$ and fitting the action evaluation model $Q _ { \phi }$ .
+
+Although the operator $\mathcal { T } _ { \mu } ^ { \pi }$ is inspired by the multi-step estimation operator $\mathcal { T } _ { \mathrm { v e m } }$ proposed by Ma et al. (2022). They have notable differences in theoretical properties. First, $\mathcal { T } _ { \mathrm { v e m } }$ can only apply to deterministic environments, while our method also applies to stochastic settings. Second, unlike $\mathcal { T } _ { \mathrm { v e m } }$ $\mathcal { T } _ { \mu } ^ { \pi }$ does not share the same fixed point with $\tau ^ { \pi }$ . We compare two methods in detail in Appendix G.
+
+# 5 RELATED WORK
+
+Reducing extrapolation error in offline RL. Offline RL typically requires careful trade-offs between maximizing expected returns and staying close to the behavior policy. Once the learned policy deviates from the behavior policy, extrapolation error will be introduced in dynamic programming, leading to performance degrade (Fujimoto et al., 2019). Several works propose to address this issue by introducing either policy regularization on the distributional discrepancy with the behavior policy (Fujimoto et al., 2019; Kumar et al., 2019; Wu et al., 2019; Fujimoto & Gu, 2021), or value pessimism about unseen actions (Kumar et al., 2020; Kostrikov et al., 2021). Another line of research directly extracts policy from the dataset through weighted regression, hoping to avoid selecting unseen actions (Peng et al., 2019; Nair et al., 2020; Wang et al., 2020). However, some recent works observe that the trade-off techniques described above are not sufficient to reduce extrapolation error, and propose to learn Q-functions through expectile regression without ever querying policy-generated actions (Kostrikov et al., 2022; Ma et al., 2022). Unlike them, We find that limited policy expressivity is the main reason that introduces extrapolation error in previous weighted regression methods, and use an expressive policy model to help reduce extrapolation error.
+
+Dynamic programming over long horizons. Simply extracting policies from behavior Q-functions can yield good performance in many D4RL tasks because it avoids dynamic programming and therefore the accompanied extrapolation error (Peng et al., 2019; Chen et al., 2020; Brandfonbrener et al., 2021). However, Kostrikov et al. (2022) shows this method performs poorly in tasks that require stitching together successful subtrajectories (e.g., Maze-like environments). Such tasks are also challenging for methods based on one-step bootstrapping because they might require hundreds of steps to reach the reward signal, with the reward discounted and estimation error accumulated along the way. Episodic memory-based methods address this problem by storing labeled experience in the dataset, and plans strictly within the trajectory to update evaluations of every decision (Blundell et al., 2016; Hu et al., 2021; Ma et al., 2022). The in-sample planning scheme allows dynamic programming over long horizons to suppress the accumulation of extrapolation error, which inspires our method.
+
+Table 1: Evaluation numbers of SfBC. Scores are normalized according to Fu et al. (2020). Numbers within 5 percent of the maximum in every individual task are highlighted in boldface. Experiment and evaluation details are provided in Appendix B. We report scores with 15 diffusion steps.   
+
+<table><tr><td>Dataset</td><td>Environment</td><td>SfBC (Ours)</td><td>IQL</td><td>VEM</td><td>AWR</td><td>BAIL</td><td>BCQ</td><td>CQL</td><td>DT</td><td>Diffuser</td></tr><tr><td>Medium-Expert</td><td>HalfCheetah</td><td>92.6±0.5</td><td>86.7</td><td>-</td><td>52.7</td><td>72.2</td><td>64.7</td><td>62.4</td><td>86.8</td><td>79.8</td></tr><tr><td>Medium-Expert</td><td>Hopper</td><td>108.6 ± 2.1</td><td>91.5</td><td>-</td><td>27.1</td><td>106.2</td><td>100.9</td><td>98.7</td><td>107.6</td><td>107.2</td></tr><tr><td>Medium-Expert</td><td>Walker</td><td>109.8± 0.2</td><td>109.6</td><td>-</td><td>53.8</td><td>107.2</td><td>57.5</td><td>111.0</td><td>108.1</td><td>108.4</td></tr><tr><td>Medium</td><td>HalfCheetah</td><td>45.9 ± 2.2</td><td>47.4</td><td>47.4</td><td>37.4</td><td>30.0</td><td>40.7</td><td>44.4</td><td>42.6</td><td>44.2</td></tr><tr><td>Medium</td><td>Hopper</td><td>57.1 ± 4.1</td><td>66.3</td><td>56.6</td><td>35.9</td><td>62.2</td><td>54.5</td><td>58.0</td><td>67.6</td><td>58.5</td></tr><tr><td>Medium</td><td>Walker</td><td>77.9 ± 2.5</td><td>78.3</td><td>74.0</td><td>17.4</td><td>73.4</td><td>53.1</td><td>79.2</td><td>74.0</td><td>79.7</td></tr><tr><td>Medium-Replay</td><td>HalfCheetah</td><td>37.1 ± 1.7</td><td>44.2</td><td>1</td><td>40.3</td><td>40.3</td><td>38.2</td><td>46.2</td><td>36.6</td><td>42.2</td></tr><tr><td>Medium-Replay</td><td>Hopper</td><td>86.2 ±9.1</td><td>94.7</td><td>1</td><td>28.4</td><td>94.7</td><td>33.1</td><td>48.6</td><td>82.7</td><td>96.8</td></tr><tr><td>Medium-Replay</td><td>Walker</td><td>65.1±5.6</td><td>73.9</td><td>-</td><td>15.5</td><td>58.8</td><td>15.0</td><td>26.7</td><td>66.6</td><td>61.2</td></tr><tr><td colspan="2">Average (Locomotion)</td><td>75.6</td><td>76.9</td><td>-</td><td>34.3</td><td>71.6</td><td>51.9</td><td>63.9</td><td>74.7</td><td>75.3</td></tr><tr><td>Default</td><td>AntMaze-umaze</td><td>92.0±2.1</td><td>87.5</td><td>87.5</td><td>56.0</td><td>85.0</td><td>78.9</td><td>74.0</td><td>59.2</td><td>-</td></tr><tr><td>Diverse</td><td>AntMaze-umaze</td><td>85.3±3.6</td><td>62.2</td><td>78.0</td><td>70.3</td><td>76.7</td><td>55.0</td><td>84.0</td><td>53.0</td><td>-</td></tr><tr><td>Play</td><td>AntMaze-medium</td><td>81.3± 2.6</td><td>71.2</td><td>78.0</td><td>0.0</td><td>15.0</td><td>0.0</td><td>61.2</td><td>0.0</td><td>-</td></tr><tr><td>Diverse</td><td>AntMaze-medium</td><td>82.0±3.1</td><td>70.0</td><td>77.0</td><td>0.0</td><td>23.3</td><td>0.0</td><td>53.7</td><td>0.0</td><td>-</td></tr><tr><td>Play</td><td>AntMaze-large</td><td>59.3±14.3</td><td>39.6</td><td>57.0</td><td>0.0</td><td>0.0</td><td>6.7</td><td>15.8</td><td>0.0</td><td>-</td></tr><tr><td>Diverse</td><td>AntMaze-large</td><td>45.5 ± 6.6</td><td>47.5</td><td>58.0</td><td>0.0</td><td>8.3</td><td>2.2</td><td>14.9</td><td>0.0</td><td>-</td></tr><tr><td colspan="2">Average (AntMaze)</td><td>74.2</td><td>63.0</td><td>72.6</td><td>21.0</td><td>46.7</td><td>23.8</td><td>50.6</td><td>18.7</td><td>-</td></tr><tr><td colspan="2">Average (Maze2d)</td><td>74.0</td><td>50.0</td><td>-</td><td>10.8</td><td>-</td><td>9.1</td><td>7.7</td><td>-</td><td>119.5</td></tr><tr><td colspan="2">Average (FrankaKitchen)</td><td>57.1</td><td>53.3</td><td>-</td><td>8.7</td><td>-</td><td>11.7</td><td>48.2</td><td>-</td><td>-</td></tr><tr><td>Both-side</td><td>Bidirectional-Car</td><td>100.0±0.0</td><td>15.7</td><td>0.0</td><td>0.0</td><td>52.0</td><td>88.0</td><td>42.3</td><td>33.3</td><td>-</td></tr><tr><td>Single-side</td><td>Bidirectional-Car</td><td>100.0±0.0</td><td>100.0</td><td>100.0</td><td>96.3</td><td>100.0</td><td>100.0</td><td>100.0</td><td>100.0</td><td>1</td></tr></table>
+
+Generative models for behavior modeling. Cloning diverse behaviors in a continuous action space requires powerful generative models. In offline RL, several works (Fujimoto et al., 2019; Kumar et al., 2019; Wu et al., 2019; Zhou et al., 2021; Chen et al., 2022) have tried using generative models such as Gaussians or VAEs to model the behavior policy. However, the learned behavior model only serves as an explicit distributional constraint for another policy during training. In broader RL research, generative adversarial networks (Goodfellow et al., 2020), masked autoencoders (Germain et al., 2015), normalizing flows (Dinh et al., 2016), and energy-based models (Du & Mordatch, 2019) have also been used for behavior modeling (Ho & Ermon, 2016; Ghasemipour et al., 2021; Singh et al., 2020; Liu et al., 2020). Recently, diffusion models (Ho et al., 2020) have achieved great success in generating diverse and high-fidelity image samples (Dhariwal & Nichol, 2021). However, exploration of its application in behavior modeling is still limited. Janner et al. (2022) proposes to solve offline tasks by iteratively denoising trajectories, while our method uses diffusion models for single-step decision-making. Concurrently with our work, Wang et al. (2022) also studies applying diffusion models to offline RL to improve policy expressivity. However, they use diffusion modeling as an implicit regularization during training of the desired policy instead of an explicit policy prior.
+
+# 6 EXPERIMENTS
+
+# 6.1 EVALUATIONS ON D4RL BENCHMARKS
+
+In Table 1, we compare the performance of SfBC to multiple offline RL methods in several D4RL (Fu et al., 2020) tasks. MuJoCo locomotion is a classic benchmark where policy-generated datasets only cover a narrow part of the state-action space, so avoiding querying out-of-sample actions is critical (Fujimoto et al., 2019; Kumar et al., 2020). The Medium dataset of this benchmark is generated by a single agent, while the Medium-Expert and the Medium-Replay dataset are generated by a mixture of policies. AntMaze is about an ant robot navigating itself in a maze, which requires both low-level robot control and high-level navigation. Since the datasets consist of undirected trajectories, solving AntMaze typically requires the algorithm to have strong “stitching” ability $\operatorname { F u }$ et al., 2020). Different environments contain mazes of different sizes, reflecting different complexity. Maze2d is very similar to AntMaze except that it’s about a ball navigating in a maze instead of an ant robot. FrankaKitchen are robot-arm manipulation tasks. We only focus on the analysis of MuJoCo locomotion and AntMaze tasks due to the page limit. Our choices of referenced baselines are detailed in Appendix E.
+
+![](images/5ccb55e50c6b9c1c1aaa47903932d170d0373889f47f0f41c704d887fd08c4d7.jpg)  
+Figure 3: Visualizations of actions taken by different RL agents in the Bidirectional-Car task. The ground truth corresponds to an agent which always takes the best actions, which is either 1.0 or -1.0. White space indicates suboptimal decisions. Green bounding boxes indicate possible initial states.
+
+Overall, SfBC outperforms most existing methods by large margins in complex tasks with sparse rewards such as AntMaze. We notice that VEM also achieves good results in AntMaze tasks and both methods share an implicit in-sample planning scheme, indicating that episodic planning is effective in improving algorithms’ stitching ability and thus beneficial in Maze-like environments. In easier locomotion tasks, SfBC provides highly competitive results compared with state-of-the-art algorithms. It can be clearly shown that performance gain is large in datasets generated by a mixture of distinctive policies (Medium-Expert) and is relatively small in datasets that are highly uniform (Medium). This is reasonable because SfBC is motivated to better model diverse behaviors.
+
+# 6.2 LEARNING FROM DIVERSE BEHAVIORS
+
+In this section, we analyze the benefit of modeling behavior policy using highly expressive generative models. Although SfBC outperforms baselines in many D4RL tasks. The improvement is mainly incremental, but not decisive. We attribute this to the lack of multiple optimal solutions in existing benchmarks. To better demonstrate the necessity of introducing an expressive generative model, we design a simple task where a heterogeneous dataset is collected in an environment that allows two distinctive optimal policies.
+
+Bidirectional-Car task. As depicted in Figure 1, we consider an environment where a car is placed in the middle of two endpoints. The car chooses an action in the range [-1,1] at each step, representing throttle, to influence the direction and speed of the car. The speed of the car will monotonically increase based on the absolute value of throttle. The direction of the car is determined by the sign of the current throttle. Equal reward will be given on the arrival of either endpoint within the rated time. It can be inferred with ease that, in any state, the optimal decision should be either 1 or -1, which is not a unimodal distribution. The collected dataset also contains highly diverse behaviors, with an approximately equal number of trajectories ending at both endpoints. For the comparative study, we collect another dataset called “Single-Side” where the only difference from the original one is that we remove all trajectories ending at the left endpoint from the dataset.
+
+![](images/5931429b8cb989542636c12780e781be3a1e3874cf5805238821078ca51eb61f.jpg)  
+Figure 4: Ablation studies of the value iteration number $K$ in MuJoCo Locomotion and Antmaze domains. $K = 1$ represents algorithms that use vanilla returns ${ \cal R } _ { n } ^ { ( 0 ) }$ as Q-learning targets without the implicit planning technique. All results are averaged over 4 independent random seeds.
+
+We test our method against several baselines, with the results given in Table 1. Among all referenced methods, SfBC is the only one that can always arrive at either endpoint within rated time in the Bidirectional-Car environment, whereas most methods successfully solve the “Single-Side” task. To gain some insight into why this happens, we illustrate the decisions made by an SfBC agent and other RL agents in the 2-dimensional state space. As is shown in Figure 3, the SfBC agent selects actions of high absolute values at nearly all states, while other unimodal actors fail to pick either one of the optimal actions when presented with two distinctive high-rewarding options. Therefore, we conclude that an expressive policy is necessary for performing diverse behavior learning.
+
+# 6.3 ABLATION STUDIES
+
+Diffusion vs. other generative models. Our first ablation study aims to evaluate 3 variants of SfBC which are respectively based on diffusion models (Ho et al., 2020), Gaussian probabilistic models, and latent-based models (VAEs, Kingma & Welling (2014)). The three variants use exactly the same training framework with the only difference being the behavior modeling method. As is shown in Table 4 in Appendix D, the diffusion-based policy outperforms the other two variants by a clear margin in most experiments, especially in tasks with heterogeneous datasets (e.g., Medium-Expert), indicating that diffusion models are fit for “high-fidelity” behavior modeling.
+
+Implicit in-sample planning. To study the importance of implicit in-sample planning on the performance of SfBC, we first visualize the estimated state values learned at different iterations of Q-learning in an AntMaze environment (Figure 2). We can see that implicit planning helps to iteratively stitch together successful subtrajectories and provides optimistic action evaluations. Then we aim to study how the value iteration number $K$ affects the performance of the algorithm in various environments. As shown in Figure 4, we compare the performance of $K$ in the range $\{ 1 , 2 , 3 , 4 , 5 \}$ and find that implicit planning is beneficial in complex tasks like AntMaze-Medium and AntMaze-Large. However, it is less important in MuJoCo-locomotion tasks. This finding is consistent with a prior work (Brandfonbrener et al., 2021).
+
+# 7 CONCLUSION
+
+In this work, we address the problem of limited policy expressivity in previous weighted regression methods by decoupling the policy model into a behavior model and an action evaluation model. Such decoupling allows us to use a highly expressive diffusion model for high-fidelity behavior modeling, which is further combined with a planning-based operator to reduce extrapolation error. Our method enables learning from a heterogeneous dataset in a continuous action space while avoiding selecting out-of-sample actions. Experimental results on the D4RL benchmark show that our approach outperforms state-of-the-art algorithms in most tasks. With this work, we hope to draw attention to the application of high-capacity generative models in offline RL.
+
+# REPRODUCIBILITY
+
+To ensure that our work is reproducible, we submit the source code as supplementary material. We also provide the pseudo-code of our algorithm in Appendix A and implementation details of our algorithm in Appendix B.
+
+# ACKNOWLEDGEMENT
+
+We thank Shiyu Huang, Yichi Zhou, and Hao Hu for discussing. This work was supported by the National Key Research and Development Program of China (2020AAA0106000, 2020AAA0106302, 2021YFB2701000), NSFC Projects (Nos. 62061136001, 62076147, U19B2034, U1811461, U19A2081, 61972224), BNRist (BNR2022RC01006), Tsinghua Institute for Guo Qiang, and the High Performance Computing Center, Tsinghua University.
+
+# REFERENCES
+
+Charles Blundell, Benigno Uria, Alexander Pritzel, Yazhe Li, Avraham Ruderman, Joel Z Leibo, Jack Rae, Daan Wierstra, and Demis Hassabis. Model-free episodic control. arXiv preprint arXiv:1606.04460, 2016.
+
+David Brandfonbrener, Will Whitney, Rajesh Ranganath, and Joan Bruna. Offline rl without off-policy evaluation. Advances in Neural Information Processing Systems, 34:4933–4946, 2021.
+
+Lili Chen, Kevin Lu, Aravind Rajeswaran, Kimin Lee, Aditya Grover, Misha Laskin, Pieter Abbeel, Aravind Srinivas, and Igor Mordatch. Decision transformer: Reinforcement learning via sequence modeling. In Advances in Neural Information Processing Systems, 2021.
+
+Xi Chen, Ali Ghadirzadeh, Tianhe Yu, Jianhao Wang, Yuan Gao, Wenzhe Li, Liang Bin, Chelsea Finn, and Chongjie Zhang. Lapo: Latent-variable advantage-weighted policy optimization for offline reinforcement learning. In Advances in Neural Information Processing Systems, 2022.
+
+Xinyue Chen, Zijian Zhou, Zheng Wang, Che Wang, Yanqiu Wu, and Keith Ross. BAIL: Bestaction imitation learning for batch deep reinforcement learning. Advances in Neural Information Processing Systems, 33, 2020.
+
+Alfredo V Clemente, Humberto N Castejon, and Arjun Chandra. Efficient parallel methods for deep ´ reinforcement learning. arXiv preprint arXiv:1705.04862, 2017.
+
+Prafulla Dhariwal and Alexander Quinn Nichol. Diffusion models beat GANs on image synthesis. In Advances in Neural Information Processing Systems, 2021.
+
+Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real nvp. arXiv preprint arXiv:1605.08803, 2016.
+
+John R Dormand and Peter J Prince. A family of embedded runge-kutta formulae. Journal of computational and applied mathematics, 6(1):19–26, 1980.
+
+Yilun Du and Igor Mordatch. Implicit generation and modeling with energy based models. Advances in Neural Information Processing Systems, 32, 2019.
+
+Justin Fu, Aviral Kumar, Ofir Nachum, George Tucker, and Sergey Levine. D4rl: Datasets for deep data-driven reinforcement learning. arXiv preprint arXiv:2004.07219, 2020.
+
+Scott Fujimoto and Shixiang Shane Gu. A minimalist approach to offline reinforcement learning. Advances in neural information processing systems, 34:20132–20145, 2021.
+
+Scott Fujimoto, David Meger, and Doina Precup. Off-policy deep reinforcement learning without exploration. In Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pp. 2052–2062. PMLR, 09–15 Jun 2019.
+
+Mathieu Germain, Karol Gregor, Iain Murray, and Hugo Larochelle. Made: Masked autoencoder for distribution estimation. In International conference on machine learning, pp. 881–889. PMLR, 2015.
+
+Seyed Kamyar Seyed Ghasemipour, Dale Schuurmans, and Shixiang Shane Gu. Emaq: Expected-max q-learning operator for simple yet effective offline and online rl. In International Conference on Machine Learning, pp. 3682–3691. PMLR, 2021.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks. Communications of the ACM, 63(11):139–144, 2020.
+
+Jonathan Ho and Stefano Ermon. Generative adversarial imitation learning. In Advances in neural information processing systems, volume 29, pp. 4565–4573, 2016.
+
+Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In Advances in Neural Information Processing Systems, 2020.
+
+Hao Hu, Jianing Ye, Zhizhou Ren, Guangxiang Zhu, and Chongjie Zhang. Generalizable episodic memory for deep reinforcement learning. arXiv preprint arXiv:2103.06469, 2021.
+
+Michael Janner, Yilun Du, Joshua Tenenbaum, and Sergey Levine. Planning with diffusion for flexible behavior synthesis. In International Conference on Machine Learning, 2022.
+
+Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings, 2014.
+
+Durk P Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improved variational inference with inverse autoregressive flow. Advances in neural information processing systems, 29, 2016.
+
+Ilya Kostrikov, Rob Fergus, Jonathan Tompson, and Ofir Nachum. Offline reinforcement learning with fisher divergence critic regularization. In International Conference on Machine Learning, pp. 5774–5783. PMLR, 2021.
+
+Ilya Kostrikov, Ashvin Nair, and Sergey Levine. Offline reinforcement learning with implicit Q-learning. In International Conference on Learning Representations, 2022.
+
+Aviral Kumar, Justin Fu, George Tucker, and Sergey Levine. Stabilizing off-policy q-learning via bootstrapping error reduction. CoRR, abs/1906.00949, 2019. URL http://arxiv.org/abs/ 1906.00949.
+
+Aviral Kumar, Aurick Zhou, George Tucker, and Sergey Levine. Conservative Q-learning for offline reinforcement learning. arXiv preprint arXiv:2006.04779, 2020.
+
+Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Manfred Otto Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. ICLR, 2016.
+
+Minghuan Liu, Tairan He, Minkai Xu, and Weinan Zhang. Energy-based imitation learning. arXiv preprint arXiv:2004.09395, 33, 2020.
+
+Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. arXiv preprint arXiv:2206.00927, 2022.
+
+James Lucas, George Tucker, Roger Grosse, and Mohammad Norouzi. Understanding posterior collapse in generative latent variable models, 2019. URL https://openreview.net/ forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ r1xaVLUYuE.
+
+Xiaoteng Ma, Yiqin Yang, Hao Hu, Jun Yang, Chongjie Zhang, Qianchuan Zhao, Bin Liang, and Qihan Liu. Offline reinforcement learning with value-based episodic memory. In International Conference on Learning Representations, 2022.
+
+Ashvin Nair, Abhishek Gupta, Murtaza Dalal, and Sergey Levine. Awac: Accelerating online reinforcement learning with offline datasets. arXiv preprint arXiv:2006.09359, 2020.
+
+Xue Bin Peng, Aviral Kumar, Grace Zhang, and Sergey Levine. Advantage-weighted regression: Simple and scalable off-policy reinforcement learning. arXiv preprint arXiv:1910.00177, 2019.
+
+Jan Peters, Katharina Mulling, and Yasemin Altun. Relative entropy policy search. In Twenty-Fourth AAAI Conference on Artificial Intelligence, 2010.
+
+Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen. Hierarchical textconditional image generation with clip latents. arXiv preprint arXiv:2204.06125, 2022.
+
+Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al. Photorealistic text-to-image diffusion models with deep language understanding. arXiv preprint arXiv:2205.11487, 2022.
+
+David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin A. Riedmiller. Deterministic policy gradient algorithms. In Proceedings of the 31th International Conference on Machine Learning, ICML 2014, Beijing, China, 21-26 June 2014, pp. 387–395, 2014.
+
+Avi Singh, Huihan Liu, Gaoyue Zhou, Albert Yu, Nicholas Rhinehart, and Sergey Levine. Parrot: Data-driven behavioral priors for reinforcement learning. arXiv preprint arXiv:2011.10024, 2020.
+
+Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International Conference on Machine Learning, pp. 2256–2265. PMLR, 2015.
+
+Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In International Conference on Learning Representations, 2021a.
+
+Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021b. URL https://openreview.net/forum? id=PxTIG12RRHS.
+
+Richard S. Sutton and Andrew G. Barto. Introduction to Reinforcement Learning. MIT Press, Cambridge, MA, USA, 1st edition, 1998. ISBN 0262193981.
+
+Richard S Sutton, David McAllester, Satinder Singh, and Yishay Mansour. Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems, 12, 1999.
+
+Zhendong Wang, Jonathan J Hunt, and Mingyuan Zhou. Diffusion policies as an expressive policy class for offline reinforcement learning. arXiv preprint arXiv:2208.06193, 2022.
+
+Ziyu Wang, Alexander Novikov, Konrad Zolna, Josh S Merel, Jost Tobias Springenberg, Scott E Reed, Bobak Shahriari, Noah Siegel, Caglar Gulcehre, Nicolas Heess, and Nando de Freitas. Critic regularized regression. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 7768–7778. Curran Associates, Inc., 2020.
+
+Jiayi Weng, Huayu Chen, Dong Yan, Kaichao You, Alexis Duburcq, Minghao Zhang, Yi Su, Hang Su, and Jun Zhu. Tianshou: A highly modularized deep reinforcement learning library. Journal of Machine Learning Research, 23(267):1–6, 2022. URL http://jmlr.org/papers/v23/ 21-1127.html.
+
+Yifan Wu, George Tucker, and Ofir Nachum. Behavior regularized offline reinforcement learning. arXiv preprint arXiv:1911.11361, 2019.
+
+Wenxuan Zhou, Sujay Bajracharya, and David Held. Plas: Latent action space for offline reinforcement learning. In Conference on Robot Learning, pp. 1719–1735. PMLR, 2021.
+
+# A ALGORITHM OVERVIEW
+
+Initialize the score-based model $s _ { \theta }$ , the action evaluation model $Q _ { \phi }$   
+Calculate vanilla discounted returns ${ \cal R } _ { n } ^ { ( 0 ) }$ for every state-action pair in dataset ${ \mathcal { D } } ^ { \mu }$   
+// Training the behavior model   
+for each gradient step do Sample $B$ data points $( s , a )$ from ${ \mathcal { D } } ^ { \mu }$ , $B$ Gaussian noises $\epsilon$ from $\mathcal { N } ( 0 , \pmb { I } )$ and $B$ time $t$ from $\mathcal { U } ( 0 , T )$ Perturb $\textbf { \em a }$ according to $\mathbf { \sigma } _ { \mathbf { \lambda } } \mathbf { a } _ { t } : = \alpha _ { t } \mathbf { \pmb { a } } + \sigma _ { t } \mathbf { \epsilon } $ Update $\theta \gets \lambda _ { s } \nabla _ { \theta } \sum [ \| \sigma _ { t } \mathbf { s } _ { \theta } ( { \pmb a } _ { t } , { \pmb s } , t ) + { \pmb \epsilon } \| _ { 2 } ^ { 2 } ]$   
+end for   
+// Training the action evaluation model iteratively   
+for iteration $k = 1$ to $K$ do Initialize training parameters $\phi$ of the action evaluation model $Q _ { \phi }$ for each gradient step do Sample $B$ data points $\Big ( s , \pmb { a } , R ^ { ( k - 1 ) } \Big )$ from ${ \mathcal { D } } ^ { \mu }$ Update $\begin{array} { r } { \phi  \phi - \lambda _ { Q } \nabla _ { \phi } \sum [ \| Q _ { \phi } ( \pmb { s } , \pmb { a } ) - R ^ { ( k - 1 ) } \| _ { 2 } ^ { 2 } ] } \end{array}$ end for // Update the $\mathcal { Q }$ -training targets as in Algorithm 2 $R ^ { ( k ) } = \operatorname { P l a n n i n g } ( { \mathcal { D } } ^ { \mu } , \mu _ { \theta } , Q _ { \phi } )$   
+end for
+
+# Algorithm 2 Implicit In-sample Planning
+
+Input a behavior dataset ${ \mathcal { D } } ^ { \mu }$ (sequentially ordered), a learned behavior policy $\mu _ { \theta }$ , a critic model $Q _ { \phi }$   
+// Evaluate every state in dataset according to Equation 13 with M Monte Carlo samples (parallelized)   
+for each minibatch $\left\{ s _ { n } \right\}$ splitted from ${ \mathcal { D } } ^ { \mu }$ do   
+Sample $\mathbf { M }$ actions $\hat { \pmb { a } } _ { n } ^ { 1 : M }$ from $\mu _ { \boldsymbol { \theta } } \bigl ( \cdot | \boldsymbol { s } _ { n } \bigr )$ , and calculate Q-values $\hat { R } _ { n } ^ { 1 : M } = Q _ { \phi } ( \pmb { s } _ { n } , \hat { \pmb { a } } _ { n } ^ { 1 : M } )$   
+Calculate state value $\begin{array} { r } { V _ { n } = \sum _ { m } \left[ \exp \left( \alpha \hat { R } _ { n } ^ { m } \right) \hat { R } _ { n } ^ { m } \right] / \sum _ { m } \exp \left( \alpha \hat { R } _ { n } ^ { m } \right) } \end{array}$   
+end for   
+// Performing implicit in-sample planning recursively   
+for timestep $n = \| \mathcal { D } ^ { \mu } \|$ to 0 do $R _ { n } = r _ { n } + \gamma \operatorname* { m a x } ( R _ { n + 1 } , V _ { n + 1 } )$ if $n$ is not the last episode step, else $r _ { n }$   
+end for   
+Output the new Q-training targets $\{ R _ { n } \}$
+
+![](images/351aeb06cb756d8f39099fac7da77ecabdf89c63a0a35201b4c9e4cfa9cdab09.jpg)  
+Figure 5: An algorithm overview of SfBC.
+
+# B EXPERIMENTAL DETAILS
+
+# B.1 IMPLEMENTATION DETAILS OF SFBC
+
+Network Architecture. SfBC includes a conditional scored-based model which estimates the score function of the behavior action distribution, and an action evaluation model which outputs the Qvalues of given state-action pairs. The architecture of the behavior model resembles U-Nets, but with spatial convolutions changed to simple dense connections, inspired by Janner et al. (2022). For the action evaluation model, we use a 2-layer MLP with 256 hidden units and SiLU activation functions. The same network architecture is applied across all tasks except for AntMaze-Large, where we add an extra layer of 512 hidden units for the action evaluation model.
+
+![](images/dfbe008ebdfc778adf78a99c9baa032e3f2b93a2ece93fd3b7e739d049fe1b4a.jpg)  
+Figure 6: The network architecture of the behavior model.
+
+Behavior training. In all experiments, we use the Adam optimizer and a batch size of 4096. The conditional scored-based model is trained for 500 data epochs with a learning rate of 1e-4. For the data perturbation method, we use Variance Preserving (VP) SDE as introduced in Song et al. (2021b), where we have $\begin{array} { r } { \mathrm { d } \pmb { x } = - \frac { 1 } { 2 } \beta ( t ) \pmb { x } \mathrm { d } t + \sqrt { \beta ( t ) } \mathrm { d } \mathbf { w } } \end{array}$ , such that we have $\begin{array} { r } { f ( { \pmb x } , t ) = - \frac { 1 } { 2 } \beta ( t ) { \pmb x } } \end{array}$ and $g ( t ) = \sqrt { \beta ( t ) }$ in Equation 6, and also:
+
+$$
+p _ { t 0 } ( \pmb { x } _ { t } | \pmb { x } _ { 0 } ) = \mathcal { N } ( \pmb { x } _ { t } | \alpha _ { t } \pmb { x } _ { 0 } , \sigma _ { t } ^ { 2 } I ) = \mathcal { N } ( \pmb { x } _ { t } | e ^ { - \frac { 1 } { 2 } \int _ { 0 } ^ { t } \beta ( s ) \mathrm { d } s } \pmb { x } _ { 0 } , [ 1 - e ^ { - \int _ { 0 } ^ { t } \beta ( s ) \mathrm { d } s } ] I ) .
+$$
+
+Following the default settings in Song et al. (2021b), we set $\beta ( t ) = ( \beta _ { \operatorname* { m a x } } - \beta _ { \operatorname* { m i n } } ) t + \beta _ { \operatorname* { m i n } }$ , with $\beta _ { \mathrm { m i n } }$ being 0.1 and $\beta _ { \mathrm { m a x } }$ being 20.
+
+Action evaluation via in-sample planning. The action evaluation model is trained for 100 data epochs with a learning rate of 1e-3 for each value iteration. We use $K = 2$ value iterations for all MuJoCo tasks, $K = 4$ for Antmaze-umaze tasks, and $K = 5$ for other Antmaze tasks. In each iteration, new Q-targets will be recalculated according to Equation 12 and Equation 13 based on the latest policy. We use Monte Carlo methods and importance sampling to estimate $V _ { n } ^ { ( k - 1 ) }$ in Equation 13:
+
+$$
+\begin{array} { l } { { { \cal V } _ { n } ^ { ( k - 1 ) } = \mathbb { E } _ { a \sim \pi ( \cdot \vert s _ { n } ) } Q _ { \phi } ( s _ { n } , a ) } } \\ { { { \mathrm { ~ } } } } \\ { { { \mathrm { ~ } } } } \\ { { { \displaystyle ~ \approx \sum _ { M } ( \cdot \vert s _ { n } ) \frac { \exp { \left( \alpha Q _ { \phi } ( s _ { n } , a ) \right) } } { Z ( s _ { n } ) } Q _ { \phi } ( s _ { n } , a ) } } } \\ { { { \mathrm { ~ } } } } \\ { { { \displaystyle ~ \approx \sum _ { M } \left[ \frac { \exp { \left( \alpha Q _ { \phi } ( s _ { n } , a ) \right) } } { \sum _ { M } \exp { \left( \alpha Q _ { \phi } ( s _ { n } , a ) \right) } } Q _ { \phi } ( s _ { n } , a ) \right] } , } } \end{array}
+$$
+
+with the inverse temperature $\alpha$ set to 20 and the Monte Carlo sample number set to 16 in all tasks. Note that at the beginning of each value iteration, we normalize Q-targets stored in the dataset and reinitialize the training parameters of the action evaluation model. Different from most prior works (Fujimoto et al., 2019; Kumar et al., 2020; Ma et al., 2022; Kostrikov et al., 2022), we do not use either ensembled networks or target networks to stabilize Q-learning.
+
+Diffusion sampling. To draw action samples from the behavior model, we use a 3rd-order specialized diffusion ODE solver proposed by Lu et al. (2022) to solve the inverse ODE problem in Equation 6. We use a diffusion step of $D = 1 5$ for all reported results in Table 1, which is significantly less than the typical 35-50 diffusion steps required if using ordinary RK45 ODE solver (Dormand & Prince, 1980). We also compare the performance and runtime of diffusion steps in the range $\{ 5 , 1 0 , 1 5 , 2 5 \}$ with the results reported in Table 2. Generally, we find that 10-25 diffusion steps perform similarly well in MuJoco Locomotion tasks and 15-25 diffusion steps perform similarly well in Antmaze tasks.
+
+Evaluation. Following the evaluation metric proposed by Fu et al. (2020), we run all Antmaze and MuJoCo experiments over 4 trials (different random seeds) and other experiments over 3 trials. For each trial, performance is averaged on another 100 test seeds for Antmaze tasks and 20 test seeds for other tasks at regular intervals (5 data epochs). During algorithm evaluation, we select actions in a deterministic way. Specifically, the action with the highest Q-value within $M$ behavior candidates will be selected for environment inference during evaluation. In MuJoCo Locomotion tasks, we average four actions with the highest Q-values among all candidates and find this technique helps to stabilize performance. We set the candidate number $M$ to 32 in all experiments.
+
+Runtime. We test the runtime of our algorithm on a RTX 2080Ti GPU. For algorithm training, the runtime cost of training the behavior model is 10.5 hours for 600 epochs, and the runtime cost of training the action evaluation model is about 31 minutes for each value iteration, (usually 2-5 iterations, 1M data points considered). For a concrete example, it roughly takes 155 minutes to train the action evaluation model $\scriptstyle ( \mathrm { K } = 5 )$ and 10.5 hours to train the behavior model for the “halfcheetah-medium” task.
+
+As for the evaluation runtime, theoretically, SfBC requires at least $D$ times of network inference time compared with non-diffusion methods $\mathcal { D } = 1$ ), $D$ being the diffusion steps. To accelerate algorithm evaluation, we implement a parallel evaluation scheme similar to Clemente et al. (2017); Weng et al. (2022) that could allow evaluating the algorithm under multiple test seeds at the same time, allowing us to significantly reduce the evaluation runtime by utilizing the parallel computing power of GPUs (Figure 7).
+
+Table 2: Ablation studies of the diffusion steps. The runtime is reported for the ”halfcheetah-medium” task on a RTX 2080Ti GPU. 1 episode stands for 1000 environment steps.   
+
+<table><tr><td>Diffusion Steps D</td><td>5 steps</td><td>10 steps</td><td>15 steps</td><td>25 steps</td></tr><tr><td>Performance (Locomotion)</td><td>2.3</td><td>72.9</td><td>75.6</td><td>74.4</td></tr><tr><td>Performance (Antmaze)</td><td>5.5</td><td>65.7</td><td>74.2</td><td>73.0</td></tr><tr><td>Runtime (1 episode, # envs=1)</td><td>22.3 s</td><td>38.0 s</td><td>50.0 s</td><td>93.0 s</td></tr><tr><td>Runtime (1 episode,# envs=20)</td><td>1.5 s</td><td>2.5 s</td><td>3.2 s</td><td>5.0s</td></tr></table>
+
+![](images/0ecfa857ffa65d11bb7760e0df71643b4c94597833b5073f3a71f7d5ec95dda1.jpg)  
+Figure 7: Evaluation runtime of SfBC.
+
+B.2 IMPLEMENTATION DETAILS FOR ABLATION STUDIES
+
+SfBC $^ +$ VAEs/Gaussians. For the VAE-based behavior model, we use exactly the same network architecture and training loss as Fujimoto et al. (2019) and train the behavior model for $3 0 0 \mathrm { k }$ iterations at a learning rate of 3e-4. For the Gaussian-based policy model, we follow Nair et al. (2020) and use a 4-layer MLP with 256 hidden units and ReLU activation functions. The sampled action from the parameterized Gaussian distribution is squashed to range $[ - 1 , 1 ]$ by a Tanh activation function. The Gaussian behavior model is trained by directly maximizing the log-likelihood of the dataset distribution for $3 0 0 \mathrm { k }$ iterations at a learning rate of 3e-4. Other experiment settings are consistent with the diffusion-based method.
+
+SfBC - Planning. Removing the planning-based procedure from SfBC is equivalent to performing SfBC for only one iteration, which only learns a behavior Q-function purely from vanilla returns. Other than this, we use the same network architecture and training paradigm as SfBC.
+
+# B.3 SOURCES OF REFERENCED BASELINE NUMBERS
+
+For IQL (Kostrikov et al., 2022), D4RL performance numbers are reported in its original paper, except for Maze2d tasks, which we reference Janner et al. (2022). The performance in the Bidirectional-Car task is based on a PyTorch reimplementation of the algorithm (https: //github.com/gwthomas/IQL-PyTorch). We use the same hyperparameters as in the original paper for MuJoCo Locomotion tasks.
+
+For VEM (Ma et al., 2022) and Diffuser (Janner et al., 2022), all D4RL performance numbers come from their respective papers. Performance numbers of VEM in the Bidirectional-Car task are based on a slightly modified version of the algorithm’s official codebase (https://github. com/YiqinYang/VEM). Since the performance of VEM is very sensitive to a hyperparameter $\tau$ in their algorithm. We evaluate $\tau \in \{ 0 . \bar { 1 } , 0 . 2 , . . . , 0 . 9 \}$ and report the best-performing choice.
+
+For AWR (Peng et al., 2019), BCQ (Fujimoto et al., 2019) and CQL (Kumar et al., 2020), all their D4RL performance numbers come from Fu et al. (2020). Their performance numbers in the Bidirectional-Car task are based on three independent codebases: https://github. com/Farama-Foundation/D4RL-Evaluations for AWR, https://github.com/ sfujim/BCQ for BCQ and https://github.com/young-geng/CQL for CQL. We mostly use the default hyperparameters in their respective codebases.
+
+For BAIL (Chen et al., 2020), all reported performance numbers come from our experiments based on a slightly modified version of its official codebase (https://github.com/lanyavik/BAIL). Note that BAIL proposes a technique to replace oracle returns with augmented returns in MuJoCo Locomotion tasks, whereas we omit using this technique because it cannot be easily applied to other offline tasks. Other than this, we use default settings in the original codebase.
+
+For DT (Chen et al., 2021), D4RL performance numbers are reported in DT’s paper, except for AntMaze tasks, which we reference Kostrikov et al. (2022). The performance numbers in the Bidirectional-Car task are based on the algorithm’s official codebase (https://github.com/ kzl/decision-transformer). We use the same hyperparameters as they did for MuJoCo Locomotion tasks.
+
+# C THEORETICAL ANALYSIS
+
+In this section, we provide some theoretical analysis of our planning-based operator:
+
+$$
+\begin{array} { r } { \mathcal { T } _ { \mu } ^ { \pi } Q ( s , \pmb { a } ) : = \operatorname* { m a x } _ { n \geq 0 } \{ ( \mathcal { T } ^ { \mu } ) ^ { n } \mathcal { T } ^ { \pi } Q ( s , \pmb { a } ) \} . } \end{array}
+$$
+
+First, we provide the following proposition to discuss the contraction property of $\mathcal { T } _ { \mu } ^ { \pi }$ and the bound of its fixed point.
+
+Proposition 1. We have the following properties of $\mathcal { T } _ { \mu } ^ { \pi }$
+
+1) $\mathcal { T } _ { \mu } ^ { \pi }$ owns monotonicity, i.e., for $\forall Q _ { 1 } \leq Q _ { 2 }$ , we have ${ \mathcal T } _ { \mu } ^ { \pi } Q _ { 1 } \leq { \mathcal T } _ { \mu } ^ { \pi } Q _ { 2 }$ .
+
+2) $\mathcal { T } _ { \mu } ^ { \pi }$ is at least a $\gamma$ -contraction.
+
+$^ 3$ ) Assume the fixed point of $\mathcal { T } _ { \mu } ^ { \pi }$ is $\tilde { Q }$ , then we have $Q ^ { \pi } ( s , a ) \leq \tilde { Q } ( s , a ) \leq Q ^ { * } ( s , a )$ holds for $\forall s , a$ here $Q ^ { \pi } , Q ^ { * }$ are the fixed points of $\tau ^ { * }$ and $\mathcal { T } ^ { \pi }$ respectively.
+
+Proof.
+
+1) For $\forall Q _ { 1 } \leq Q _ { 2 } , \forall s , \mathbf { a } , \forall n \in \mathbb { N }$ , we have
+
+Thus we have
+
+$$
+\mathcal { T } _ { \mu } ^ { \pi } Q _ { 1 } ( s , a ) = \operatorname* { m a x } _ { n \geq 0 } \{ ( \mathscr { T } ^ { \mu } ) ^ { n } \mathscr { T } ^ { \pi } Q _ { 1 } ( s , a ) \} \leq \operatorname* { m a x } _ { n \geq 0 } \{ ( \mathscr { T } ^ { \mu } ) ^ { n } \mathscr { T } ^ { \pi } Q _ { 2 } ( s , a ) \} = \mathscr { T } _ { \mu } ^ { \pi } Q _ { 2 } ( s , a ) .
+$$
+
+2) For $\forall Q _ { 1 } , Q _ { 2 } , \forall s , a$ , we have
+
+$$
+\begin{array} { r l } & { \left| { \cal T } _ { \mu } ^ { \pi } Q _ { 1 } ( s , a ) - { \cal T } _ { \mu } ^ { \pi } Q _ { 2 } ( s , a ) \right| = \displaystyle \left| \operatorname* { m a x } _ { n \geq 0 } \{ ( { \cal T } ^ { \mu } ) ^ { n } { \cal T } ^ { \pi } Q _ { 1 } ( s , a ) \} - \operatorname* { m a x } _ { n \geq 0 } \{ ( { \cal T } ^ { \mu } ) ^ { n } { \cal T } ^ { \pi } Q _ { 2 } ( s , a ) \} \right| } \\ & { \qquad \leq \operatorname* { m a x } _ { n \geq 0 } \{ | ( { \cal T } ^ { \mu } ) ^ { n } { \cal T } ^ { \pi } Q _ { 1 } ( s , a ) - ( { \cal T } ^ { \mu } ) ^ { n } { \cal T } ^ { \pi } Q _ { 2 } ( s , a ) | \} } \\ & { \qquad \leq \operatorname* { m a x } _ { n \geq 0 } \{ \gamma ^ { n + 1 } \| Q _ { 1 } - Q _ { 2 } \| _ { \infty } \} } \\ & { \qquad = \gamma \| Q _ { 1 } - Q _ { 2 } \| _ { \infty } . } \end{array}
+$$
+
+Consequently, $\mathcal { T } _ { \mu } ^ { \pi }$ is at least a $\gamma$ -contraction.
+
+3) For $\forall s , a$ , we first prove $Q ^ { \pi } ( s , \pmb { a } ) \leq \tilde { Q } ( s , \pmb { a } )$ . For $\forall m \in \mathbb { N }$ , we have
+
+$$
+\begin{array} { r l } & { \displaystyle { \cal Q } ^ { \pi } ( s , { \boldsymbol a } ) = \mathcal { T } ^ { \pi } { \cal Q } ^ { \pi } ( s , { \boldsymbol a } ) \leq \mathcal { T } _ { \mu } ^ { \pi } { \cal Q } ^ { \pi } ( s , { \boldsymbol a } ) = \mathcal { T } _ { \mu } ^ { \pi } \mathcal { T } ^ { \pi } { \cal Q } ^ { \pi } ( s , { \boldsymbol a } ) } \\ & { \leq \mathcal { T } _ { \mu } ^ { \pi } \mathcal { T } _ { \mu } ^ { \pi } { \cal Q } ^ { \pi } ( s , { \boldsymbol a } ) \quad \mathrm { ( b y ~ m o n o t o n i c i t y ~ o f ~ } \mathcal { T } _ { \mu } ^ { \pi } ) } \\ & { \leq \ldots \leq ( \mathcal { T } _ { \mu } ^ { \pi } ) ^ { m } { \cal Q } ^ { \pi } ( s , { \boldsymbol a } ) , } \\ { \mathrm { T h u s } \quad { \cal Q } ^ { \pi } ( s , { \boldsymbol a } ) \leq \displaystyle \operatorname* { l i m } _ { m  \infty } ( \mathcal { T } _ { \mu } ^ { \pi } ) ^ { m } { \cal Q } ^ { \pi } ( s , { \boldsymbol a } ) = \tilde { \boldsymbol { Q } } ( s , { \boldsymbol a } ) . } \end{array}
+$$
+
+Now we prove that $\tilde { Q } ( s , a ) \leq Q ^ { * } ( s , a )$ . We have
+
+$$
+\begin{array} { r l } & { \mathcal { T } ^ { \pi } Q ^ { * } ( s , a ) = \mathcal { R } ( s , { a } ) + \gamma \mathbb { E } _ { s ^ { \prime } } \mathbb { E } _ { a ^ { \prime } \sim \hat { \pi } ( \cdot \vert s ^ { \prime } ) } Q ^ { * } ( s ^ { \prime } , { a } ^ { \prime } ) } \\ & { \qquad \leq \mathcal { R } ( s , a ) + \gamma \mathbb { E } _ { s ^ { \prime } } \operatorname* { m a x } _ { a ^ { \prime } } Q ^ { * } ( s ^ { \prime } , a ^ { \prime } ) = Q ^ { * } ( s , a ) . } \end{array}
+$$
+
+Similarly, we have ${ \mathcal { T } } ^ { \mu } Q ^ { * } ( s , a ) \leq Q ^ { * } ( s , a )$ .
+
+Then for $\forall n , m \in \mathbb { N }$ ,
+
+$$
+\begin{array} { r l r } & { \begin{array} { l l } & { Q ^ { * } ( s , a ) \geq T ^ { \mu } Q ^ { * } ( s , a ) \geq ( T ^ { \mu } ) ^ { 2 } Q ^ { * } ( s , a ) } & { \mathrm { ( b y ~ m o n o t o n i c i t y ~ o f ~ } T ^ { \mu } ) } \\ & { \qquad \geq \ldots \geq ( T ^ { \mu } ) ^ { n } Q ^ { * } ( s , a ) } \\ & { \qquad \geq ( T ^ { \mu } ) ^ { n } { \mathcal T } ^ { \pi } Q ^ { * } ( s , a ) , } & { \mathrm { ( b y ~ m o n o t o n i c i t y ~ o f ~ } ( T ^ { \mu } ) ^ { n } ) } \end{array} } \\ & { } \\ & { \begin{array} { r l r } { \mathrm { \ h u s } } & { Q ^ { * } ( s , a ) \geq { \mathcal T } _ { \mu } ^ { \pi } Q ^ { * } ( s , a ) } \\ & { \qquad \geq { \mathcal T } _ { \mu } ^ { \pi } { \mathcal T } _ { \mu } ^ { \pi } Q ^ { * } ( s , a ) } & { \mathrm { ( b y ~ m o n o t o n i c i t y ~ o f ~ } { \mathcal T } _ { \mu } ^ { \pi } ) } \\ & { \qquad \geq \ldots \geq ( { \mathcal T } _ { \mu } ^ { \pi } ) ^ { m } Q ^ { * } ( s , a ) , } & \\ { \mathrm { \ h u s } } & { Q ^ { * } ( s , a ) \geq \operatorname* { l i m } _ { m \to \infty } ( { \mathcal T } _ { \mu } ^ { \pi } ) ^ { m } Q ^ { * } ( s , a ) = \tilde { Q } ( s , a ) . } \end{array} } \end{array}
+$$
+
+Moreover, similar to the analysis in Ma et al. (2022), we provide the following proposition to show that at the beginning of the training when the current Q function estimates $Q ( s , a )$ is significantly pessimistic, our $\mathcal { T } _ { \mu } ^ { \pi }$ provides a relatively optimistic update and can contract the estimation error more quickly.
+
+Proposition 2. In practice, we consider $\begin{array} { r } { { \cal T } _ { \mu } ^ { \pi } Q ( s , a ) : = \operatorname* { m a x } _ { 0 \leq n \leq N } \{ ( { \cal T } ^ { \mu } ) ^ { n } { \cal T } ^ { \pi } Q ( s , a ) \} } \end{array}$ . Then we have:
+
+$$
+\begin{array} { r } { | { \cal T } _ { \mu } ^ { \pi } Q ( s , a ) - Q ^ { * } ( s , a ) | \leq { \gamma } ^ { n ^ { * } ( s , a ) } \| Q - \tilde { Q } _ { n ^ { * } } \| _ { \infty } + \| \tilde { Q } _ { n ^ { * } } - Q ^ { * } \| _ { \infty } , \quad \forall s , a , } \end{array}
+$$
+
+here $\begin{array} { r } { n ^ { \ast } ( \pmb { \mathscr { s } } , \pmb { a } ) = \arg \operatorname* { m a x } _ { 0 \leq n \leq N } \{ ( \mathscr { T } ^ { \mu } ) ^ { n } \mathscr { T } ^ { \pi } Q ( \pmb { \mathscr { s } } , \pmb { a } ) \} } \end{array}$ and ${ \tilde { Q } } _ { n ^ { * } }$ is the fixed point of $( \mathcal { T } ^ { \mu } ) ^ { n ^ { * } ( s , a ) } \mathcal { T } ^ { \pi }$
+
+Proof. We can use the triangle inequality to prove this result
+
+$$
+\begin{array} { r l } & { \ | T _ { \mu } ^ { \pi } Q ( s , a ) - Q ^ { * } ( s , a ) | } \\ & { = | ( T ^ { \mu } ) ^ { n ^ { * } ( s , a ) } T ^ { \pi } Q ( s , a ) - Q ^ { * } ( s , a ) | } \\ & { \leq | ( T ^ { \mu } ) ^ { n ^ { * } ( s , a ) } T ^ { \pi } Q ( s , a ) - ( T ^ { \mu } ) ^ { n ^ { * } ( s , a ) } T ^ { \pi } \tilde { Q } _ { n ^ { * } } ( s , a ) | + | ( T ^ { \mu } ) ^ { n ^ { * } ( s , a ) } T ^ { \pi } \tilde { Q } _ { n ^ { * } } ( s , a ) - Q ^ { * } ( s , a ) | } \\ & { = | ( T ^ { \mu } ) ^ { n ^ { * } ( s , a ) } T ^ { \pi } Q ( s , a ) - ( T ^ { \mu } ) ^ { n ^ { * } ( s , a ) } T ^ { \pi } \tilde { Q } _ { n ^ { * } } ( s , a ) | + | \tilde { Q } _ { n ^ { * } } ( s , a ) - Q ^ { * } ( s , a ) | } \\ & { \leq \gamma ^ { n ^ { * } ( s , a ) } \| Q - \tilde { Q } _ { n ^ { * } } \| _ { \infty } + \| \tilde { Q } _ { n ^ { * } } - Q ^ { * } \| _ { \infty } . } \end{array}
+$$
+
+When $Q$ is significantly lower than $\tilde { Q } _ { n ^ { * } } , Q ^ { * }$ , $\| { \tilde { Q } } _ { n ^ { * } } - Q ^ { * } \| _ { \infty }$ is often conspicuously lower than $\| Q - { \tilde { Q } } _ { n ^ { * } } \| _ { \infty }$ and $n ^ { * } ( s , { \pmb a } )$ is relatively large (this often happens at the beginning of the training since the initial $\mathrm { \bf Q }$ estimates are often near zero and thus pessimistic). At this time, based on this proposition, our operator $\mathcal { T } _ { \mu } ^ { \pi }$ , could contract the estimation error with a rate of around $\gamma ^ { n ^ { * } ( s , \pmb { a } ) }$ , which could significantly reduce extrapolation iterations required.
+
+D MISSING PERFORMANCE NUMBERS   
+
+<table><tr><td>Dataset</td><td>Environment</td><td>SfBC (Ours)</td><td>IQL</td><td>VEM</td><td>AWR</td><td>BAIL</td><td>BCQ</td><td>CQL</td><td>DT</td><td>Diffuser</td></tr><tr><td>Sparse</td><td>Maze2d-umaze</td><td>73.9 ± 6.6</td><td>47.4</td><td>-</td><td>1.0</td><td>-</td><td>12.8</td><td>5.7</td><td>-</td><td>113.9</td></tr><tr><td>Sparse</td><td>Maze2d-medium</td><td>73.8±2.9</td><td>34.9</td><td>■</td><td>7.6</td><td>-</td><td>8.3</td><td>5.0</td><td></td><td>121.5</td></tr><tr><td>Sparse</td><td>Maze2d-large</td><td>74.4 ± 1.7</td><td>58.6</td><td>=</td><td>23.7</td><td>=</td><td>6.2</td><td>12.5</td><td></td><td>123.0</td></tr><tr><td colspan="2">Average (Maze2d)</td><td>74.0</td><td>50.0</td><td></td><td>10.8</td><td>■</td><td>9.1</td><td>7.7</td><td>=</td><td>119.5</td></tr><tr><td>Complete</td><td>FrankaKitchen</td><td>77.9 ± 0.6</td><td>62.5</td><td></td><td>0.0</td><td>■</td><td>8.11</td><td>43.8</td><td></td><td>=</td></tr><tr><td>Partial</td><td>FrankaKitchen</td><td>47.9 ± 4.1</td><td>46.3</td><td></td><td>15.4</td><td>=</td><td>18.9</td><td>49.8</td><td></td><td></td></tr><tr><td>Mixed</td><td>FrankaKitchen</td><td>45.4 ± 1.6</td><td>51.0</td><td></td><td>10.6</td><td>■</td><td>8.1</td><td>51.0</td><td></td><td></td></tr><tr><td colspan="2">Average (FrankaKitchen)</td><td>57.1</td><td>53.3</td><td>-</td><td>8.7</td><td>-</td><td>11.7</td><td>48.2</td><td>■</td><td>■</td></tr></table>
+
+Table 3: Additional performance numbers of SfBC in Maze2d and FrankaKitchen tasks. We report the mean and standard deviation over three seeds for SfBC. Scores are normalized according to $\mathrm { F u }$ et al. (2020).   
+Table 4: Ablations of generative modeling methods and the implicit planning method. We report the mean and standard deviation over four seeds for the main experiment and three seeds for other experiments. Scores are normalized according to Fu et al. (2020).   
+
+<table><tr><td>Dataset</td><td>Environment</td><td>SfBC</td><td>SfBC +Gaussian</td><td>SfBC + VAE</td><td>SfBC - Planning</td></tr><tr><td>Medium-Expert</td><td>HalfCheetah</td><td>92.6 ± 0.5</td><td>79.4 ± 1.4</td><td>85.2 ± 2.9</td><td>91.4± 0.6</td></tr><tr><td>Medium-Expert</td><td>Hopper</td><td>108.6± 2.1</td><td>107.8± 7.8</td><td>92.0 ± 7.3</td><td>109.0 ±1.0</td></tr><tr><td>Medium-Expert</td><td>Walker</td><td>109.8±0.2</td><td>71.5 ± 1.5</td><td>109.3± 2.5</td><td>109.4 ± 0.9</td></tr><tr><td>Medium</td><td>HalfCheetah</td><td>45.9 ± 2.2</td><td>42.0± 0.2</td><td>43.4± 0.1</td><td>42.4±0.2</td></tr><tr><td>Medium</td><td>Hopper</td><td>57.1 ± 4.1</td><td>58.1 ± 1.5</td><td>65.6 ± 3.3</td><td>60.1 ± 4.2</td></tr><tr><td>Medium</td><td>Walker</td><td>77.9 ± 2.5</td><td>82.4± 1.1</td><td>79.1 ± 2.5</td><td>80.3 ± 0.9</td></tr><tr><td>Medium-Replay</td><td>HalfCheetah</td><td>37.1 ± 1.7</td><td>36.2±1.2</td><td>42.4± 0.5</td><td>37.5 ± 0.6</td></tr><tr><td>Medium-Replay</td><td>Hopper</td><td>86.2 ± 9.1</td><td>67.8 ± 6.5</td><td>58.6± 4.8</td><td>58.6 ± 1.3</td></tr><tr><td>Medium-Replay</td><td>Walker</td><td>65.1± 5.6</td><td>65.8± 4.4</td><td>62.2 ± 4.3</td><td>62.6 ± 2.2</td></tr><tr><td colspan="2">Average</td><td>75.6</td><td>67.9</td><td>70.9</td><td>72.3</td></tr><tr><td>Default</td><td>AntMaze-umaze</td><td>92.0± 2.1</td><td>93.3± 2.4</td><td>91.6± 2.4</td><td>96.7 ± 4.7</td></tr><tr><td>Diverse</td><td>AntMaze-umaze</td><td>85.3±3.6</td><td>88.3±2.4</td><td>78.3 ± 4.7</td><td>80.0 ± 10.8</td></tr><tr><td>Play</td><td>AntMaze-medium</td><td>81.3 ± 2.6</td><td>80.0 ± 4.1</td><td>68.3± 2.4</td><td>35.0 ± 4.1</td></tr><tr><td>Diverse</td><td>AntMaze-medium</td><td>82.0±3.1</td><td>85.0 ± 7.1</td><td>65.0 ± 7.1</td><td>33.3 ± 6.2</td></tr><tr><td>Play</td><td>AntMaze-large</td><td>59.3 ± 14.3</td><td>43.3 ± 7.1</td><td>35.0 ±8.2</td><td>8.3±8.5</td></tr><tr><td>Diverse</td><td>AntMaze-large</td><td>45.5±6.6</td><td>26.7±8.5</td><td>20.0±0.0</td><td>6.7 ± 4.7</td></tr><tr><td colspan="2">Average</td><td>74.2</td><td>69.4</td><td>59.7</td><td>43.3</td></tr></table>
+
+# E CHOICES OF REFERENCED BASELINES
+
+Referenced baselines methods of SfBC can be roughly divided into four categories: 1. Policy regression methods that require dynamic programming such as IQL (Kostrikov et al., 2022) and VEM (Ma et al., 2022). 2. Policy regression methods that use vanilla returns as regression weights such as AWR (Peng et al., 2019) and BAIL (Chen et al., 2020). 3. Adaptations of existing off-policy algorithms with policy regularization such as BCQ (Fujimoto et al., 2019) and CQL (Kumar et al., 2020). 4. Sequence modeling methods such as DT (Chen et al., 2021) and Diffuser (Janner et al., 2022). Here we further highlight several methods which bear some resemblance to our approach: Both IQL and SfBC aim to entirely avoid selecting out-of-sample actions, except that IQL uses weighted regression while SfBC does not; VEM also uses an implicit in-sample planning scheme similar to ours; BCQ also uses a generative model (VAE) for behavior modeling, but only to assist the learning of another policy model; Diffuser, like SfBC, is also a diffusion-based algorithm, but uses approximated guided sampling at trajectory level instead of importance sampling at step level.
+
+![](images/935150bd852360484e51b19e1ce6727e2c76bc36a21b9032e1e204adc1c1241e.jpg)  
+Figure 8: Training curves of SfBC for MuJoCo and Antmaze tasks with different diffusion steps. We report the mean and standard deviation over four seeds for all experiments. Note that the parameters of the critic model are initialized at the beginning of each value iteration (every 100 data epochs).
+
+# G CONNECTIONS TO PRIOR WORKS
+
+In this section, we discuss in more detail the connections between SfBC and two prior works, namely VEM (Ma et al., 2022) and EMAQ (Ghasemipour et al., 2021).
+
+# G.1 VEM
+
+Our in-sample planning-based Q-operator $\mathcal { T } _ { \mu } ^ { \pi }$ bears some similarity to the multi-step estimation operator $\mathcal { T } _ { \mathrm { v e m } }$ proposed by Ma et al. (2022). A simplified version of $\mathcal { T } _ { \mathrm { v e m } }$ is defined as:
+
+$$
+{ \mathcal { T } } _ { \mathrm { v e m } } V ( s ) : = \operatorname* { m a x } _ { n \geq 0 } \{ ( { \mathcal { T } } ^ { \mu } ) ^ { n } { \mathcal { T } } _ { \mu } ^ { \tau } V ( s ) \} ,
+$$
+
+which is is built on $\mathcal { T } _ { \mu } ^ { \tau }$ , an expectile-based V-learning operator proposed by VEM:
+
+$$
+\Gamma _ { \mu } ^ { \tau } V ( s ) : = \mathbb { E } _ { a \sim \mu ( \cdot | s ) } \left\{ \begin{array} { l l } { \tau [ r ( s , a ) + \gamma V ( s ^ { \prime } ) ] + ( 1 - \tau ) V ( s ) } & { \mathrm { ~ i f ~ } \quad r ( s , a ) + \gamma V ( s ^ { \prime } ) \geq V ( s ) } \\ { ( 1 - \tau ) [ r ( s , a ) + \gamma V ( s ^ { \prime } ) ] + \tau V ( s ) } & { \mathrm { ~ i f ~ } \quad r ( s , a ) + \gamma V ( s ^ { \prime } ) < V ( s ) } \end{array} \right\}
+$$
+
+here $\tau \in [ 0 , 1 )$ is a hyperparameter that helps interpolate the Bellman expectation operator $\mathcal { T } ^ { \mu }$ $\tau = 0 . 5 )$ and the Bellman optimality operator $\tau ^ { * }$ $\tau  1 . 0$ ). $V ( \cdot )$ is an arbitrary scalar function. $\mathcal { T } _ { \mu } ^ { \tau }$ has some nice properties such as monotonicity $( \mathcal T _ { \mu } ^ { \tau _ { 2 } } V ( \pmb { s } ) > \mathcal T _ { \mu } ^ { \tau _ { 1 } } V ( \pmb { s } )$ always holds for any $V$ given $\tau _ { 2 } > \tau _ { 1 }$ ). With these properties, Ma et al. (2022) derives that $\mathcal { \tilde { T } } _ { \mathrm { v e m } }$ and $\mathcal { T } _ { \mu } ^ { \tau }$ share the same fixed point.
+
+However, VEM cannot be applied to stochastic environments because Equation 27 requires comparing $V ( s )$ and $r ( s , a ) + \gamma V ( s ^ { \prime } )$ . While $V ( s )$ is a scalar given $\pmb { s }$ , $r ( s , { \pmb a } ) + \gamma V ( { \pmb s } ^ { \prime } )$ is a random variable since $\pmb { s } ^ { \prime } \sim P ( \cdot | \pmb { s } , \pmb { a } )$ . To fix this problem, VEM simply assumes that the environment is deterministic, namely $r ( s , a )$ and $P ( \cdot | s , a )$ are all Dirac.
+
+Compared with VEM, our in-sample planning-based $\mathrm { Q }$ -operator $\mathcal { T } _ { \mu } ^ { \pi }$ is not dependent on the expectilebased $\mathrm { v } .$ -operator $\mathcal { T } _ { \mu } ^ { \tau }$ , but uses an hypothetically improved policy $\pi > \mu$ for optimistic planning:
+
+$$
+\begin{array} { r } { \mathcal { T } _ { \mu } ^ { \pi } Q ( s , \pmb { a } ) : = \underset { n \geq 0 } { \operatorname* { m a x } } \{ ( \mathcal { T } ^ { \mu } ) ^ { n } \mathcal { T } ^ { \pi } Q ( s , \pmb { a } ) \} , } \end{array}
+$$
+
+which does not require the environment to be deterministic. A disadvantage of using $\mathcal { T } ^ { \pi }$ to replace $\smash { \mathcal { T } _ { \mu } ^ { \tau } }$ is that we no longer have the monotonicity property (e.g., $\mathcal T ^ { \pi } Q ( s , { \pmb a } ) > \mathcal T ^ { \mu } Q ( s , { \pmb a } )$ always holds for any $Q$ ). However, we can still derive that the fixed point of $\mathcal { T } _ { \mu } ^ { \pi }$ is bounded between $Q ^ { \pi }$ and $Q ^ { * }$ (See Appendix C for detailed results and proofs).
+
+# G.2 EMAQ
+
+The high-level idea of the selecting-from-behavior-candidates approach bears some resemblance to the Expected-Max Q-Learning (EMaQ) algorithm proposed by (Ghasemipour et al., 2021). EMaQ is built upon BCQ (Fujimoto et al., 2019), which computes the training target in Q-Learning by:
+
+$$
+\mathcal { T } _ { \mathrm { B C Q } } ^ { * } Q ( s , a ) : = r ( s , a ) + \gamma \operatorname* { m a x } _ { a ^ { \prime } \sim \mu _ { \theta } ( \cdot \vert s ^ { \prime } ) } \{ Q ( s ^ { \prime } , a ^ { \prime } + \xi _ { \phi } ( s ^ { \prime } , a ^ { \prime } ) ) \} ,
+$$
+
+where $\xi _ { \phi } ( s , a )$ is an explicitly constrained perturbation network that helps relax the constraint of behavior policy $\mu$ . The core motivation for EMaQ is to remove the perturbation model $\xi _ { \phi } ( \pmb { s } , \pmb { a } )$ , by taking max over $_ \mathrm { N }$ Q-function evaluations:
+
+$$
+\mathcal { T } _ { \mathrm { E M a Q } } Q ( s , a ) : = r ( s , a ) + \gamma \mathbb { E } _ { \{ a _ { i } ^ { \prime } \} ^ { N } \sim \mu _ { \theta } ( \cdot \vert s ^ { \prime } ) } \big [ \operatorname* { m a x } _ { a _ { i } ^ { \prime } \in \{ a _ { i } ^ { \prime } \} ^ { N } } Q ( s ^ { \prime } , a _ { i } ^ { \prime } ) \big ] .
+$$
+
+For EMaQ, $N$ serves as a hyperparameter to interpolate $\mathcal { T } ^ { \mu }$ and $\tau ^ { * }$ . When $N = 1$ , $\tau _ { \mathrm { E M a Q } }$ becomes $\mathcal { T } ^ { \mu }$ . When $N \to \infty$ , $\tau _ { \mathrm { E M a Q } }$ approaches $\tau ^ { * }$ because $\{ { \pmb a } _ { i } ^ { \prime } \} ^ { N }$ nearly covers the whole action space.
+
+In contrast, for SfBC, the hyperparameter $N$ is the number of Monte Carlo samples used to estimate the training Q-targets:
+
+$$
+\begin{array} { r l r } {  { \mathcal { T } ^ { \pi } Q ( s , a ) = r ( s , a ) + \gamma \mathbb { E } _ { a ^ { \prime } \sim \pi ( \cdot \vert s ^ { \prime } ) } Q ( s ^ { \prime } , a ^ { \prime } ) } } \\ & { } & { = r ( s , a ) + \gamma \mathbb { E } _ { a ^ { \prime } \sim \mu ( \cdot \vert s ^ { \prime } ) } \frac { \exp { ( \alpha Q ( s ^ { \prime } , a ^ { \prime } ) ) } } { Z ( s ^ { \prime } ) } Q ( s ^ { \prime } , a ^ { \prime } ) } \\ & { } & { \approx r ( s , a ) + \gamma \sum _ { N } [ \frac { \exp { ( \alpha Q ( s ^ { \prime } , a ^ { \prime } ) ) } } { \sum _ { N } \exp { ( \alpha Q ( s ^ { \prime } , a ^ { \prime } ) ) } } Q ( s ^ { \prime } , a ^ { \prime } ) ] } \end{array}
+$$

parse/dev/LKEYuYNOqx/LKEYuYNOqx.md

@@ -0,0 +1,381 @@
+# GENIE: Higher-Order Denoising Diffusion Solvers
+
+Tim Dockhorn1,2,3,∗ Arash Vahdat1 Karsten Kreis1
+
+1NVIDIA 2University of Waterloo 3Vector Institute
+
+tim.dockhorn@uwaterloo.ca, {avahdat,kkreis}@nvidia.com
+
+# Abstract
+
+Denoising diffusion models (DDMs) have emerged as a powerful class of generative models. A forward diffusion process slowly perturbs the data, while a deep model learns to gradually denoise. Synthesis amounts to solving a differential equation (DE) defined by the learnt model. Solving the DE requires slow iterative solvers for high-quality generation. In this work, we propose Higher-Order Denoising Diffusion Solvers (GENIE): Based on truncated Taylor methods, we derive a novel higher-order solver that significantly accelerates synthesis. Our solver relies on higher-order gradients of the perturbed data distribution, that is, higher-order score functions. In practice, only Jacobian-vector products (JVPs) are required and we propose to extract them from the first-order score network via automatic differentiation. We then distill the JVPs into a separate neural network that allows us to efficiently compute the necessary higher-order terms for our novel sampler during synthesis. We only need to train a small additional head on top of the first-order score network. We validate GENIE on multiple image generation benchmarks and demonstrate that GENIE outperforms all previous solvers. Unlike recent methods that fundamentally alter the generation process in DDMs, our GENIE solves the true generative DE and still enables applications such as encoding and guided sampling. Project page and code: https://nv-tlabs.github.io/GENIE.
+
+# 1 Introduction
+
+Denoising diffusion models (DDMs) offer both state-of-the-art synthesis quality and sample diversity in combination with a robust and scalable learning objective. DDMs have been used for image [1–5] and video [6, 7] synthesis, super-resolution [8, 9], deblurring [10, 11], image editing and inpainting [5, 12–14], text-to-image synthesis [15–17], conditional and semantic image generation [18–22], imageto-image translation [14, 23, 24] and for inverse problems in medical imaging [25–31]. They also enable high-quality speech synthesis [32–37], 3D shape generation [38–42], molecular modeling [43– 46], maximum likelihood training [47–50], and more [51–56]. In DDMs, a diffusion process gradually perturbs the data towards random noise, while a deep neural network learns to denoise. Formally, the problem reduces to learning the score function, i.e., the gradient of the log-density of the perturbed data. The (approximate) inverse of the forward diffusion can be described by an ordinary or a stochastic differential equation (ODE or SDE, respectively), defined by the learned score function, and can therefore be used for generation when starting from random noise [47, 57].
+
+A crucial drawback of DDMs is that the generative ODE or SDE is typically difficult to solve, due to the complex score function. Therefore, efficient and tailored samplers are required for fast synthesis. In this work, building on the generative ODE [47, 57, 58], we rigorously derive a novel second-order ODE solver using truncated Taylor methods [59]. These higher-order methods require higher-order gradients of the ODE—in our case this includes higher-order gradients of the log-density of the perturbed data, i.e., higher-order score functions. Because such higher-order scores are usually not available, existing works typically use simple first-order solvers or samplers with low accuracy [1, 57, 58, 60], higher-order methods that rely on suboptimal finite difference or other approximations [61–63], or alternative approaches [64–66] for accelerated sampling. Here, we fundamentally avoid such approximations and directly model the higher-order gradient terms: Importantly, our novel Higher-Order Denoising Diffusion Solver (GENIE) relies on Jacobian-vector products (JVPs) involving second-order scores. We propose to calculate these JVPs by automatic differentiation of the regular learnt first-order scores. For computational efficiency, we then distill the entire higher-order gradient of the ODE, including the JVPs, into a separate neural network. In practice, we only need to add a small head to the first-order score network to predict the components of the higher-order ODE gradient. By directly modeling the JVPs we avoid explicitly forming high-dimensional higher-order scores. Intuitively, the higher-order terms in GENIE capture the local curvature of the ODE and enable larger steps when iteratively solving the generative ODE (Fig. 1).
+
+![](images/2fa56efdc3917106559f22cc915f86ecab674151e8c74d8a2bfbd5feb5c5f0e1.jpg)  
+Figure 1: Our novel Higher-Order Denoising Diffusion Solver (GENIE) relies on the second truncated Taylor method (TTM) to simulate a (re-parametrized) Probability Flow ODE for sampling from denoising diffusion models. The second TTM captures the local curvature of the ODE’s gradient field and enables more accurate extrapolation and larger step sizes than the first TTM (Euler’s method), which previous methods such as DDIM [58] utilize.
+
+Experimentally, we validate GENIE on multiple image modeling benchmarks and achieve state-ofthe-art performance in solving the generative ODE of DDMs with few synthesis steps. In contrast to recent methods that fundamentally modify the generation process of DDMs by training conditional GANs [67] or by distilling the full sampling trajectory [68, 69], GENIE solves the true generative ODE. Therefore, we also show that we can still encode images in the DDM’s latent space, as required for instance for image interpolation, and use techniques such as guided sampling [4, 57, 70].
+
+We make the following contributions: (i) We introduce GENIE, a novel second-order ODE solver for fast DDM sampling. (ii) We propose to extract the required higher-order terms from the first-order score model by automatic differentiation. In contrast to existing works, we explicitly work with higher-order scores without finite difference approximations. To the best of our knowledge, GENIE is the first method that explicitly uses higher-order scores for generative modeling with DDMs. (iii) We propose to directly model the necessary JVPs and distill them into a small neural network. (iv) We outperform all previous solvers and samplers for the generative differential equations of DDMs.
+
+# 2 Background
+
+We consider continuous-time DDMs [1, 57, 71] whose forward process can be described by
+
+$$
+p _ { t } ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 } ) = \mathcal { N } ( \mathbf { x } _ { t } ; \alpha _ { t } \mathbf { x } _ { 0 } , \sigma _ { t } ^ { 2 } I ) ,
+$$
+
+where $\mathbf { x } _ { 0 } \sim p _ { 0 } ( \mathbf { x } _ { 0 } )$ is drawn from the empirical data distribution and $\mathbf { x } _ { t }$ refers to diffused data samples at time $t \in [ 0 , 1 ]$ along the diffusion process. The functions $\alpha _ { t }$ and $\sigma _ { t }$ are generally chosen such that the logarithmic signal-to-noise ratio [48] $\begin{array} { r } { \log { \frac { \alpha _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } } } \end{array}$ decreases monotonically with $t$ and the data diffuses towards random noise, i.e., $p _ { 1 } ( \mathbf { x } _ { 1 } ) \approx \mathcal { N } ( \mathbf { x } _ { 1 } ; \mathbf { 0 } , \bar { I } )$ . We use variance-preserving [57] diffusion processes for which $\sigma _ { t } ^ { 2 } = 1 - \alpha _ { t } ^ { 2 }$ (however, all methods introduced in this work are applicable to more general DDMs). The diffusion process can then be expressed by the (variance-preserving) SDE
+
+$$
+\begin{array} { r } { d \mathbf { x } _ { t } = - \frac { 1 } { 2 } \beta _ { t } \mathbf { x } _ { t } d t + \sqrt { \beta _ { t } } d \mathbf { w } _ { t } , } \end{array}
+$$
+
+where diffusi $\begin{array} { r } { \beta _ { t } = - \frac { d } { d t } \log \alpha _ { t } ^ { 2 } } \end{array}$ , f $\mathbf { x } _ { 0 } \sim p _ { 0 } ( \mathbf { x } _ { 0 } )$ and rts th $\mathbf { w } _ { t }$ is a standard Wiener process. A corresponding reverseorward diffusion is given by [57, 72, 73]
+
+$$
+\begin{array} { r } { d \mathbf { x } _ { t } = - \frac { 1 } { 2 } \beta _ { t } \left[ \mathbf { x } _ { t } + 2 \nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } ) \right] d t + \sqrt { \beta _ { t } } d \mathbf { w } _ { t } , } \end{array}
+$$
+
+and this reverse-time generative SDE is marginally equivalent to the generative ODE [47, 57]
+
+$$
+\begin{array} { r } { d \mathbf { x } _ { t } = - \frac { 1 } { 2 } \beta _ { t } \left[ \mathbf { x } _ { t } + \nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } ) \right] d t , } \end{array}
+$$
+
+where $\nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } )$ is the score function. Eq. (4) is referred to as the Probability Flow ODE [57], an instance of continuous Normalizing flows [74, 75]. To generate samples from the DDM, one can sample $\mathbf { x } _ { 1 } \sim \mathcal { N } ( \mathbf { x } _ { 1 } ; \mathbf { 0 } , I )$ and numerically simulate either the Probability Flow ODE or the generative SDE, replacing the unknown score function by a learned score model $s _ { \theta } ( \mathbf { x } _ { t } , t ) \approx \nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } )$ .
+
+The DDIM solver [58] has been particularly popular to simulate DDMs due to its speed and simplicity. It has been shown that DDIM is Euler’s method applied to an ODE based on a re-parameterization of the Probability Flow ODE [58, 69]: Defining $\begin{array} { r } { \gamma _ { t } = \sqrt { \frac { 1 - \alpha _ { t } ^ { 2 } } { \alpha _ { t } ^ { 2 } } } } \end{array}$ and $\bar { \mathbf { x } } _ { t } = \mathbf { x } _ { t } \sqrt { 1 + \gamma _ { t } ^ { 2 } }$ , we have
+
+$$
+\frac { d \bar { \mathbf { x } } _ { t } } { d \gamma _ { t } } = \sqrt { 1 + \gamma _ { t } ^ { 2 } } \frac { d \mathbf { x } _ { t } } { d t } \frac { d t } { d \gamma _ { t } } + \mathbf { x } _ { t } \frac { \gamma _ { t } } { \sqrt { 1 + \gamma _ { t } ^ { 2 } } } = - \frac { \gamma _ { t } } { \sqrt { 1 + \gamma _ { t } ^ { 2 } } } \nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } ) ,
+$$
+
+where we inserted Eq. (4) for $\frac { d { \bf x } _ { t } } { d t }$ and used $\begin{array} { r } { \beta ( t ) \frac { d t } { d \gamma _ { t } } = \frac { 2 \gamma _ { t } } { \gamma _ { t } ^ { 2 } + 1 } } \end{array}$ . Letting $\begin{array} { r } { s _ { \theta } ( \mathbf { x } _ { t } , t ) : = - \frac { \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) } { \sigma _ { t } } } \end{array}$ denote a parameterization of the score model, the approximate generative DDIM ODE is then given by
+
+$$
+d \bar { \mathbf { x } } _ { t } = \epsilon _ { \theta } \left( \mathbf { x } _ { t } , t \right) d \gamma _ { t } ,
+$$
+
+where we used $\begin{array} { r } { \sigma _ { t } = \sqrt { 1 - \alpha _ { t } ^ { 2 } } = \frac { \gamma _ { t } } { \sqrt { \gamma _ { t } ^ { 2 } + 1 } } } \end{array}$ (see App. A for a more detailed derivation of Eq. (6)). The model $\epsilon _ { \theta } ( \mathbf { x } _ { t } , t )$ can be learned by minimizing the score matching objective [1, 76]
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } \mathbb { E } _ { t \sim \mathcal { U } [ t _ { \mathrm { c u t o f f } } , 1 ] , \mathbf { x } _ { 0 } \sim p ( \mathbf { x } _ { 0 } ) , \epsilon \sim \mathcal { N } ( \mathbf { 0 } , I ) } \left[ g ( t ) \lVert \epsilon - \epsilon _ { \boldsymbol { \theta } } ( \mathbf { x } _ { t } , t ) \rVert _ { 2 } ^ { 2 } \right] , \quad \mathbf { x } _ { t } = \alpha _ { t } \mathbf { x } _ { 0 } + \sigma _ { t } \epsilon ,
+$$
+
+for small $0 < t _ { \mathrm { c u t o f f } } \ll 1$ . As is standard practice, we set $g ( t ) = 1$ . Other weighting functions $g ( t )$ are possible; for example, setting $\begin{array} { r } { g ( t ) = \frac { \beta _ { t } } { 2 \sigma _ { t } ^ { 2 } } } \end{array}$ recovers maximum likelihood learning [47–50].
+
+# 3 Higher-Order Denoising Diffusion Solver
+
+As discussed in Sec. 2, the so-known DDIM solver [58] is simply Euler’s method applied to the DDIM ODE (cf. Eq. (6)). In this work, we apply a higher-order method to the DDIM ODE, building on the truncated Taylor method (TTM) [59]. The $p$ -th TTM is simply the $p$ -th order Taylor polynomial applied to an ODE. For example, for the general $\begin{array} { r } { \frac { d \mathbf { y } } { d t } = \pmb { f } ( \mathbf { y } , t ) } \end{array}$ , the $p$ -th TTM reads as
+
+$$
+\mathbf { y } _ { t _ { n + 1 } } = \mathbf { y } _ { t _ { n } } + h _ { n } { \frac { d \mathbf { y } } { d t } } | _ { ( \mathbf { y } _ { t _ { n } } , t _ { n } ) } + \cdot \cdot \cdot + { \frac { 1 } { p ! } } h _ { n } ^ { p } { \frac { d ^ { p } \mathbf { y } } { d t ^ { p } } } | _ { ( \mathbf { y } _ { t _ { n } } , t _ { n } ) } ,
+$$
+
+where $h _ { n } = t _ { n + 1 } - t _ { n }$ (see App. B.1 for a truncation error analysis with respect to the exact ODE solution). Note that the first TTM is simply Euler’s method. Applying the second TTM to the DDIM ODE results in the following scheme:
+
+$$
+\bar { \mathbf { x } } _ { t _ { n + 1 } } = \bar { \mathbf { x } } _ { t _ { n } } + h _ { n } \epsilon _ { \theta } ( \mathbf { x } _ { t _ { n } } , t _ { n } ) + \frac { 1 } { 2 } h _ { n } ^ { 2 } \frac { d \epsilon _ { \theta } } { d \gamma _ { t } } | _ { ( \mathbf { x } _ { t _ { n } } , t _ { n } ) } ,
+$$
+
+where $h _ { n } = \gamma _ { t _ { n + 1 } } - \gamma _ { t _ { n } }$ . Recall that $\begin{array} { r } { \gamma _ { t } = \sqrt { \frac { 1 - \alpha _ { t } ^ { 2 } } { \alpha _ { t } ^ { 2 } } } } \end{array}$ , where the function $\alpha _ { t }$ is a time-dependent hyperparameter of the DDM. The total derivative $\begin{array} { r } { d _ { \gamma _ { t } } \epsilon _ { \theta } : = \frac { d \epsilon _ { \theta } } { d \gamma _ { t } } } \end{array}$ can be decomposed as follows
+
+$$
+d _ { \gamma _ { t } } \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) = \frac { \partial \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) } { \partial \mathbf { x } _ { t } } \frac { d \mathbf { x } _ { t } } { d \gamma _ { t } } + \frac { \partial \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) } { \partial t } \frac { d t } { d \gamma _ { t } } ,
+$$
+
+where ∂ϵθ(xt,t) denotes the Jacobian of ϵθ(xt, t) and
+
+$$
+\frac { d \mathbf { x } _ { t } } { d \gamma _ { t } } = \frac { \partial \mathbf { x } _ { t } } { \partial \bar { \mathbf { x } } _ { t } } \frac { d \bar { \mathbf { x } } _ { t } } { d \gamma _ { t } } + \frac { \partial \mathbf { x } _ { t } } { \partial \gamma _ { t } } = \frac { 1 } { \sqrt { \gamma _ { t } ^ { 2 } + 1 } } \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) - \frac { \gamma _ { t } } { 1 + \gamma _ { t } ^ { 2 } } \mathbf { x } _ { t } .
+$$
+
+If not explicitly stated otherwise, we refer to the second TTM applied to the DDIM ODE, i.e., the scheme in Eq. (9), as Higher-Order Denoising Diffusion Solver (GENIE). Intuitively, the higher-order gradient terms used in the second TMM model the local curvature of the ODE. This translates into a Taylor formula-based extrapolation that is quadratic in time (cf. Eqs. (8) and (9)) and more accurate than linear extrapolation, as in Euler’s method, thereby enabling larger time steps (see Fig. 1 for a visualization). In App. B, we also discuss the application of the third TTM to the DDIM ODE. We emphasize that TTMs are not restricted to the DDIM ODE and could just as well be applied to the Probability Flow ODE [57] (also see App. B) or neural ODEs [74, 75] more generally.
+
+![](images/6bb6f978f610d0b9ad0c8360c80c2ee271995905931f2e32afad6cfc621481de.jpg)  
+Figure 2: Modeling a complex 2D toy distribution: Samples in $( b )$ and (c) are generated via DDIM and GENIE, respectively, with 25 solver steps using the analytical score function of the ground truth distribution.
+
+The Benefit of Higher-Order Methods: We showcase the benefit of higher-order methods on a 2D toy distribution (Fig. 2a) for which we know the score function as well as all higher-order derivatives necessary for GENIE analytically. We generate 1k different accurate “ground truth” trajectories $\mathbf { x } _ { t }$ using DDIM with $1 0 \mathrm { k }$ steps. We compare these “ground truth” trajectories to single steps of DDIM and GENIE for varying step sizes $\Delta t$ . We then measure the mean $L _ { 2 }$ -distance of the single steps $\hat { \mathbf { x } } _ { t } ( \Delta t )$ to the “ground truth” trajectories $\mathbf { x } _ { t }$ , and we repeat this experiment for three starting points $t \in \{ 0 . 1 , 0 . 2 , 0 . 5 \}$ . We see (Fig. 3 (top)) that GENIE can use larger step sizes to stay within a certain error tolerance for all starting points $t$ . We further show samples for DDIM and GENIE, using 25 solver steps, in Fig. 2. DDIM has the undesired behavior of sampling low-density regions between modes, whereas GENIE looks like a slightly noisy version of the ground truth distribution (Fig. 2a).
+
+Comparison to Multistep Methods: Linear multistep methods are an alternative higher-order method to solve ODEs. Liu et al. [63] applied the well-established Adams–Bashforth [AB, 77] method to the DDIM ODE. AB methods can be derived from TTMs by approximating higher-order derivatives $\frac { d ^ { p } \mathbf { y } } { d t ^ { p } }$ using the finite difference method [78]. For example, the second AB method is obtained from the second TTM by replacing $\textstyle { \frac { d ^ { 2 } \mathbf { y } } { d t ^ { 2 } } }$ with the first-order forward difference approximation $( f ( { \bf y } _ { t _ { n } } , t _ { n } ) - f ( { \bf y } _ { t _ { n - 1 } } , t _ { n - 1 } ) ) / h _ { n - 1 }$ . In Fig. 3 (bottom), we visualize the mean $L _ { 2 }$ -norm of the difference $\xi _ { t } ( \Delta t )$ between the analytical derivative $d _ { \gamma _ { t } } \epsilon _ { \theta }$ and its first-order forward difference approximation for varying step sizes $\Delta t$ for the 2D toy distribution. The approximation is especially poor at small $t$ for which the score function becomes complex (App. E for details on all toy experiments).
+
+![](images/2ad9de1841582ccc50eb5d0fa23d979238a408c542c4fd3cc8dc0127a2d94204.jpg)  
+Figure 3: Top: Single step error using analytical score function. Bottom: Norm of difference $\xi _ { t } ( \Delta t )$ between analytical and approximate derivative computed via finite difference method.
+
+# 3.1 Learning Higher-Order Derivatives
+
+The above observations inspire to apply GENIE to DDMs of more complex and high-dimensional data such as images. Regular DDMs learn a model $\epsilon _ { \theta }$ for the first-order score; however, the higher-order gradient terms required for GENIE (cf. Eq. (10)) are not immediately available to us, unlike in the toy example above. Let us insert Eq. (11) into Eq. (10) and analyze the required terms more closely:
+
+$$
+d _ { \gamma _ { t } } \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) = \frac { 1 } { \sqrt { \gamma _ { t } ^ { 2 } + 1 } } \underbrace { \frac { \partial \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) } { \partial \mathbf { x } _ { t } } \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) } _ { \mathrm { J V P _ { 1 } } } - \frac { \gamma _ { t } } { 1 + \gamma _ { t } ^ { 2 } } \underbrace { \frac { \partial \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) } { \partial \mathbf { x } _ { t } } \mathbf { x } _ { t } } _ { \mathrm { J V P _ { 2 } } } + \frac { \partial \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) } { \partial t } \frac { d t } { d \gamma _ { t } } .
+$$
+
+We see that the full derivative decomposes into two JVP terms and one simpler time derivative term. The term $\frac { \partial \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) } { \partial \mathbf { x } _ { t } }$ plays a crucial role in Eq. (12). It can be expressed as
+
+$$
+\frac { \partial \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) } { \partial \mathbf { x } _ { t } } = - \sigma _ { t } \frac { \partial \mathbf { s } _ { \theta } ( \mathbf { x } _ { t } , t ) } { \partial \mathbf { x } _ { t } } \approx - \sigma _ { t } \nabla _ { \mathbf { x } _ { t } } ^ { \top } \nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } ) ,
+$$
+
+which means that GENIE relies on second-order score functions $\nabla _ { \mathbf { x } _ { t } } ^ { \top } \nabla _ { \mathbf { x } _ { t } } \log p _ { t } ( \mathbf { x } _ { t } )$ under the hood.
+
+Given a DDM, that is, given $\epsilon _ { \theta }$ , we could compute the derivative $d _ { \gamma _ { t } } \epsilon _ { \theta }$ for the GENIE scheme in Eq. (9) using automatic differentiation (AD). This would, however, make a single step of GENIE at least twice as costly as DDIM, because we would need a forward pass through the $\epsilon _ { \theta }$ network to compute $\epsilon _ { \theta } ( \mathbf { x } _ { t } , t )$ itself, and another pass to compute the JVPs and the time derivative in Eq. (12). These forward passes cannot be parallelized, since the vector-part of $\mathrm { J V P _ { 1 } }$ in Eq. (12) involves $\epsilon _ { \theta }$ itself, and needs to be known before computing the JVP. To accelerate sampling, this overhead is too expensive.
+
+Gradient Distillation: To avoid this overhead, we propose to first distill $d _ { \gamma _ { t } } \epsilon _ { \theta }$ into a separate neural network. During distillation training, we can use the slow AD-based calculation of $d _ { \gamma _ { t } } \epsilon _ { \theta }$ , but during synthesis we call the trained neural network. We build on the observation that the internal representations of the neural network modeling $\epsilon _ { \theta }$ (in our case a U-Net [79] architecture) can be used for downstream tasks [80, 81]: specifically, we provide the last feature layer from the $\epsilon _ { \theta }$ network together with its time embedding as well as $\mathbf { x } _ { t }$ and the output $\epsilon _ { \theta } ( \mathbf { x } _ { t } , t )$ to a small prediction head $k _ { \psi } ( \mathbf { x } _ { t } , t )$ that models the different terms in Eq. (12) (see Fig. 4). The overhead generated by $k _ { \psi }$ is small, for instance less than $2 \%$ for our CIFAR-10 model (also see Sec. 5), and we found this approach to provide excellent performance. Note that in principle we could also train an independent deep neural network, which does not make use of the internal representations of $\epsilon _ { \theta }$ and could therefore theoretically be run in parallel to the $\epsilon _ { \theta }$ model. We justify using small prediction heads over independent neural networks because AD-based distillation training is slow: in each training iteration we first need to call the $\epsilon _ { \theta }$ network, then calculate the JVP terms, and only then can we call the distillation model. By modeling $d _ { \gamma _ { t } } \epsilon _ { \theta }$ via small prediction heads, while reusing the internal representation of the score model, we can make training relatively fast: we only need to train $k _ { \psi }$ for up to 50k iterations. In contrast, training score models from scratch takes roughly an order of magnitude more iterations. We leave training of independent networks to predict $d _ { \gamma _ { t } } \epsilon _ { \theta }$ to future work.
+
+![](images/84e196cce59effec39df1f43f06ce007ed2847fd24211471c891706b6af15d94.jpg)  
+Figure 4: Our distilled model $k _ { \psi }$ that predicts the gradient $d _ { \gamma _ { t } } \epsilon _ { \theta }$ is implemented as a small additional output head on top of the first-order score model $\epsilon _ { \theta }$ . Purple layers are used both in $\epsilon _ { \theta }$ and $k _ { \psi }$ ; green layers are specific for $\epsilon _ { \theta }$ and $k _ { \psi }$ .
+
+Mixed Network Parameterization: We found that learning $d _ { \gamma _ { t } } \epsilon _ { \theta }$ directly as single output of a neural network can be challenging. Assuming a single data point distribution $p _ { 0 } ( \mathbf { x } _ { 0 } ) = \delta ( \mathbf { x } _ { 0 } = \mathbf { 0 } )$ , for which we know the diffused score function and all higher-order derivatives analytically, we found that the terms in Eq. (12) all behave very differently within the $t \in [ 0 , 1 ]$ interval (for instance, the prefactor of $\mathrm { J V P _ { 1 } }$ in Eq. (12) approaches 1 as $t  0$ , while $\mathrm { J V P _ { 2 } }$ ’s prefactor vanishes). As outlined in detail in App. C.2.3, this simple single data point assumption implies an effective mixed network parameterization, an approach inspired by the “mixed score parametrizations” in Vahdat et al. [49] and Dockhorn et al. [60]. In particular, we model
+
+$$
+k _ { \psi } = - \frac { 1 } { \gamma _ { t } } k _ { \psi } ^ { ( 1 ) } + \frac { \gamma _ { t } } { 1 + \gamma _ { t } ^ { 2 } } k _ { \psi } ^ { ( 2 ) } + \frac { 1 } { \gamma _ { t } ( 1 + \gamma _ { t } ^ { 2 } ) } k _ { \psi } ^ { ( 3 ) } \approx d _ { \gamma _ { t } } \epsilon _ { \theta } ,
+$$
+
+where $k _ { \psi } ^ { ( i ) } ( \mathbf { x } _ { t } , t ) , i \in \{ 1 , 2 , 3 \}$ , are different output channels of the neural network (i.e. the additional head on top of the $\epsilon _ { \theta }$ network). The three terms in Eq. (14) exactly correspond to the three terms of Eq. (12), in the same order. We show the superior performance of this parametrization in Sec. 5.3.
+
+Learning Objective: Ideally, we would like our model $k _ { \psi }$ to match $d _ { \gamma _ { t } } \epsilon _ { \theta }$ exactly, for all $t \in [ 0 , T ]$ and $\mathbf { x } _ { t }$ in the diffused data distribution, which the generative ODE trajectories traverse. This suggests a simple (weighted) $L _ { 2 }$ -loss, similar to regular score matching losses for DDMs [1, 57]:
+
+$$
+\operatorname* { m i n } _ { \psi } \mathbb { E } _ { t \sim \mathcal { U } [ t _ { \mathrm { c u t o f f } } , 1 ] , \mathbf { x } _ { 0 } \sim p ( \mathbf { x } _ { 0 } ) , \epsilon \sim \mathcal { N } ( \mathbf { 0 } , I ) } \left[ g _ { \mathrm { d } } ( t ) \| k _ { \psi } ( \alpha _ { t } \mathbf { x } _ { 0 } + \sigma _ { t } \epsilon , t ) - d _ { \gamma _ { t } } \epsilon _ { \theta } ( \alpha _ { t } \mathbf { x } _ { 0 } + \sigma _ { t } \epsilon , t ) \| _ { 2 } ^ { 2 } \right]
+$$
+
+for diffused data points $\alpha _ { t } \mathbf { x } _ { 0 } + \sigma _ { t } \mathbf { \epsilon } \mathbf { \epsilon }$ and $g _ { \mathrm { d } } ( t ) = \gamma _ { t } ^ { 2 }$ to counteract the $1 / \gamma _ { t }$ in the first and third terms of Eq. (14). This leads to a roughly constant loss over different time values $t$ . During training we compute $d _ { \gamma _ { t } } \epsilon _ { \theta }$ via AD; however, at inference time we use the learned prediction head $k _ { \psi }$ to approximate $d _ { \gamma _ { t } } \epsilon _ { \theta }$ . In App. C.2.4, we provide pseudo code for training and sampling with heads $k _ { \psi }$ . Note that our distillation objective is consistent and principled: if $k _ { \psi }$ matches $d _ { \gamma _ { t } } \epsilon _ { \theta }$ exactly, the resulting GENIE algorithm recovers the second TTM exactly (extended discussion in App. B.4).
+
+Alternative Learning Approaches: As shown in Eq. (13), GENIE relies on second-order score functions. Recently, Meng et al. [82] directly learnt such higher-order scores with higher-order score matching objectives. Directly applying these techniques has the downside that we would need to explicitly form the higher-order score terms $\nabla _ { \mathbf { x } _ { t } } ^ { \top } \epsilon _ { \theta } ( \mathbf { x } _ { t } , \dot { t } )$ , which are very high-dimensional for data such as images. Low-rank approximations are possible, but potentially insufficient for high performance. In our approach, we are avoiding this complication by directly modeling the lower-dimensional JVPs. We found that the methods from Meng et al. [82] can be modified to provide higher-order score matching objectives for the JVP terms required for GENIE and we briefly explored this (see App. D). However, our distillation approach with AD-based higher-order gradients worked much better. Nevertheless, this is an interesting direction for future research. To the best of our knowledge, GENIE is the first solver for the generative differential equations of DDMs that directly uses higher-order scores (in the form of the distilled JVPs) for generative modeling without finite difference or other approximations.
+
+# 4 Related Work
+
+Accelerated Sampling from DDMs. Several previous works address the slow sampling of DDMs: One line of work reduces and readjusts the timesteps [3, 64] used in time-discretized DDMs [1, 71]. This can be done systematically by grid search [32] or dynamic programming [83]. Bao et al. [65] speed up sampling by defining a new DDM with optimal reverse variances. DDIM [58], discussed in Sec. 2, was also introduced as a method to accelerate DDM synthesis. Further works leverage modern ODE and SDE solvers for fast synthesis from (continuous-time) DDMs: For instance, higher-order Runge–Kutta methods [57, 84] and adaptive step size SDE solvers [62] have been used. These methods are not optimally suited for the few-step synthesis regime, in which GENIE shines; see also Sec. 5. Most closely related to our work is Liu et al. [63], which simulates the DDIM ODE [58] using a higher-order linear multistep method [77]. As shown in Sec. 3, linear multistep methods can be considered an approximation of the TTMs used in GENIE. Furthermore, Tachibana et al. [61] solve the generative SDE via a higher-order Itô–Taylor method [59] and in contrast to our work, they propose to use an “ideal derivative trick” to approximate higher-order score functions. In App. B.2, we show that applying this ideal derivative approximation to the DDIM ODE does not have any effect: the “ideal derivatives” are zero by construction. Note that in GENIE, we in fact use the DDIM ODE, rather than, for example, the regular Probability Flow ODE [57], as the base ODE for GENIE.
+
+Alternatively, sampling from DDMs can also be accelerated via learning: For instance, Watson et al. [66] learn parameters of a generalized family of DDMs by optimizing for perceptual output quality. Luhman and Luhman [68] and Salimans and Ho [69] distill a DDIM sampler into a student model, which enables sampling in as few as a single step. Xiao et al. [67] replace DDMs’ Gaussian samplers with expressive generative adversarial networks, similarly allowing for few-step synthesis. GENIE can also be considered a learning-based approach, as we distill a derivative of the generative ODE into a separate neural network. However, in contrast to the mentioned methods, GENIE still solves the true underlying generative ODE, which has major advantages: for instance, it can still be used easily for classifier-guided sampling [4, 57, 70] and to efficiently encode data into latent space—a prerequisite for likelihood calculation [47, 57] and editing applications [17]. Note that the learnt sampler [66] defines a proper probabilistic generalized DDM; however, it isn’t clear how it relates to the generative SDE or ODE and therefore how compatible the method is with applications such as classifier guidance.
+
+Other approaches to accelerate DDM sampling change the diffusion itself [60, 85, 86] or train DDMs in the latent space of a Variational Autoencoder [49]. GENIE is complementary to these methods.
+
+Higher-Order ODE Gradients beyond DDMs. TTMs [78] and other methods that leverage higherorder gradients are also applied outside the scope of DDMs. For instance, higher-order derivatives can play a crucial role when developing solvers [87] and regularization techniques [88, 89] for neural ODEs [74, 75]. Outside the field of machine learning, higher-order TTMs have been widely studied, for example, to develop solvers for stiff [90] and non-stiff [90, 91] systems.
+
+Concurrent Works. Zhang and Chen [92] motivate the DDIM ODE from an exponential integrator perspective applied to the Probability Flow ODE and propose to apply existing solvers from the numerical ODE literature, namely, Runge–Kutta and linear multistepping, to the DDIM ODE directly. Lu et al. [93] similarly recognize the semi-linear structure of the Probability Flow ODE, derive dedicated solvers, and introduce new step size schedulers to accelerate DDM sampling. Karras et al. [94] propose new fast solvers, both deterministic and stochastic, specifically designed for the differential equations arising in DDMs. Both Zhang et al. [95] and Karras et al. [94] realize that the DDIM ODE has “straight line solution trajectories” for spherical normal data and single data points—this exactly corresponds to our derivation that the higher-order terms in the DDIM ODE are zero in such a setting (see App. B.2). Bao et al. [96] learn covariance matrices for DDM sampling using prediction heads somewhat similar to the ones in GENIE; in App. G.1, we thoroughly discuss the differences between GENIE and the method proposed in Bao et al. [96].
+
+# 5 Experiments
+
+Datasets: We run experiments on five datasets: CIFAR-10 [97] (resolution 32), LSUN Bedrooms [98] (128), LSUN Church-Outdoor [98] (128), (conditional) ImageNet [99] (64), and AFHQv2 [100] (512). On AFHQv2 we only consider the subset of cats; referred to as “Cats” in the remainder of this work.
+
+Architectures: Except for CIFAR-10 (we use a checkpoint by Song et al. [57]), we train our own score models using architectures introduced by previous works [1, 4]. The architecture of our prediction heads is based on (modified) BigGAN residual blocks [57, 101]. To minimize computational overhead, we only use a single residual block. See App. C for training and architecture details.
+
+Evaluation: We measure sample quality via Fréchet Inception Distance [FID, 102] (see App. F.1).
+
+Synthesis Strategy: We simulate the DDIM ODE from $t { = } 1$ up to $t { = } 1 0 ^ { - 3 }$ using evaluation times following a quadratic function (quadratic striding [58]). For variance-preserving DDMs, it can be beneficial to denoise the ODE solver output at the cutoff $t { = } 1 0 ^ { - 3 }$ , i.e., $\begin{array} { r } { \mathbf { x } _ { 0 } = \frac { \mathbf { x } _ { t } - \sigma _ { t } \epsilon _ { \theta } \left( \mathbf { x } _ { t } , t \right) } { \alpha _ { t } } } \end{array}$ [57, 103]. Note that the denoising step involves a score model evaluation, and therefore “loses” a function evaluation that could otherwise be used as an additional step in the ODE solver. To this end, denoising the output of the ODE solver is left as a hyperparameter of our synthesis strategy.
+
+Analytical First Step (AFS): Every additional neural network call becomes crucial in the low number of function evaluations (NFEs) regime. We found that we can improve the performance of GENIE and all other methods evaluated on our checkpoints by replacing the learned score with the (analytical) score of $\mathcal { N } ( \mathbf { 0 } , I ) \approx p _ { t = 1 } ( \mathbf { x } _ { t } )$ in the first step of the ODE solver. The “gained” function evaluation can then be used as an additional step in the ODE solver. Similarly to the denoising step mentioned above, AFS is treated as a hyperparameter of our Synthesis Strategy. AFS details in App. F.2.
+
+Accounting for Computational Overhead: GENIE has a slightly increased computational overhead compared to other solvers due to the prediction head $k _ { \psi }$ . The computational overhead is increased by $1 . 4 7 \%$ , $2 . 8 3 \%$ , $1 4 . 0 \%$ , and $1 4 . 4 \%$ on CIFAR-10, ImageNet, LSUN Bedrooms, and LSUN ChurchOutdoor, respectively (see also App. C.2.5). This additional overhead is always accounted for implicitly: we divide the NFEs by the computational overhead and round to the nearest integer. For example, on LSUN Bedrooms, we compare baselines with 10/15 NFEs to GENIE with 9/13 NFEs.
+
+# 5.1 Image Generation
+
+In Fig. 5 we compare our method to the most competitive baselines. In particular, on the same score model checkpoints, we compare GENIE with DDIM [58], S-PNDM [63], and F-PNDM [63]. For these four methods, we only include the best result over the two hyperparameters discussed above, namely, the denoising step and AFS (see App. F.6 for tables with all results). We also include three competitive results from the literature [64–66] that use different checkpoints and sampling strategies: for each method, we include the best result for their respective set of hyperparameters. We do not compare in this figure with Knowledge Distillation [KD, 68], Progressive Distillation [PG, 69] and Denoising Diffusion GANs [DDGAN, 67] as they do not solve the generative ODE/SDE and use fundamentally different sampling approaches with drawbacks discussed in Sec. 4.
+
+For $\mathrm { N F E s } \in \{ 1 0 , 1 5 , 2 0 , 2 5 \}$ , GENIE outperforms all baselines (on the same checkpoint) on all four datasets (see detailed results in App. F.6 and GENIE image samples in App. F.7). On CIFAR-10 and (conditional) ImageNet, GENIE also outperforms these baselines for NFE $\mathord { \mathrm { : } } = 5$ , whereas DDIM outperforms GENIE slightly on the LSUN datasets (see tables in App. F.6). GENIE also performs better than the three additional baselines from the literature (which use different checkpoints and sampling strategies) with the exception of the Learned Sampler [LS, 66] on LSUN Bedrooms for $\mathrm { N F E s } { = } 2 0$ . Though LS uses a learned striding schedule on LSUN Bedrooms (whereas GENIE simply uses quadratic striding), the LS’s advantage is most likely due to the different checkpoint. In Tab. 1, we investigate the effect of optimizing the striding schedule, via learning (LS) or grid search (DDIM & GENIE), on CIFAR-10 and find that its significance decreases rapidly with increased NFEs (also see App. F.6 for details). In Tab. 1, we also show additional baseline results; however, we do not include commonly-used adaptive step size solvers in Fig. 5, as they are arguably not well-suited for this low NFE regime: for example, on the same CIFAR-10 checkpoint we use for GENIE, the adaptive SDE solver introduced in Jolicoeur-Martineau et al. [62] obtains an FID of 82.4 at 48 NFEs. Also on the same checkpoint, the adaptive Runge–Kutta 4(5) [84] method applied to the ProbabilityFlow ODE achieves an FID of 13.1 at 38 NFEs (solver tolerances set to $1 0 ^ { - 2 }$ ).
+
+![](images/78b15833e9ea19268d437b4335596de5b81475277c99e91652b7c2adfe6f619a.jpg)
+
+![](images/da7e06b595c1bdc0ecf48721fa115cd3178fce040cfaf2a8f547c48b1067c9eb.jpg)  
+Figure 5: Unconditional performance on four popular benchmark datasets. The first four methods use the same score model checkpoints, whereas the last three methods all use different checkpoints. (†): numbers are taken from literature.   
+Table 1: Unconditional CIFAR-10 generative performance (measured in FID). Methods above the middle line use the same score model checkpoint; methods below all use different ones. (†): numbers are taken from literature. (\*): methods either learn an optimal striding schedule (Learned Sampler) or do a small grid search over striding schedules (DDIM & GENIE); also see App. F.6
+
+<table><tr><td>Method</td><td>NFEs=5</td><td>NFEs=10</td><td>NFEs=15</td><td>NFEs=20</td><td>NFEs=25</td></tr><tr><td>GENIE(ours) (*)</td><td>11.2</td><td>5.28</td><td>4.49</td><td>3.94</td><td>3.64</td></tr><tr><td>GENIE (ours)</td><td>13.9</td><td>5.97</td><td>4.49</td><td>3.94</td><td>3.67</td></tr><tr><td>DDIM[58] (*)</td><td>27.6</td><td>11.2</td><td>7.35</td><td>5.87</td><td>5.16</td></tr><tr><td>DDIM[58]</td><td>29.7</td><td>11.2</td><td>7.35</td><td>5.87</td><td>5.16</td></tr><tr><td>S-PNDM[63]</td><td>35.9</td><td>10.3</td><td>6.61</td><td>5.20</td><td>4.51</td></tr><tr><td>F-PNDM[63]</td><td>N/A</td><td>N/A</td><td>10.3</td><td>5.96</td><td>4.73</td></tr><tr><td>Euler-Maruyama</td><td>325</td><td>230</td><td>164</td><td>112</td><td>80.3</td></tr><tr><td>FastDDIM [64] (t)</td><td>1</td><td>9.90</td><td>-</td><td>5.05</td><td>-</td></tr><tr><td>Learned Sampler 66] (t/*)</td><td>12.4</td><td>7.86</td><td>5.90</td><td>4.72</td><td>4.25</td></tr><tr><td>Learned Sampler [66] (t)</td><td>14.3</td><td>8.15</td><td>5.94</td><td>4.89</td><td>4.47</td></tr><tr><td>Analytic DDIM [65] (t)</td><td>-</td><td>14.0</td><td>-</td><td>-</td><td>5.71</td></tr><tr><td>CLD-SGM[60]</td><td>334</td><td>306</td><td>236</td><td>162</td><td>106</td></tr><tr><td>VESDE-PC[57]</td><td>461</td><td>461</td><td>461</td><td>461</td><td>462</td></tr></table>
+
+The results in Fig. 5 suggest that higher-order gradient information, as used in GENIE, can be efficiently leveraged for image synthesis. Despite using small prediction heads our distillation seems to be sufficiently accurate: for reference, replacing the distillation heads with the derivatives computed via AD, we obtain FIDs of 9.22, 4.11, 3.54, 3.46 using 10, 20, 30, and 40 NFEs, respectively (NFEs adjusted assuming an additional computational overhead of $100 \%$ ). As discussed in Sec. 3, linear multistep methods such as S-PNDM [63] and F-PNDM [63] can be considered (finite difference) approximations to TTMs as used in GENIE. These approximations can be inaccurate for large timesteps, which potentially explains their inferior performance when compared to GENIE. When compared to DDIM, the superior performance of GENIE seems to become less significant for large NFE: this is in line with the theory, as higher-order gradients contribute less for smaller step sizes (see the GENIE scheme in Eq. (9)). Approaches such as FastDDIM [64] and AnalyticDDIM [65], which adapt variances and discretizations of discrete-time DDMs, are useful; however, GENIE suggests that rigorous higher-order ODE solvers leveraging the continuous-time DDM formalism are still more powerful. To the best of our knowledge, the only methods that outperform GENIE abandon this ODE or SDE formulation entirely and train NFE-specific models [67, 69] which are optimized for the single use-case of image synthesis.
+
+# 5.2 Guidance and Encoding
+
+As discussed in Sec. 4, one major drawback of approaches such as KD [68], PG [69] and DDGAN [67] is that they abandon the ODE/SDE formalism, and cannot easily use methods such as classifier(-free) guidance [57, 70] or perform image encoding. However, these techniques can play an important role in synthesizing photorealistic images from DDMs [3, 4, 15, 17], as well as for image editing tasks [12, 17].
+
+Classifier-Free Guidance [70]: We replace the unconditional model $\epsilon _ { \theta } ( \mathbf { x } _ { t } , t )$ with $\hat { \epsilon } _ { \theta } ( \mathbf { x } _ { t } , t , c , w ) = ( 1 + w ) \epsilon _ { \theta } ( \mathbf { x } _ { t } , t , c ) -$ $w \epsilon _ { \theta } ( \mathbf { x } _ { t } , t )$ in the DDIM ODE (cf Eq. (6)), where $\epsilon _ { \theta } ( \mathbf { x } _ { t } , t , c )$ is a conditional model and $w > 1 . 0$ is the “guidance scale”. GENIE then requires the derivative
+
+![](images/a7c81a8ba04762e096723835d21c426ac67299bce0e01d2dca2db0977bd15121.jpg)  
+Figure 6: Sample quality as a function of guidance scale on ImageNet.
+
+![](images/3c429bd17bd930a87cae4e149e69b8a4cd2c559750055bce4e4f24be870a5954.jpg)  
+Figure 8: Encoding and subsequent decoding on LSUN Church-Outdoor. Left: Visual reconstruction. Right: $L _ { 2 }$ -distance to reference in Inception feature space [104], averaged over 100 images.
+
+$$
+\begin{array} { r } { d _ { \gamma _ { t } } \hat { \epsilon } _ { \theta } ( \mathbf { x } _ { t } , t , c , w ) = ( 1 + w ) d _ { \gamma _ { t } } \epsilon _ { \theta } ( \mathbf { x } _ { t } , t , c ) - w d _ { \gamma _ { t } } \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) . } \end{array}
+$$
+
+for guidance. Hence, we need to distill $d _ { \gamma _ { t } } \mathbf { \epsilon } _ { \theta } ( \mathbf { x } _ { t } , t , c )$ and $d _ { \gamma _ { t } } \mathbf { \epsilon } _ { \theta } ( \mathbf { x } _ { t } , t )$ , for which we could also share parameters [70]. We compare GENIE with DDIM on ImageNet in Fig. 6. GENIE clearly outperforms DDIM, in particular for few NFEs, and GENIE also synthesizes high-quality images (see Fig. 7).
+
+Image Encoding: We can use GENIE also to solve the generative ODE in reverse to encode given images. Therefore, we compare GENIE to DDIM on the “encode-decode” task, analyzing reconstructions for different NFEs (used twice for encoding and decoding): We find that GENIE reconstructs images much more accurately (see Fig. 8). For more details on this experiment as well as the guidance experiment above, see App. F.4 and App. F.3, respectively. We also show latent space interpolations for both GENIE and DDIM in App. F.5.
+
+# 5.3 Ablation Studies
+
+Table 2: CIFAR-10 ablation studies (measured in FID).   
+
+<table><tr><td>Ablation</td><td>NFEs=5</td><td>NFEs=10</td><td>NFEs=15</td><td>NFEs=20</td><td>NFEs=25</td></tr><tr><td>Standard</td><td>13.9</td><td>6.04</td><td>4.49</td><td>3.94</td><td>3.67</td></tr><tr><td>No mixed</td><td>14.7</td><td>6.32</td><td>4.82</td><td>4.31</td><td>4.10</td></tr><tr><td>No weighting</td><td>14.8</td><td>7.45</td><td>5.89</td><td>5.17</td><td>4.80</td></tr><tr><td>Bigger model</td><td>13.7</td><td>5.58</td><td>4.46</td><td>4.05</td><td>3.77</td></tr></table>
+
+We perform ablation studies over architecture and training objective for the prediction heads used in GENIE: In Tab. 2, “No mixed” refers to learning $d _ { \gamma _ { t } } \epsilon _ { \theta }$ directly as single network output without mixed network
+
+parameterization; “No weighting” refers to setting $g _ { \mathrm { d } } ( t ) = 1$ in Eq. (15); “Standard” uses both the mixed network parameterization and the weighting function $g _ { \mathrm { d } } ( t ) \stackrel { \cdot } { = } \gamma _ { t } ^ { 2 }$ . We can see that having both the mixed network parametrization and the weighting function is clearly beneficial. We also tested deeper networks in the prediction heads: for “Bigger model” we increased the number of residual blocks from one to two. The performance is roughly on par with “Standard”, and we therefore opted for the smaller head due to the lower computational overhead.
+
+# 5.4 Upsampling
+
+Table 3: Cats (upsampler) generative performance (measured in FID).   
+
+<table><tr><td>Method</td><td>NFEs=5</td><td>NFEs=10</td><td>NFEs=15</td></tr><tr><td>GENIE (ours)</td><td>5.53</td><td>4.90</td><td>4.83</td></tr><tr><td>DDIM [58]</td><td>9.47</td><td>6.64</td><td>5.85</td></tr><tr><td>S-PNDM[63]</td><td>14.6</td><td>11.0</td><td>8.83</td></tr><tr><td>F-PNDM [[63]</td><td>N/A</td><td>N/A</td><td>11.7</td></tr></table>
+
+Cascaded diffusion model pipelines [2] and DDM-based super-resolution [8] have become crucial ingredients in DDMs for large-scale image generation [105]. Hence, we also explore the applicability of GENIE in this setting. We train a $1 2 8 \times 1 2 8$ base model as well as a $1 2 8 \times 1 2 8 \to 5 1 2 \times 5 1 2$ diffusion upsampler [2, 8] on
+
+Cats. In Tab. 3, we compare the generative performance of GENIE to other fast samplers for the upsampler (in isolation). We find that GENIE performs very well on this task: with only five NFEs GENIE outperforms all other methods at $\mathrm { N F E s } { = } 1 5$ . We show upsampled samples for GENIE with $\mathrm { N F E s } { = } 5$ in Fig. 9. For more quantitative and qualitative results, we refer to App. F.6 and App. F.7, respectively. Training and inference details for the score model and the GENIE prediction head, for both base model and upsampler, can be found in App. C.
+
+![](images/1f8da56958a3893eba42484c8c83ad8c44ce8fe9311b671d4095b82218275ccc.jpg)  
+Figure 9: High-resolution images generated with the $1 2 8 \times 1 2 8 \to 5 1 2 \times 5 1 2$ GENIE upsampler using only five neural network calls. For the two images at the top, the upsampler is conditioned on test images from the Cats dataset. For the two images at the bottom, the upsampler is conditioned on samples from the $1 2 8 \times 1 2 8$ GENIE base model (generated using 25 NFEs); an upsampler neural network evaluation is roughly four times as expensive as a base model evaluation.
+
+# 6 Conclusions
+
+We introduced GENIE, a higher-order ODE solver for DDMs. GENIE improves upon the commonly used DDIM solver by capturing the local curvature of its ODE’s gradient field, which allows for larger step sizes when solving the ODE. We further propose to distill the required higher-order derivatives into a small prediction head—which we can efficiently call during inference—on top of the first-order score network. A limitation of GENIE is that it is still slightly slower than approaches that abandon the differential equation framework of DDMs altogether, which, however, comes at the considerable cost of preventing applications such as guided sampling. To overcome this limitation, future work could leverage even higher-order gradients to accelerate sampling from DDMs even further (also see App. G.2).
+
+Broader Impact. Fast synthesis from DDMs, the goal of GENIE, can potentially make DDMs an attractive method for promising interactive generative modeling applications, such as digital content creation or real-time audio synthesis, and also reduce DDMs’ environmental footprint by decreasing the computational load during inference. Although we validate GENIE on image synthesis, it could also be utilized for other tasks, which makes its broader societal impact application-dependent. In that context, it is important that practitioners apply an abundance of caution to mitigate impacts given generative modeling can also be used for malicious purposes, discussed for instance in Vaccari and Chadwick [106], Nguyen et al. [107], Mirsky and Lee [108].
+
+# Acknowledgements
+
+This work was funded by NVIDIA. Tim Dockhorn acknowledges additional funding from the Vector Institute Research Grant, not in direct support of this work. We thank Yaoliang Yu for discussions.
+
+# References
+
+[1] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising Diffusion Probabilistic Models. In Advances in Neural Information Processing Systems, 2020.   
+[2] Jonathan Ho, Chitwan Saharia, William Chan, David J Fleet, Mohammad Norouzi, and Tim Salimans. Cascaded Diffusion Models for High Fidelity Image Generation. arXiv:2106.15282, 2021. [3] Alexander Quinn Nichol and Prafulla Dhariwal. Improved Denoising Diffusion Probabilistic Models. In International Conference on Machine Learning, 2021.   
+[4] Prafulla Dhariwal and Alex Nichol. Diffusion Models Beat GANs on Image Synthesis. In Neural Information Processing Systems, 2021. [5] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High-Resolution Image Synthesis with Latent Diffusion Models. arXiv:2112.10752, 2021.   
+[6] Ruihan Yang, Prakhar Srivastava, and Stephan Mandt. Diffusion Probabilistic Modeling for Video Generation. arXiv:2203.09481, 2022.   
+[7] Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J. Fleet. Video Diffusion Models. arXiv:2204.03458, 2022.   
+[8] Chitwan Saharia, Jonathan Ho, William Chan, Tim Salimans, David J Fleet, and Mohammad Norouzi. Image Super-Resolution via Iterative Refinement. arXiv:2104.07636, 2021. [9] Haoying Li, Yifan Yang, Meng Chang, Huajun Feng, Zhihai Xu, Qi Li, and Yueting Chen. SRDiff: Single Image Super-Resolution with Diffusion Probabilistic Models. arXiv:2104.14951, 2021.   
+[10] Jay Whang, Mauricio Delbracio, Hossein Talebi, Chitwan Saharia, Alexandros G. Dimakis, and Peyman Milanfar. Deblurring via Stochastic Refinement. arXiv:2112.02475, 2021.   
+[11] Bahjat Kawar, Michael Elad, Stefano Ermon, and Jiaming Song. Denoising Diffusion Restoration Models. arXiv:2201.11793, 2022.   
+[12] Chenlin Meng, Yang Song, Jiaming Song, Jiajun Wu, Jun-Yan Zhu, and Stefano Ermon. SDEdit: Image Synthesis and Editing with Stochastic Differential Equations. arXiv:2108.01073, 2021.   
+[13] Andreas Lugmayr, Martin Danelljan, Andres Romero, Fisher Yu, Radu Timofte, and Luc Van Gool. RePaint: Inpainting using Denoising Diffusion Probabilistic Models. arXiv:2201.09865, 2022.   
+[14] Chitwan Saharia, William Chan, Huiwen Chang, Chris A. Lee, Jonathan Ho, Tim Salimans, David J. Fleet, and Mohammad Norouzi. Palette: Image-to-Image Diffusion Models. arXiv:2111.05826, 2021.   
+[15] Alex Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob McGrew, Ilya Sutskever, and Mark Chen. GLIDE: Towards Photorealistic Image Generation and Editing with TextGuided Diffusion Models. arXiv:2112.10741, 2021.   
+[16] Omri Avrahami, Dani Lischinski, and Ohad Fried. lended Diffusion for Text-driven Editing of Natural Images. arXiv:2111.14818, 2021.   
+[17] Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen. Hierarchical TextConditional Image Generation with CLIP Latents. arXiv:2204.06125, 2022.   
+[18] Jooyoung Choi, Sungwon Kim, Yonghyun Jeong, Youngjune Gwon, and Sungroh Yoon. ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2021.   
+[19] Kushagra Pandey, Avideep Mukherjee, Piyush Rai, and Abhishek Kumar. DiffuseVAE: Efficient, Controllable and High-Fidelity Generation from Low-Dimensional Latents. arXiv:2201.00308, 2022.   
+[20] Konpat Preechakul, Nattanat Chatthee, Suttisak Wizadwongsa, and Supasorn Suwajanakorn. Diffusion Autoencoders: Toward a Meaningful and Decodable Representation. arXiv:2111.15640, 2021.   
+[21] Xihui Liu, Dong Huk Park, Samaneh Azadi, Gong Zhang, Arman Chopikyan, Yuxiao Hu, Humphrey Shi, Anna Rohrbach, and Trevor Darrell. More Control for Free! Image Synthesis with Semantic Diffusion Guidance. arXiv:2112.05744, 2021.   
+[22] Georgios Batzolis, Jan Stanczuk, Carola-Bibiane Schönlieb, and Christian Etmann. Conditional Image Generation with Score-Based Diffusion Models. arXiv:2111.13606, 2021.   
+[23] Hiroshi Sasaki, Chris G. Willcocks, and Toby P. Breckon. UNIT-DDPM: UNpaired Image Translation with Denoising Diffusion Probabilistic Models. arXiv:2104.05358, 2021.   
+[24] Xuan Su, Jiaming Song, Chenlin Meng, and Stefano Ermon. Dual Diffusion Implicit Bridges for Image-to-Image Translation. arXiv:2203.08382, 2022.   
+[25] Yang Song, Liyue Shen, Lei Xing, and Stefano Ermon. Solving Inverse Problems in Medical Imaging with Score-Based Generative Models. In International Conference on Learning Representations, 2022.   
+[26] Cheng Peng, Pengfei Guo, S. Kevin Zhou, Vishal Patel, and Rama Chellappa. Towards performant and reliable undersampled MR reconstruction via diffusion model sampling. arXiv:2203.04292, 2022.   
+[27] Yutong Xie and Quanzheng Li. Measurement-conditioned Denoising Diffusion Probabilistic Model for Under-sampled Medical Image Reconstruction. arXiv:2203.03623, 2022.   
+[28] Guanxiong Luo, Martin Heide, and Martin Uecker. MRI Reconstruction via Data Driven Markov Chain with Joint Uncertainty Estimation. arXiv:2202.01479, 2022.   
+[29] Hyungjin Chung and Jong Chul Ye. Score-based diffusion models for accelerated MRI. arXiv:2110.05243, 2021.   
+[30] Dewei Hu, Yuankai K. Tao, and Ipek Oguz. Unsupervised Denoising of Retinal OCT with Diffusion Probabilistic Model. arXiv:2201.11760, 2022.   
+[31] Hyungjin Chung, Byeongsu Sim, and Jong Chul Ye. Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contraction. arXiv:2112.05146, 2021.   
+[32] Nanxin Chen, Yu Zhang, Heiga Zen, Ron J Weiss, Mohammad Norouzi, and William Chan. WaveGrad: Estimating Gradients for Waveform Generation. In International Conference on Learning Representations, 2021.   
+[33] Zhifeng Kong, Wei Ping, Jiaji Huang, Kexin Zhao, and Bryan Catanzaro. DiffWave: A Versatile Diffusion Model for Audio Synthesis. In International Conference on Learning Representations, 2021.   
+[34] Myeonghun Jeong, Hyeongju Kim, Sung Jun Cheon, Byoung Jin Choi, and Nam Soo Kim. Diff-TTS: A Denoising Diffusion Model for Text-to-Speech. arXiv preprint arXiv:2104.01409, 2021.   
+[35] Nanxin Chen, Yu Zhang, Heiga Zen, Ron J. Weiss, Mohammad Norouzi, Najim Dehak, and William Chan. WaveGrad 2: Iterative Refinement for Text-to-Speech Synthesis. arXiv:2106.09660, 2021.   
+[36] Vadim Popov, Ivan Vovk, Vladimir Gogoryan, Tasnima Sadekova, and Mikhail Kudinov. Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speech. In International Conference on Machine Learning, 2021.   
+[37] Songxiang Liu, Dan Su, and Dong Yu. DiffGAN-TTS: High-Fidelity and Efficient Text-to-Speech with Denoising Diffusion GANs. arXiv:2201.11972, 2022.   
+[38] Shitong Luo and Wei Hu. Diffusion Probabilistic Models for 3D Point Cloud Generation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021.   
+[39] Ruojin Cai, Guandao Yang, Hadar Averbuch-Elor, Zekun Hao, Serge Belongie, Noah Snavely, and Bharath Hariharan. Learning Gradient Fields for Shape Generation. In Proceedings of the European Conference on Computer Vision (ECCV), 2020.   
+[40] Linqi Zhou, Yilun Du, and Jiajun Wu. 3D Shape Generation and Completion through Point-Voxel Diffusion. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2021.   
+[41] Zhaoyang Lyu, Zhifeng Kong, Xudong XU, Liang Pan, and Dahua Lin. A Conditional Point DiffusionRefinement Paradigm for 3D Point Cloud Completion. In International Conference on Learning Representations, 2022.   
+[42] Emilien Dupont, Hyunjik Kim, S. M. Ali Eslami, Danilo Rezende, and Dan Rosenbaum. From data to functa: Your data point is a function and you should treat it like one. arXiv:2201.12204, 2022.   
+[43] Chence Shi, Shitong Luo, Minkai Xu, and Jian Tang. Learning gradient fields for molecular conformation generation. In International Conference on Machine Learning, 2021.   
+[44] Minkai Xu, Lantao Yu, Yang Song, Chence Shi, Stefano Ermon, and Jian Tang. GeoDiff: A Geometric Diffusion Model for Molecular Conformation Generation. In International Conference on Learning Representations, 2022.   
+[45] Tian Xie, Xiang Fu, Octavian-Eugen Ganea, Regina Barzilay, and Tommi S. Jaakkola. Crystal Diffusion Variational Autoencoder for Periodic Material Generation. In International Conference on Learning Representations, 2022.   
+[46] Jaehyeong Jo, Seul Lee, and Sung Ju Hwang. Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations. arXiv:2202.02514, 2022.   
+[47] Yang Song, Conor Durkan, Iain Murray, and Stefano Ermon. Maximum Likelihood Training of ScoreBased Diffusion Models. In Neural Information Processing Systems (NeurIPS), 2021.   
+[48] Diederik P Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational Diffusion Models. In Advances in Neural Information Processing Systems, 2021.   
+[49] Arash Vahdat, Karsten Kreis, and Jan Kautz. Score-based Generative Modeling in Latent Space. In Neural Information Processing Systems (NeurIPS), 2021.   
+[50] Chin-Wei Huang, Jae Hyun Lim, and Aaron Courville. A Variational Perspective on Diffusion-Based Generative Models and Score Matching. In Neural Information Processing Systems (NeurIPS), 2021.   
+[51] Chenhao Niu, Yang Song, Jiaming Song, Shengjia Zhao, Aditya Grover, and Stefano Ermon. Permutation invariant graph generation via score-based generative modeling. In International Conference on Artificial Intelligence and Statistics, 2020.   
+[52] Jongmin Yoon, Sung Ju Hwang, and Juho Lee. Adversarial Purification with Score-based Generative Models. In International Conference on Machine Learning, 2021.   
+[53] Yusuke Tashiro, Jiaming Song, Yang Song, and Stefano Ermon. CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation. In Advances in Neural Information Processing Systems, 2021.   
+[54] Valentin De Bortoli, Emile Mathieu, Michael Hutchinson, James Thornton, Yee Whye Teh, and Arnaud Doucet. Riemannian Score-Based Generative Modeling. arXiv:2202.02763, 2022.   
+[55] Pedro Sanchez and Sotirios A. Tsaftaris. Diffusion Causal Models for Counterfactual Estimation. In First Conference on Causal Learning and Reasoning, 2022.   
+[56] Weili Nie, Brandon Guo, Yujia Huang, Chaowei Xiao, Arash Vahdat, and Anima Anandkumar. Diffusion Models for Adversarial Purification. In International Conference on Machine Learning (ICML), 2022.   
+[57] Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-Based Generative Modeling through Stochastic Differential Equations. In International Conference on Learning Representations, 2021.   
+[58] Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising Diffusion Implicit Models. In International Conference on Learning Representations, 2021.   
+[59] Peter E. Kloeden and Eckhard Platen. Numerical Solution of Stochastic Differential Equations. Springer, Berlin, 1992.   
+[60] Tim Dockhorn, Arash Vahdat, and Karsten Kreis. Score-Based Generative Modeling with CriticallyDamped Langevin Diffusion. In International Conference on Learning Representations, 2022.   
+[61] Hideyuki Tachibana, Mocho Go, Muneyoshi Inahara, Yotaro Katayama, and Yotaro Watanabe. Itô- Taylor Sampling Scheme for Denoising Diffusion Probabilistic Models using Ideal Derivatives. arXiv:2112.13339, 2021.   
+[62] Alexia Jolicoeur-Martineau, Ke Li, Rémi Piché-Taillefer, Tal Kachman, and Ioannis Mitliagkas. Gotta Go Fast When Generating Data with Score-Based Models. arXiv:2105.14080, 2021.   
+[63] Luping Liu, Yi Ren, Zhijie Lin, and Zhou Zhao. Pseudo Numerical Methods for Diffusion Models on Manifolds. In International Conference on Learning Representations, 2022.   
+[64] Zhifeng Kong and Wei Ping. On Fast Sampling of Diffusion Probabilistic Models. arXiv:2106.00132, 2021.   
+[65] Fan Bao, Chongxuan Li, Jun Zhu, and Bo Zhang. Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models. In International Conference on Learning Representations, 2022.   
+[66] Daniel Watson, William Chan, Jonathan Ho, and Mohammad Norouzi. Learning Fast Samplers for Diffusion Models by Differentiating Through Sample Quality. In International Conference on Learning Representations, 2022.   
+[67] Zhisheng Xiao, Karsten Kreis, and Arash Vahdat. Tackling the Generative Learning Trilemma with Denoising Diffusion GANs. In International Conference on Learning Representations, 2022.   
+[68] Eric Luhman and Troy Luhman. Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed. arXiv:2101.02388, 2021.   
+[69] Tim Salimans and Jonathan Ho. Progressive Distillation for Fast Sampling of Diffusion Models. In International Conference on Learning Representations, 2022.   
+[70] Jonathan Ho and Tim Salimans. Classifier-Free Diffusion Guidance. In NeurIPS 2021 Workshop on Deep Generative Models and Downstream Applications, 2021.   
+[71] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep Unsupervised Learning using Nonequilibrium Thermodynamics. In International Conference on Machine Learning, 2015.   
+[72] Brian DO Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12(3):313–326, 1982.   
+[73] Ulrich G Haussmann and Etienne Pardoux. Time Reversal of Diffusions. The Annals of Probability, pages 1188–1205, 1986.   
+[74] Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural Ordinary Differential Equations. Advances in Neural Information Processing Systems, 2018.   
+[75] Will Grathwohl, Ricky T. Q. Chen, Jesse Bettencourt, Ilya Sutskever, and David Duvenaud. FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models. International Conference on Learning Representations, 2019.   
+[76] Pascal Vincent. A Connection Between Score Matching and Denoising Autoencoders. Neural Computation, 23(7):1661–1674, 2011.   
+[77] John Charles Butcher. Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, 2016.   
+[78] Peter E Kloeden and Eckhard Platen. Numerical Solution of Stochastic Differential Equations. Springer, 1992.   
+[79] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-Net: Convolutional Networks for Biomedical Image Segmentation. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 234–241. Springer, 2015.   
+[80] Jooyoung Choi, Jungbeom Lee, Chaehun Shin, Sungwon Kim, Hyunwoo Kim, and Sungroh Yoon. Perception Prioritized Training of Diffusion Models. arXiv:2204.00227, 2022.   
+[81] Dmitry Baranchuk, Andrey Voynov, Ivan Rubachev, Valentin Khrulkov, and Artem Babenko. LabelEfficient Semantic Segmentation with Diffusion Models. In International Conference on Learning Representations, 2022.   
+[82] Chenlin Meng, Yang Song, Wenzhe Li, and Stefano Ermon. Estimating High Order Gradients of the Data Distribution by Denoising. Advances in Neural Information Processing Systems, 34, 2021.   
+[83] Daniel Watson, Jonathan Ho, Mohammad Norouzi, and William Chan. Learning to Efficiently Sample from Diffusion Probabilistic Models. arXiv:2106.03802, 2021.   
+[84] J. R. Dormand and P. J. Prince. A family of embedded Runge–Kutta formulae. Journal of Computational and Applied Mathematics, 6(1):19–26, 1980.   
+[85] Eliya Nachmani, Robin San Roman, and Lior Wolf. Non Gaussian Denoising Diffusion Models. arXiv:2106.07582, 2021. [86] Max WY Lam, Jun Wang, Rongjie Huang, Dan Su, and Dong Yu. Bilateral Denoising Diffusion Models. arXiv:2108.11514, 2021.   
+[87] Franck Djeumou, Cyrus Neary, Eric Goubault, Sylvie Putot, and Ufuk Topcu. Taylor-Lagrange Neural Ordinary Differential Equations: Toward Fast Training and Evaluation of Neural ODEs. arXiv:2201.05715, 2022. [88] Jacob Kelly, Jesse Bettencourt, Matthew J Johnson, and David K Duvenaud. Learning differential equations that are easy to solve. Advances in Neural Information Processing Systems, 33:4370–4380, 2020. [89] Chris Finlay, Jörn-Henrik Jacobsen, Levon Nurbekyan, and Adam Oberman. How to train your neural ODE: the world of Jacobian and kinetic regularization. In International conference on machine learning, pages 3154–3164. PMLR, 2020. [90] YF Chang and George Corliss. ATOMFT: solving ODEs and DAEs using Taylor series. Computers & Mathematics with Applications, 28(10-12):209–233, 1994. [91] George Corliss and YF Chang. Solving ordinary differential equations using Taylor series. ACM Transactions on Mathematical Software (TOMS), 8(2):114–144, 1982. [92] Qinsheng Zhang and Yongxin Chen. Fast Sampling of Diffusion Models with Exponential Integrator. arXiv:2204.13902, 2022. [93] Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps. arXiv:2206.00927, 2022. [94] Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the Design Space of DiffusionBased Generative Models. arXiv:2206.00364, 2022. [95] Qinsheng Zhang, Molei Tao, and Yongxin Chen. gDDIM: Generalized denoising diffusion implicit models. arXiv:2206.05564, 2022. [96] Fan Bao, Chongxuan Li, Jiacheng Sun, Jun Zhu, and Bo Zhang. Estimating the Optimal Covariance with Imperfect Mean in Diffusion Probabilistic Models. arXiv:2206.07309, 2022. [97] Alex Krizhevsky. Learning Multiple Layers of Features from Tiny Images. Technical report, University of Toronto, 2009.   
+[98] Fisher Yu, Ari Seff, Yinda Zhang, Shuran Song, Thomas Funkhouser, and Jianxiong Xiao. LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop. arXiv preprint arXiv:1506.03365, 2015. [99] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. ImageNet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.   
+[100] Yunjey Choi, Youngjung Uh, Jaejun Yoo, and Jung-Woo Ha. StarGAN v2: Diverse Image Synthesis for Multiple Domains. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8188–8197, 2020.   
+[101] Andrew Brock, Jeff Donahue, and Karen Simonyan. Large Scale GAN Training for High Fidelity Natural Image Synthesis. In International Conference on Learning Representations, 2019.   
+[102] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.   
+[103] Alexia Jolicoeur-Martineau, Rémi Piché-Taillefer, Ioannis Mitliagkas, and Remi Tachet des Combes. Adversarial score matching and improved sampling for image generation. In International Conference on Learning Representations, 2021.   
+[104] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the Inception Architecture for Computer Vision. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 2818–2826, 2016.   
+[105] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al. Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding. arXiv:2205.11487, 2022.   
+[106] Cristian Vaccari and Andrew Chadwick. Deepfakes and Disinformation: Exploring the Impact of Synthetic Political Video on Deception, Uncertainty, and Trust in News. Social Media $^ +$ Society, 6(1): 2056305120903408, 2020.   
+[107] Thanh Thi Nguyen, Quoc Viet Hung Nguyen, Cuong M. Nguyen, Dung Nguyen, Duc Thanh Nguyen, and Saeid Nahavandi. Deep Learning for Deepfakes Creation and Detection: A Survey. arXiv:1909.11573, 2021.   
+[108] Yisroel Mirsky and Wenke Lee. The Creation and Detection of Deepfakes: A Survey. ACM Comput. Surv., 54(1), 2021.   
+[109] Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral Normalization for Generative Adversarial Networks. In International Conference on Learning Representations (ICLR), 2018.   
+[110] Diederik P Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization. In International Conference on Learning Representations, 2015.   
+[111] Mehdi SM Sajjadi, Olivier Bachem, Mario Lucic, Olivier Bousquet, and Sylvain Gelly. Assessing Generative Models via Precision and Recall. Advances in Neural Information Processing Systems, 31, 2018.   
+[112] Tuomas Kynkäänniemi, Tero Karras, Samuli Laine, Jaakko Lehtinen, and Timo Aila. Improved Precision and Recall Metric for Assessing Generative Models. Advances in Neural Information Processing Systems, 32, 2019.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Sec. 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Sec. 6.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] We did not derive novel theoretical results. We rather propose a novel higher-order solver for sampling from denoising diffusion generative models.   
+(b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] We will provide code and instructions to reproduce all experiments in the near future.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We provide all model training and evaluation details in the Appendix, including all hyperparameters.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Following standard conventions in the related generative modeling literature, we do not report error bars. Furthermore, we avoid running similar setups repeatedly to save computational resources.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Please see App. F.8.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] On CIFAR10, we use a checkpoint provided by Song et al. [57]. For baseline comparisons, we run publicly available code: in particular, we make use of the repositories https://github.com/yang-song/score_sde_pytorch and https://github. com/nv-tlabs/CLD-SGM from previous, publicly available papers, which we cite (Song et al. [57] and Dockhorn et al. [60], respectively). We also cite all used datasets: CIFAR-10 [97], LSUN Bedrooms [98], LSUN Church-Outdoor [98], ImageNet [99], and AFHQv2 [100].   
+(b) Did you mention the license of the assets? [Yes] In the Appendix, we mention the licenses of all codes we are running for baseline comparisons and evaluation. See App. F.1.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] We are only using publicly available datasets, which have been widely used in the generative modeling literature as standard benchmarks (CIFAR-10, ImageNet, LSUN Church-Outdoor, LSUN Bedrooms, AFHQv2) and avoid using face image datasets.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/Y6A4-R_Hgsw/Y6A4-R_Hgsw.md

@@ -0,0 +1,321 @@
+# Toward a realistic model of speech processing in the brain with self-supervised learning
+
+Juliette Millet∗ 1,2,3 Charlotte Caucheteux∗ 1,4 Pierre Orhan2 Yves Boubenec2   
+Alexandre Gramfort4 Ewan Dunbar2,5 Christophe Pallier6 Jean-Rémi King1,2
+
+∗These authors contributed equally 1Meta AI, Paris, France 2Ecole Normale Supérieure, PSL University, Paris, France 3LPI, Université de Paris cité, Paris, France 4Université Paris-Saclay, Inria, CEA, Palaiseau, France 5University of Toronto, Toronto, Canada 6Cognitive Neuroimaging Unit, INSERM, Gif-sur-Yvette, France
+
+{jumi,ccaucheteux,jeanremi}@meta.com
+
+# Abstract
+
+Several deep neural networks have recently been shown to generate activations similar to those of the brain in response to the same input. These algorithms, however, remain largely implausible: they require (1) extraordinarily large amounts of data, (2) unobtainable supervised labels, (3) textual rather than raw sensory input, and / or (4) implausibly large memory (e.g. thousands of contextual words). These elements highlight the need to identify algorithms that, under these limitations, would suffice to account for both behavioral and brain responses. Focusing on speech processing, we here hypothesize that self-supervised algorithms trained on the raw waveform constitute a promising candidate. Specifically, we compare a recent self-supervised model, wav2vec 2.0, to the brain activity of 412 English, French, and Mandarin individuals recorded with functional Magnetic Resonance Imaging (fMRI), while they listened to approximately one hour of audio books. First, we show that this algorithm learns brain-like representations with as little as 600 hours of unlabelled speech – a quantity comparable to what infants can be exposed to during language acquisition. Second, its functional hierarchy aligns with the cortical hierarchy of speech processing. Third, different training regimes reveal a functional specialization akin to the cortex: wav2vec 2.0 learns sound-generic, speech-specific and language-specific representations similar to those of the prefrontal and temporal cortices. Fourth, we confirm the similarity of this specialization with the behavior of 386 additional participants. These elements, resulting from the largest neuroimaging benchmark to date, show how self-supervised learning can account for a rich organization of speech processing in the brain, and thus delineate a path to identify the laws of language acquisition which shape the human brain.
+
+# 1 Introduction
+
+The performance of deep neural networks has taken off over the past decade. Algorithms trained on object classification, text translation, and speech recognition are starting to reach human-level performance [Xu et al., 2020]. Furthermore, the representations generated by these algorithms have repeatedly been shown to correlate with those of the brain [Kriegeskorte, 2015, Yamins and DiCarlo, 2016, Kietzmann et al., 2018, Kell and McDermott, 2019, Cichy and Kaiser, 2019, Toneva and
+
+Wehbe, 2019, Millet and King, 2021, Caucheteux and King, 2022], suggesting that these algorithms converge to brain-like computations.
+
+Such convergence, however, should not obscure the major differences that remain between these deep learning models and the brain. In particular, the above comparisons derive from models trained with (1) extraordinarily large amounts of data (40GB for GPT-2 [Radford et al., 2019], the equivalent of multiple lifetimes of reading), (2) supervised labels, which is rarely the case for humans (e.g. [Yamins and DiCarlo, 2016]), (3) data in a textual rather than a raw sensory format, and/or (4) considerable memory (e.g., language models typically have parallel access to thousands of context words to process text). These differences highlight the pressing necessity to identify architectures and learning objectives which, subject to these four constraints, would be sufficient to account for both behavior and brain responses.
+
+Here, we hypothesize that the latest self-supervised architectures trained on raw sensory data constitute promising candidates [Borgholt et al., 2022, Bardes et al., 2021, Baevski et al., 2020]. We focus on wav2vec 2.0 [Baevski et al., 2020], an architecture that stacks convolutional and transformer layers to predict a quantization of the latent representations of speech waveforms. We train wav2vec 2.0 on $6 0 0 \mathrm { h }$ of effective speech – a quantity roughly comparable to what infants are exposed to during early language acquisition (speech only makes up a small fraction of infants’ daily experience) [Dupoux, 2018, Hart and Risley, 1992, Gilkerson et al., 2017].
+
+We use standard encoding analyses [Naselaris et al., 2011, Huth et al., 2016, Yamins and DiCarlo, 2016, Kell et al., 2018] (Figure 1) to compare this model to the brains of 412 healthy volunteers (351 English speakers, 28 French speakers, and 33 Mandarin speakers) recorded with functional magnetic resonance imaging (fMRI) while they passively listened to approximately one hour of audio books in their native language [Nastase et al., 2020, Li et al., 2021] (8.5 hours of distinct audio materials in total).
+
+To better understand the similarities between wav2vec 2.0 and the brain, we compare brain activity to each layer of this model, as well as to several variants, namely (1) a random (untrained) wav2vec 2.0 model, (2) a model trained on $6 0 0 \mathrm { h }$ of non-speech sounds, (3) a model trained on $6 0 0 \mathrm { h }$ of nonnative speech (for example, a model trained on English speech and mapped onto the brain responses to French-speaking participants), (4) a model trained on $6 0 0 \mathrm { h }$ of native speech (for example, a model trained on English speech and mapped onto the brain responses to English participants), and (5) a model trained directly on speech-to-text (i.e., a supervised learning scheme) on the native language of the participants.
+
+Our results provide four main contributions. First, self-supervised learning leads wav2vec 2.0 to learn latent representations of the speech waveform similar to those of the human brain. Second, the functional hierarchy of its transformer layers aligns with the cortical hierarchy of speech in the brain, and reveals the whole-brain organisation of speech processing with an unprecedented clarity. Third, the auditory-, speech-, and language-specific representations learned by the model converge to those of the human brain. Fourth, behavioral comparisons to 386 supplementary participants’ results on a speech sound discrimination task confirm this common language specialization.
+
+# 2 Methods
+
+# 2.1 Models
+
+We train several variants of wav2vec 2.0 [Baevski et al., 2020] from scratch on different speech datasets using two different learning objectives (a self-supervised and a supervised objective).
+
+# 2.1.1 Architecture
+
+Wav2vec 2.0 consists of three main modules. First, a feature encoder composed of seven blocks of temporal convolutions (output dimension 512) transforms the speech input $S$ (raw mono waveform at $1 6 \mathrm { k H z } ,$ ) into a latent representation $z$ (output dimension of 512, frequency $4 9 \mathrm { H z }$ , stride of $2 0 \mathrm { m s }$ between each frame, receptive field of $2 5 \mathrm { m s }$ ). Second, a quantization module discretizes $z$ into $q$ , a dictionary of discrete and latent representations of sounds. Third, $z$ is input to a “context network” consisting of 12 transformer blocks (model dimension 768, inner dimension 3072, and 8 attention heads), which together yield a contextualized embedding $c$ , of the same dimensionality of $q$ .
+
+![](images/0108210fe7905a463adbe8dcaff036f14b22834823e153a0740726cd7ec51d03.jpg)  
+Figure 1: Comparing speech representations in brains and deep neural networks. A. We analyze the brain activity of 412 participants recorded with functional Magnetic Resonance Imaging (fMRI) while they passively listened to audio books in their native language (French, English or Mandarin). B. After training wav2vec 2.0 [Baevski et al., 2020] with self-supervised learning $( L )$ over $6 0 0 \mathrm { h }$ of unlabelled, effective speech, we extract its activations in response to the audio books that were presented to the participants. We assess the similarity between the activations of the model $X$ and brain activity $Y$ with a standard encoding model $W$ [Nastase et al., 2020] evaluated with a cross-validated Pearson correlation $R$ . C. Examples of the true BOLD response (black) and the predicted BOLD response (red) estimated from a linear projection of the model’s activations in three voxels randomly selected from the $1 0 ^ { t h }$ percentile of best voxels identified by the noise ceiling analysis for the first $2 0 0 \mathrm { s }$ of a representative story in the test set.
+
+# 2.1.2 Learning objective
+
+Self-supervised learning. In this training paradigm, the model optimizes two losses. The first loss is contrastive and requires the model to predict the quantized representation $q$ of some masked input using $c$ , from a finite set of quantized representations drawn from the input sample. The second loss ensures that the quantized representations are diverse. See Section A.2 and [Baevski et al., 2020] for details.
+
+Supervised learning. In this training paradigm, the quantization module is discarded and a linear layer mapping $c$ to phonemes is added at the end of the pipeline. The model is randomly initialized and all layers (including the feature encoder) are trained using a Connectionist Temporal Classification (CTC) [Graves, 2012] loss to perform phone recognition. For both training paradigms, we extract the activations of each layer from both the feature encoder (outputting $z$ ) and the context network (outputting $c$ ). We extract the representations of the convolutional and transformer blocks using an input window of $1 0 \mathrm { s }$ of raw waveform (stride $= 5 \mathrm { s }$ ).
+
+# 2.1.3 Training
+
+Datasets. We successively train different wav2vec 2.0 models using each of four datasets: (i) the French and (ii) English CommonVoice corpora [Ardila et al., 2020], (iii) the MAGICDATA Mandarin Chinese Read Speech Corpus [Co., 2019], and (iv) a non-speech subset of the Audioset dataset [Gemmeke et al., 2017], which contains recordings of various acoustic scenes.
+
+![](images/a2c62bde7bf5a21d03ea09e776aa663813579eb4b66f32bdc7eb401f27369b53.jpg)  
+Figure 2: Self-supervised learning suffices for wav2vec 2.0 to generate brain-like representations of speech. A. Brain score $( R )$ assessed for each subject and voxel independently, and here averaged across subjects for clarity. Only scores significantly above chance level, as assessed using a two-sided Wilcoxon test across subjects after correction for multiple comparison are color-coded $( p < 1 0 ^ { - 1 0 } ,$ ). B. $R$ scores for the same wav2vec2 model, averaged across subjects and voxels in four brain areas typically involved during speech processing (the primary and secondary auditory cortices, the superior temporal gyrus, the superior temporal sulcus, and the infero-frontal gyrus). In grey, the brain score obtained with a randomly initialized wav2vec 2.0 architecture. Error bars are the standard errors of the mean (SEM) across subjects. The stars indicate a significant difference between the random and trained model (all $p < 1 0 ^ { \bar { - } 4 \cdot }$ ). C. $R$ scores of wav2vec 2.0 without training (top), trained with a supervised (middle) and self-supervised learning rule (bottom), on the same 600 hours of effective speech. Scores are averaged across subjects and voxels and error bars are SEM across subjects.
+
+Preprocessing. All the audio datasets were randomly subsampled to have an approximate size of 600 hours, downsampled to $1 6 \mathrm { k H z }$ and converted to mono with the Sox software1. We randomly split the datasets into a training $( 8 0 \% )$ , a validation $( 1 0 \% )$ and a test set $( 1 0 \% )$ . The audio recordings we use from the Audioset dataset are filtered so that they do not contain speech or any sounds produced by humans, such as laughter or singing. For the speech datasets, we also use their corresponding annotations (in the supervised settings). We phonemize these annotations using eSpeakNG2. The number of different phoneme symbols in these annotations is similar for French (32), English (39), and Mandarin Chinese (33).
+
+Implementation. We train all of our models using the fairseq implementation of wav2vec $2 . 0 ^ { 3 }$ using default hyperparameters. We also analyze a model whose parameters were randomly initialised (“untrained” model).
+
+We use self-supervised learning to train four models: three on the speech datasets (French, English, and Mandarin) and one on the acoustic scenes dataset. In each case, the training was performed using the same configuration file (namely, the base configuration provided in the fairseq repository for pretraining wav2vec 2.0 on LibriSpeech [Panayotov et al., 2015]). We train the models for $4 0 0 \mathrm { k }$ updates and select the ones with the best validation loss.
+
+We also use the supervised training paradigm to train three models, on the French, English, and Mandarin datasets, respectively. Each training was performed using the same configuration file, which was identical to the configuration provided in the fairseq repository for fine-tuning wav2vec 2.0 on the 960 hour Voxpopuli corpus [Wang et al., 2021], except that parameters were not frozen (freeze_finetune_updates $\ l = \ 0$ ) and learning was performed on all parameters of the models using the CTC loss (feature_grad $\mathtt { m u l t } = 0 . 1$ ). We train the models for $4 0 0 \mathrm { k }$ updates and we use the ones with the best word error rate (WER) on the validation set. The French model obtains 13.9 WER, the English model 28.6 WER, and the Mandarin model 4.6 WER, on their respective test sets.
+
+![](images/dd1bfac5692c34719c32080ed188f83def94de46fa168ed7f54764d58779f37c.jpg)  
+Figure 3: The functional hierarchy of wav2vec 2.0 maps onto the speech hierarchy in the brain. A. We compute the $R$ score for each layer of wav2vec 2.0 separately and estimate, for each voxel, the layer with highest brain score on average across subjects. Only the voxels with significant brain scores are displayed $( p < 1 0 ^ { - 1 8 } .$ ). While the first transformer layers (blue) map onto the low-level auditory cortices (A1 and A2), the deeper layers (orange and red) map onto brain regions associated with higher-level processes (e.g. STS and IFG). B. Layer-wise $R$ scores averaged across all voxels. Error bars are SEM across subjects. C. Proportion of voxels with most predictive layer $\mathbf { \dot { x } }$ -axis) in four regions typically involved in speech processing. While most voxels in the primary cortex are best predicted by the first layers of the transformer, higher-level brain areas are best predicted by deeper layers.
+
+# 2.2 Functional MRI
+
+We analyse a composite set of fMRI recordings aggregated from the Little Prince [Li et al., 2021] and the Narratives public datasets [Nastase et al., 2020].
+
+Narratives. This dataset4 contains the fMRI recordings of 345 native English-speaking participants listening to English narratives (4.6 hours of unique audio in total). The participants listened to different stories varying from 7 to $9 8 \mathrm { { m i n } }$ (mean $= 2 6 \mathrm { m i n }$ ). Following [Nastase et al., 2020], we (1) focus on fifteen representative stories and ignore the narratives that have been modified by scrambling and (2) exclude eight participants because of noisy recordings. Overall, this selection results in a curated dataset of 303 participants listening to fifteen stories ranging from 3 min to $5 6 \mathrm { { m i n } }$ , for a total of 4 hours of unique audio (36,018 words from a vocabulary of 4,004 unique words).
+
+The Little Prince. This dataset5 contains fMRI recordings of 48 English native speakers, 33 Mandarin native speakers, and 28 French native speakers listening to The Little Prince in their respective native language. The experiment itself was divided into nine runs of approximately $1 0 \mathrm { { m i n } }$ of passive listening. For each language condition, the story was read by a single native speaker. The English, Mandarin, and French audiobooks last 94, 90 and 97 minutes respectively.
+
+![](images/6e054a1d797bef13919978e0b331fba2e5de8e29aa26442c3280a7fa528d64ca.jpg)  
+Figure 4: The specialization of wav2vec 2.0’s representations follows and clarifies the acoustic, speech, and language regions in the brain. A. We first evaluate humans’ language specificity by quantifying their ability to perceive phonemes of their native or non-native languages (Section 2.4) in a ABX matching-to-sample task [Schatz, 2016] (higher is better). As expected, humans are better at matching phonemes of their native language. B. Then, we train four wav2vec 2.0 models with self-supervised learning on four datasets – non-speech acoustic scenes, English, and French, and compute their ABX accuracy on the same speech datasets as humans. The ‘random’ model is wav2vec 2.0 without any training. C. Brain score $( R )$ of each model (with an added model trained on Mandarin), averaged across voxels, in four regions of the brain (Section 2.2). D. Acoustic, speech and language specificity for each voxel. For instance, one voxel is considered specific to the ‘native’ model if its native $R$ score is higher than its ‘non-native’ $R$ score $( p < . 0 5 )$ . Only the voxels with significant $R$ scores for the untrained model are displayed $( p < 1 0 ^ { - 1 8 } ,$ ). Error bars are the SEM across phone pairs in A and B, and across subjects in C. The stars indicates a significant difference between two conditions (Section 2.3).
+
+Preprocessing. For Narratives, we did not perform additional preprocessing: we use the public preprocessing of the dataset already projected on the surface space (“fsaverage6”) without spatial smoothing (labelled “afni-nosmooth” in the data repository). In contrast, the Little Prince dataset is only provided in a volumetric space. Consequently, for each language condition separately, we subselected the cortical voxels by computing a brain mask using the average of all participants’ fMRI data realigned onto a common template brain via Freesurfer [Fischl, 2012]. These voxels are then projected onto a brain surface using nilearn’s vol_to_surf function with defaults parameters [Abraham et al., 2014] and a ‘fsaverage6‘ template surface [Fischl, 2012]. For both Narratives and The Little Prince, fMRI signals are normalized across the time dimension to have a mean of 0 and a variance of 1, for each participant, surface voxel and session independently.
+
+Brain parcellation. For the purposes of certain analyses, we group the fMRI voxels into regions of interest using the Destrieux Atlas [Destrieux et al., 2010]. This parcellation results in 75 brain regions in each hemisphere. For simplicity, we label the regions as follows: A1 and A2 represents Heschl gyrus, which is the anatomical location of the primary and secondary auditory cortices, STG and STS are the superior temporal gyrus and sulcus, and IFG is the inferior frontal gyrus.
+
+# 2.3 Brain score (R)
+
+To quantify the similarity between the network’s activations $X$ and the brain recordings $Y$ , we use a standard linear encoding model [Huth et al., 2016, Yamins and DiCarlo, 2016]. For each subject, we split the data into train and test sets using a five-fold cross-validation setting. For each train split, a linear mapping $W$ is fitted to predict the brain response $Y _ { \mathrm { t r a i n } }$ given $X _ { \mathrm { t r a i n } }$ . $W$ combines a temporal alignment function with fixed weight, and a trained penalized linear regression.
+
+Temporal alignment. The sampling frequency of the model’s activations (between 49 and $2 0 0 \mathrm { H z }$ ) differs from the sampling frequency of fMRI BOLD signals $( 0 . 5 \mathrm { H z } )$ . Furthermore, the BOLD signals have delayed responses spanning over several seconds. Thus, we first convolve the model activations with a standard hemodynamic response function (HRF) using nistats [Abraham et al., 2014] compute_regressor function with the ‘glover’ model and default parameters. This results in the convolved activations $X _ { \mathrm { t r a i n } } ^ { \prime }$ with the same sampling frequency as the fMRI $Y _ { \mathrm { t r a i n } }$ (see A.3).
+
+Penalised linear regression. Once temporally aligned, we fit an $\ell _ { 2 }$ -penalised linear regression that predicts the brain signals $Y _ { \mathrm { t r a i n } }$ given the activations $X _ { \mathrm { t r a i n } }$ . We use the RidgeCV function from scikit-learn [Pedregosa et al., 2011], with the penalization hyperparameter $\lambda$ varying between 10 and $1 0 ^ { 8 }$ (20 values scaled logarithmically) chosen independently for each dimension with a nested cross-validation over the training set (see A.4).
+
+Evaluation. We evaluate the linear mapping $W$ on the held out sets by measuring Pearson’s correlation between predicted and actual brain responses: $R = \mathrm { c o r r } \big ( Y _ { \mathrm { t e s t } } , \mathbf { \bar { W } } \cdot X _ { \mathrm { t e s t } } \big )$ . Finally, we average the correlation scores across test splits to obtain the final “brain score”. To report the average layer $k ^ { * }$ with the highest brain score for each voxel (Figure 3), while being robust to regressionto-the-mean biases, we first find the best layer $k _ { s }$ for each participant $s$ and each voxel independently and then compute a circular mean across the $N = 4 1 2$ participants and the $K = 1 9$ layers: $\begin{array} { r } { k ^ { * } = \mathrm { a n g l e } \left( \frac { 1 } { N } \sum _ { s = 1 } ^ { N } \exp \left( \frac { 2 i \pi k _ { s } } { K + 1 } \right) \right) } \end{array}$
+
+Statistics. We assess the reliability of brain scores with second-level analyses across participants thanks to a Wilcoxon signed-rank test across participants. Thus, the resulting p-values are not affected by fMRI auto-correlation within participants. We perform statistical correction for multiple comparisons with Benjamini–Hochberg False Discovery Rate (FDR) across voxels [Benjamini, 2010].
+
+# 2.4 Behavioral experiment
+
+To compare the phonetic representations of our models to those of humans, we compare the forcedchoice discrimination judgements of online participants6 to an analogous method applied to wav2vec 2.0 [Schatz, 2016]. Specifically, for each triplet of sound “ABX”, participants judged whether the stimulus X was more similar to A or B. Analogously, we computed the Euclidean distance in the most discriminative layer of wav2vec 2.0 (here transformer layer 5) to determine whether X was closer to A or B. Additional data, analyses and model-human comparison can be found in [Millet and Dunbar, 2022]. We focus on the French and English stimuli, which represent $\approx 6 { , } 0 0 0$ ABX triplets (testing 508 English and 524 French phone pairs), with 386 participants in total (193 from each language group).
+
+In Figure 4-A, we report the ABX accuracy of English- and French-speaking participants in both their native and non-native language (either English or French). We first average results per phone pair, and then average over phone pairs to obtain the ABX discrimination accuracy. Similarly, in Figure 4-B, we compute the ABX accuracy of our wav2vec 2.0 models on the same evaluation sets as the participants, using the parameters described in [Millet and Dunbar, 2022]. English and French models are evaluated on the same (‘native’) or different (‘non-native’) language stimuli as their training. The random and non-speech models are evaluated on both French and English speech stimuli.
+
+# 3 Results
+
+Wav2vec 2.0 maps onto brain responses to speech. We estimate whether the activations of wav2vec 2.0 models linearly map onto the human brain activity of 412 individuals listening to audio books in the fMRI scanner. For this, we first independently train three models with $6 0 0 \mathrm { h }$ of
+
+French, English, or Mandarin, respectively, and compute the brain scores $( R )$ with the corresponding participants. Specifically, we (1) convolve the activations $( X )$ of the model with a hemodynamic response function (HRF), (2) train a $\ell _ { 2 }$ -penalized linear regression on a training split to map them to brain activity $Y$ , and (3) compute the Pearson correlation coefficient between (i) the true fMRI activity and (ii) the predicted activations on a test split. The models’ activations significantly predict brain activity in nearly all cortical areas, reaching the highest $R$ scores in the primary and secondary auditory cortices (Figure 2-A B). These scores are significantly higher than those obtained with a randomly initialised model $( p < 1 0 ^ { - 5 0 }$ on average across voxels), and this comparison is robust across language groups (all $p < 1 0 ^ { - 5 } .$ ).
+
+Comparison of self-supervised to supervised models. Does self-supervision reach representations that are as brain-like as those obtained with supervised learning? To address this issue, we trained wav2vec 2.0 with an alternative, supervised objective, namely, predicting phonetic annotations from the same 600 hours of effective speech sounds. We then implemented the $R$ score analyses described above. The results show that self-supervised learning in fact leads to modestly but significantly better $R$ scores than supervised learning (Figure 2-C): $\Delta R = 0 . 0 0 2 , p < 1 0 ^ { 6 }$ .
+
+The hierarchy of wav2vec 2.0 maps onto the hierarchy of the cortex. To compare the speech hierarchy in the brain with the functional hierarchy learned by wav2vec 2.0, we evaluate the $R$ score of each layer of the model (Figure 3). First, we observe that convolutional layers are less predictive than transformer layers. Second, within the transformers, the hierarchy of representations aligns with the expected cortical hierarchy [Hickok and Poeppel, 2007]: while low-level areas (A1, A2) are best predicted by the first transformer layers, higher level areas (IFG, STS) are best predicted by deeper layers. Remarkably, this hierarchy extends to supplementary motor and motor areas in both hemispheres (Figure 3-A).
+
+Language specificity in phone discrimination tasks. The acoustic features underlying speech (fricatives, vowels, and so on) may also characterize non-speech sounds (the sound of tree leaves in the wind, of a stone falling, and so on). Does the model show commonalities merely with general auditory processing in the brain, or does it capture speech-specific processing? If so, does it show commonalities with brain representations that are specific to the native language of the participants, or merely to general speech processing? We first evaluate the specialization of humans’ perception to their native language using an ABX behavioral task (Section 2.4). Specifically, we compare 386 French and English participants on their ability to distinguish native and non-native phones. As expected [Bohn, 2017, Kuhl et al., 2005], participants were better at discriminating native sounds than non-native ones (across phone pairs: $\dot { p } < 1 0 ^ { - 1 8 }$ , Figure 4-A). Second, applying the same test to our self-supervised French and English models shows that, like humans, models best discriminate sounds from their ‘native’ language (i.e., the French model better distinguishes French stimuli than English ones, across phone pairs, and vice versa: $p < 0 . 0 5 )$ . Interestingly, the ABX accuracy of the model is significantly higher than participants’. This quantitative difference may be partially explained by the fact that participants – and online participants in particular – undergo fluctuating attention, and adopt strategies which can negatively impact performance [Humphreys, 1939]. Finally, as expected, the random and acoustic models obtain the worst ABX accuracy. Overall, These results confirm that $6 0 0 \mathrm { h }$ of self-supervised learning on effective speech suffices for wav2vec 2.0 to learn language-specific representations (Figure 4-B).
+
+Wav2vec 2.0 and the brain learn language specific representations. Next, we compare the brain scores of random, non-speech, non-native and native models (Figure 4-C D). First, our results show that the non-speech model attains higher $R$ scores than the random model (on average across voxels, $\Delta R = 0 . 0 0 6$ , $p = 1 0 ^ { - 3 1 }$ ) confirming the importance of learning to generate brainlike representations. Second, non-native models attain higher $R$ scores than the non-speech model $( \Delta R \bar { = } ~ 0 . 0 0 2 , p = 1 0 ^ { - 9 } )$ ), confirming that wav2vec 2.0 learns speech-specific representations of sounds when trained on speech. Finally, the native model attains higher $R$ scores than non-native models $( \Delta R = 0 . 0 0 2 , p \stackrel { - } { = } 1 0 ^ { - 1 5 } )$ .
+
+# 4 Discussion
+
+Human infants acquire language with little to no supervision: A few hundred hours of speech suffices for their young brain to learn to discretize phonemes, segment morphemes, and assemble words in the language(s) of their social group [Dupoux, 2018, Gilkerson et al., 2017]. However, the learning principle that allows this unique feat remains, to date, unknown.
+
+Here, we test whether self-supervised learning applied to a limited amount of speech effectively accounts for the organization of speech processing in the human brain as measured with fMRI. For this, we train several variants of wav2vec 2.0 [Baevski et al., 2020] with three curated datasets of French, English, and Mandarin, and compare their activations to those of a large group of French, English, and Mandarin speakers recorded with fMRI while passively listening to audio stories. Our results show that this self-supervised model learns (i) representations that linearly map onto a remarkably distributed set of cortical regions (Figure 2), (ii) a computational hierarchy that aligns with the cortical hierarchy (Figure 3), and (iii) features specific to the language of the participants (Figure 4).
+
+Towards a biologically-plausible learning principle. These results extend recent findings on the similarities between the brain and a variety of deep learning models trained with biologicallyimplausible objectives and data. First, fMRI [Kell et al., 2018, Millet and King, 2021, Thompson et al., 2021], electroencephalography [Huang et al., 2018], and multi- or single-unit responses to sounds [Koumura et al., 2019, Begus et al., 2022] have been shown to be linearly predicted by the activations of deep convolutional networks trained on supervised auditory tasks. For example, [Millet and King, 2021] showed that a supervised speech-to-text model better accounted for brain responses to speech in 102 individuals when it was trained on speech recognition rather than auditory scene classification. Similarly, [Kell et al., 2018] showed that eight participants listening to brief speech and non-speech sounds demonstrated fMRI responses in the temporal lobe that aligned with those of a deep convolutional neural network trained on a binary auditory classification task. Our results, based on up to 50 times more fMRI recordings of the entire cortex show that such representational similarities hold with a self-supervised objective [Lerner et al., 2011, Berezutskaya et al., 2017, Caucheteux et al., 2021c,b]. Second, a growing series of MEG [Toneva and Wehbe, 2019, Caucheteux and King, 2022], fMRI [Mitchell et al., 2008, Qian et al., 2016, Pereira et al., 2018, Schwartz et al., 2019, Antonello et al., 2021, Jain and Huth, 2018] and electro-physiology studies [Schrimpf et al., 2021, Goldstein et al., 2022] showed that text-based language models trained on very large corpora generate brain-like representations too. While these results suggest elements of convergence between language models and the brain [Caucheteux and King, 2022], they also remain biologically implausible: not only are these algorithms pre-equipped with abstract linguistic units such as characters and words, but they are trained on corpora that no one would ever be able to read in their lifetime. In contrast, wav2vec 2.0 is here trained with a reasonable amount of raw speech waveforms [Hart and Risley, 1992, Gilkerson et al., 2017, Dupoux, 2018]. The functional similarity between wav2vec 2.0 and the brain thus opens the way to clarify how humans learn to process speech.
+
+The emergence of a brain-like hierarchy of speech processing. The present study reveals the hierarchical organization of speech processing with remarkable clarity. First, the functional hierarchy learnt by wav2vec 2.0 is aligned with the anatomy: e.g. the superior temporal sulcus and the temporal pole are known to project to the ventral and dorsal part of the inferofrontal gyrus, respectively [Petkov et al., 2015]. Second, the identification of functional gradients within the prefrontal cortex, and down to the motor areas typically associated with larynx and mouth control [Dichter et al., 2018] reinforces the relevance of motor processes to speech perception [Kellis et al., 2010, Mugler et al., 2014, Shamma et al., 2021]. Finally, the existence of multiple levels of representations around the inferofrontal cortex is consistent with the idea that Broca’s area may be responsible for merging linguistic units [Chomsky, 2000, Friederici, 1999, Hagoort, 2005, Poeppel et al., 2012]. It should be noted, however, that our results aggregate a large cohort of individuals which could mask a more modular organization at the individual level.
+
+Interpreting the neural representations of speech. Interpreting neural representations is a notoriously difficult challenge to both AI and neuroscience. Here, we first investigate language specificity and show that the neural representations specific to the native models are primarily represented in the superior temporal sulcus and middle temporal gyrus (Figure 4D): areas known to represent phonetic features [Mesgarani et al., 2014]. However, these effect are relatively modest (Figure 4): the random model and the non-speech model reach, in STS and STG, $67 \%$ and $87 \%$ of the brain scores obtained by the “native” model, respectively. While this high baseline initially surprised us, this phenomenon could be explained by the fact that the auditory cortex is continuously bombarded by – and should thus be tuned to – non-speech input. Second, our probing analyses show that the models trained with self-supervised learning learn relevant acoustic and linguistic representations (Supplementary Figure S1). This result, consistent with Vaidya et al. [2022] and Stephenson et al. [2019], suggests that the difference of brain scores observed between the random, non-native and native models (Figure 4) may be partly driven by the corresponding spectro-temporal, phonetic, word and sentence-level representations, respectively. These elements of interpretation remain, however, scarce, and a systematic interpretation of the representations shared between wav2vec 2.0 and the brain remains necessary.
+
+Scope of the study. It is important to stress that the scope of the present study could be broadened in several ways. First, our study focuses on adult speakers, whose cultural and educational background is not representative of the population [Henrich et al., 2010]. Second, we focus on the passive listening of three languages. Third, we focus on one self-supervised learning architecture [Baevski et al., 2020], and its functional alignment with fMRI, whose temporal resolution is notoriously limited. Generalizing the present approach to more languages [Malik-Moraleda et al., 2022], a larger spectrum of children and adult participants recorded with a variety of electrophysiological and neuroimaging devices will thus be essential to confirm, precise, and/or mitigate the present findings.
+
+The remaining gap between brain and speech models. Several major gaps can be evidenced between wav2vec 2.0 and the brain. First, the transformer layers are not temporally constrained: each layer can access all elements within the contextual window. This differs from the necessarily recurrent nature of processing in the brain. Second, wav2vec 2.0 behaves differently to humans in specific tasks. In particular, it is overly-sensitive to band-pass filtering, non-robustly exploit fine temporal structures [Weerts et al., 2021] and fails to display the expected categorical responses [Millet et al., 2021]. Third, recent studies show that wav2vec 2.0 encodes significantly less semantic information than text-based models [Pasad et al., 2021, Vaidya et al., 2022]. While our analyses suggest that learning allows wav2vec 2.0 to capture some lexical features in its deep layers (Figure S1, Table S4), it remains unclear whether these layers also capture complex syntactic structures, such as recursive syntactic trees [Lakretz et al., 2021, Caucheteux et al., 2021a]. We speculate that these limitations may be due to the time scales of wav2vec 2.0 which, unlike humans, learns very short-term representations of speech. In any case, these differences likely explain why the brain scores of wav2vec 2.0 remain substantially lower than our noise-ceiling ( $1 9 \%$ on average, and up to $74 \%$ in Heschl’s gyrus and sulcus, Table S1, Figure S2).
+
+Overall, the complexity of the human brain is often thought to be incompatible with a simple theory: “Even if there were enough data available about the contents of each brain area, there probably would not be a ready set of equations to describe them, their relationships, and the ways they change over time” [Gallant, 2013]. By showing how the equations of self-supervised learning give rise to brain-like processes, this work contributes to challenge this view.
+
+# Acknowledgments
+
+This project was funded, in part, by the Bettencourt-Schueller Foundation, the Philippe Foundation, and FrontCog grant ANR-17-EURE-0017 to JRK for his work at PSL.
+
+# References
+
+Alexandre Abraham, Fabian Pedregosa, Michael Eickenberg, Philippe Gervais, Andreas Mueller, Jean Kossaifi, Alexandre Gramfort, Bertrand Thirion, and Gaël Varoquaux. Machine learning for neuroimaging with scikit-learn. Frontiers in neuroinformatics, 8:14, 2014.
+
+Richard Antonello, Javier S Turek, Vy Vo, and Alexander Huth. Low-dimensional structure in the space of language representations is reflected in brain responses. Advances in Neural Information Processing Systems, 34, 2021.
+
+Rosana Ardila, Megan Branson, Kelly Davis, Michael Kohler, Josh Meyer, Michael Henretty, Reuben Morais, Lindsay Saunders, Francis M Tyers, and Gregor Weber. Common voice: A massively-multilingual speech corpus. In LREC, 2020.
+
+Alexei Baevski, H. Zhou, Abdel rahman Mohamed, and Michael Auli. wav2vec 2.0: A framework for self-supervised learning of speech representations. ArXiv, abs/2006.11477, 2020.
+
+Adrien Bardes, Jean Ponce, and Yann LeCun. Vicreg: Variance-invariance-covariance regularization for self-supervised learning. arXiv preprint arXiv:2105.04906, 2021.
+
+Gasper Begus, Alan Zhou, and Christina Zhao. Encoding of speech in convolutional layers and the brain stem based on language experience. bioRxiv, 2022.
+
+Yoav Benjamini. Discovering the false discovery rate. Journal of the Royal Statistical Society: series $B$ (statistical methodology), 72(4):405–416, 2010.
+
+Julia Berezutskaya, Zachary V Freudenburg, Umut Güçlü, Marcel AJ van Gerven, and Nick F Ramsey. Neural tuning to low-level features of speech throughout the perisylvian cortex. Journal of Neuroscience, 37(33):7906–7920, 2017.
+
+Ocke-Schwen Bohn. Cross-language and second language speech perception. The handbook of psycholinguistics, pages 213–239, 2017.
+
+Lasse Borgholt, Jakob Drachmann Havtorn, Joakim Edin, Lars Maaløe, and Christian Igel. A brief overview of unsupervised neural speech representation learning. arXiv preprint arXiv:2203.01829, 2022.
+
+Charlotte Caucheteux and Jean-Rémi King. Brains and algorithms partially converge in natural language processing. Communications Biology, 5(1):1–10, 2022.
+
+Charlotte Caucheteux, Alexandre Gramfort, and Jean-Remi King. Disentangling syntax and semantics in the brain with deep networks. In Proceedings of the 38th International Conference on Machine Learning, pages 1336–1348. PMLR, July 2021a. ISSN: 2640-3498.
+
+Charlotte Caucheteux, Alexandre Gramfort, and Jean-Remi King. Long-range and hierarchical language predictions in brains and algorithms. arXiv:2111.14232 [cs, q-bio], November 2021b. arXiv: 2111.14232.
+
+Charlotte Caucheteux, Alexandre Gramfort, and Jean-Remi King. Model-based analysis of brain activity reveals the hierarchy of language in 305 subjects. In Findings of the Association for Computational Linguistics: EMNLP 2021, pages 3635–3644, Punta Cana, Dominican Republic, November 2021c. Association for Computational Linguistics.
+
+Charlotte Caucheteux, Alexandre Gramfort, and Jean-Rémi King. Deep language algorithms predict semantic comprehension from brain activity. Scientific Reports, 12(1):16327, September 2022. ISSN 2045-2322. doi: 10.1038/s41598-022-20460-9. Number: 1 Publisher: Nature Publishing Group.
+
+Noam Chomsky. Linguistics and brain science. Image, language, brain, pages 13–28, 2000.
+
+Radoslaw M Cichy and Daniel Kaiser. Deep neural networks as scientific models. Trends in cognitive sciences, 23(4):305–317, 2019.
+
+Beijing Magic Data Technology Co. Magic data chinese mandarin conversational speech. http://www.imagicdatatech.com/index.php/home/dataopensource/data_ info/id/101, 2019.
+
+Christophe Destrieux, Bruce Fischl, Anders Dale, and Eric Halgren. Automatic parcellation of human cortical gyri and sulci using standard anatomical nomenclature. NeuroImage, 53(1):1–15, October 2010. ISSN 1053-8119. doi: 10.1016/j.neuroimage.2010.06.010.
+
+Benjamin K Dichter, Jonathan D Breshears, Matthew K Leonard, and Edward F Chang. The control of vocal pitch in human laryngeal motor cortex. Cell, 174(1):21–31, 2018.
+
+Emmanuel Dupoux. Cognitive science in the era of artificial intelligence: A roadmap for reverseengineering the infant language-learner. Cognition, 173:43–59, 2018.
+
+Bruce Fischl. Freesurfer. Neuroimage, 62(2):774–781, 2012.
+
+Angela D Friederici. The neurobiology of language comprehension. In Language comprehension: A biological perspective, pages 265–304. Springer, 1999.
+
+Jack Gallant. in ’reading minds’. Nature, 502(7472):428, 2013.
+
+Jort F Gemmeke, Daniel PW Ellis, Dylan Freedman, Aren Jansen, Wade Lawrence, R Channing Moore, Manoj Plakal, and Marvin Ritter. Audio set: An ontology and human-labeled dataset for audio events. In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 776–780. IEEE, 2017.
+
+Jill Gilkerson, Jeffrey A Richards, Steven F Warren, Judith K Montgomery, Charles R Greenwood, D Kimbrough Oller, John HL Hansen, and Terrance D Paul. Mapping the early language environment using all-day recordings and automated analysis. American journal of speech-language pathology, 26(2):248–265, 2017.
+
+Ariel Goldstein, Zaid Zada, Eliav Buchnik, Mariano Schain, Amy Price, Bobbi Aubrey, Samuel A Nastase, Amir Feder, Dotan Emanuel, Alon Cohen, et al. Shared computational principles for language processing in humans and deep language models. Nature neuroscience, 25(3):369–380, 2022.
+
+Alex Graves. Connectionist temporal classification. In Supervised Sequence Labelling with Recurrent Neural Networks, pages 61–93. Springer, 2012.
+
+Peter Hagoort. On broca, brain, and binding: a new framework. Trends in cognitive sciences, 9(9): 416–423, 2005.
+
+Betty Hart and Todd R Risley. American parenting of language-learning children: Persisting differences in family-child interactions observed in natural home environments. Developmental psychology, 28(6):1096, 1992.
+
+Joseph Henrich, Steven J. Heine, and Ara Norenzayan. The weirdest people in the world? Behavioral and Brain Sciences, 33(2-3):61–83, June 2010. ISSN 0140-525X, 1469-1825. doi: 10.1017/S0140525X0999152X.
+
+Gregory Hickok and David Poeppel. The cortical organization of speech processing. Nature reviews neuroscience, 8(5):393–402, 2007.
+
+Nicholas Huang, Malcolm Slaney, and Mounya Elhilali. Connecting deep neural networks to physical, perceptual, and electrophysiological auditory signals. Frontiers in neuroscience, 12:532, 2018.
+
+L. G. Humphreys. Acquisition and extinction of verbal expectations in a situation analogous to conditioning. Journal of Experimental Psychology, 25:294–301, 1939. ISSN 0022-1015. doi: 10.1037/h0053555. Place: US Publisher: American Psychological Association.
+
+Alexander G. Huth, Wendy A. de Heer, Thomas L. Griffiths, Frédéric E. Theunissen, and Jack L. Gallant. Natural speech reveals the semantic maps that tile human cerebral cortex. Nature, 532 (7600):453–458, April 2016. ISSN 0028-0836, 1476-4687. doi: 10.1038/nature17637.
+
+Shailee Jain and Alexander Huth. Incorporating context into language encoding models for fmri. Advances in neural information processing systems, 31, 2018.
+
+Alexander JE Kell and Josh H McDermott. Deep neural network models of sensory systems: windows onto the role of task constraints. Current opinion in neurobiology, 55:121–132, 2019.
+
+Alexander JE Kell, Daniel LK Yamins, Erica N Shook, Sam V Norman-Haignere, and Josh H McDermott. A task-optimized neural network replicates human auditory behavior, predicts brain responses, and reveals a cortical processing hierarchy. Neuron, 98(3):630–644, 2018.
+
+Spencer Kellis, Kai Miller, Kyle Thomson, Richard Brown, Paul House, and Bradley Greger. Decoding spoken words using local field potentials recorded from the cortical surface. Journal of neural engineering, 7(5):056007, 2010.
+
+Tim C Kietzmann, Patrick McClure, and Nikolaus Kriegeskorte. Deep neural networks in computational neuroscience. BioRxiv, page 133504, 2018.
+
+Takuya Koumura, Hiroki Terashima, and Shigeto Furukawa. Cascaded tuning to amplitude modulation for natural sound recognition. Journal of Neuroscience, 39(28):5517–5533, 2019.
+
+Nikolaus Kriegeskorte. Deep neural networks: a new framework for modeling biological vision and brain information processing. Annual review of vision science, 1:417–446, 2015.
+
+Patricia K Kuhl, Barbara T Conboy, Denise Padden, Tobey Nelson, and Jessica Pruitt. Early speech perception and later language development: Implications for the" critical period". Language learning and development, 1(3-4):237–264, 2005.
+
+Yair Lakretz, Théo Desbordes, Jean-Rémi King, Benoît Crabbé, Maxime Oquab, and Stanislas Dehaene. Can RNNs learn Recursive Nested Subject-Verb Agreements? arXiv:2101.02258 [cs], January 2021. URL http://arxiv.org/abs/2101.02258. arXiv: 2101.02258.
+
+Y. Lerner, C. J. Honey, L. J. Silbert, and U. Hasson. Topographic Mapping of a Hierarchy of Temporal Receptive Windows Using a Narrated Story. Journal of Neuroscience, 31(8):2906– 2915, February 2011. ISSN 0270-6474, 1529-2401. doi: 10.1523/JNEUROSCI.3684-10.2011.
+
+Jixing Li, Shohini Bhattasali, Shulin Zhang, Berta Franzluebbers, Wen-Ming Luh, R Nathan Spreng, Jonathan R Brennan, Yiming Yang, Christophe Pallier, and John T Hale. Le petit prince: A multilingual fmri corpus using ecological stimuli. bioRxiv, 2021.
+
+Saima Malik-Moraleda, Dima Ayyash, Jeanne Gallée, Josef Affourtit, Malte Hoffmann, Zachary Mineroff, Olessia Jouravlev, and Evelina Fedorenko. An investigation across 45 languages and 12 language families reveals a universal language network. Nature Neuroscience, 25(8):1014– 1019, August 2022. ISSN 1546-1726. doi: 10.1038/s41593-022-01114-5. Number: 8 Publisher: Nature Publishing Group.
+
+Nima Mesgarani, Connie Cheung, Keith Johnson, and Edward F. Chang. Phonetic Feature Encoding in Human Superior Temporal Gyrus. Science, 343(6174):1006–1010, February 2014. ISSN 0036-8075, 1095-9203. doi: 10.1126/science.1245994. Publisher: American Association for the Advancement of Science Section: Report.
+
+Juliette Millet and Ewan Dunbar. Do self-supervised speech models develop human-like perception biases? AAAI 2022 Workshop, Self-Supervised Learning for Audio and Speech Processing, 2022.
+
+Juliette Millet and Jean-Remi King. Inductive biases, pretraining and fine-tuning jointly account for brain responses to speech. arXiv preprint arXiv:2103.01032, 2021.
+
+Juliette Millet, Ioana Chitoran, and Ewan Dunbar. Predicting non-native speech perception using the perceptual assimilation model and state-of-the-art acoustic models. In Proceedings of the 25th Conference on Computational Natural Language Learning, pages 661–673, Online, November 2021. Association for Computational Linguistics. doi: 10.18653/v1/2021.conll-1.51.
+
+Tom M Mitchell, Svetlana V Shinkareva, Andrew Carlson, Kai-Min Chang, Vicente L Malave, Robert A Mason, and Marcel Adam Just. Predicting human brain activity associated with the meanings of nouns. science, 320(5880):1191–1195, 2008.
+
+Emily M Mugler, James L Patton, Robert D Flint, Zachary A Wright, Stephan U Schuele, Joshua Rosenow, Jerry J Shih, Dean J Krusienski, and Marc W Slutzky. Direct classification of all american english phonemes using signals from functional speech motor cortex. Journal of neural engineering, 11(3):035015, 2014.
+
+Thomas Naselaris, Kendrick N Kay, Shinji Nishimoto, and Jack L Gallant. Encoding and decoding in fmri. Neuroimage, 56(2):400–410, 2011.
+
+Samuel A Nastase, Yun-Fei Liu, Hanna Hillman, Asieh Zadbood, Liat Hasenfratz, Neggin Keshavarzian, Janice Chen, Christopher J Honey, Yaara Yeshurun, Mor Regev, et al. Narratives: fmri data for evaluating models of naturalistic language comprehension. preprint. Neuroscience, December, pages 2020–06, 2020.
+
+Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev Khudanpur. Librispeech: an asr corpus based on public domain audio books. In 2015 IEEE international conference on acoustics, speech and signal processing (ICASSP), pages 5206–5210. IEEE, 2015.
+
+Ankita Pasad, Ju-Chieh Chou, and Karen Livescu. Layer-wise analysis of a self-supervised speech representation model. arXiv preprint arXiv:2107.04734, 2021.
+
+Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al. Scikit-learn: Machine learning in python. the Journal of machine Learning research, 12:2825–2830, 2011.
+
+Francisco Pereira, Bin Lou, Brianna Pritchett, Samuel Ritter, Samuel J Gershman, Nancy Kanwisher, Matthew Botvinick, and Evelina Fedorenko. Toward a universal decoder of linguistic meaning from brain activation. Nature communications, 9(1):1–13, 2018.
+
+Christopher I. Petkov, Yukiko Kikuchi, Alice E. Milne, Mortimer Mishkin, Josef P. Rauschecker, and Nikos K. Logothetis. Different forms of effective connectivity in primate frontotemporal pathways. Nature Communications, 6(1):6000, January 2015. ISSN 2041-1723. doi: 10.1038/ ncomms7000. Number: 1 Publisher: Nature Publishing Group.
+
+David Poeppel, Karen Emmorey, Gregory Hickok, and Liina Pylkkänen. Towards a new neurobiology of language. Journal of Neuroscience, 32(41):14125–14131, 2012.
+
+Peng Qian, Xipeng Qiu, and Xuanjing Huang. Bridging lstm architecture and the neural dynamics during reading. arXiv preprint arXiv:1604.06635, 2016.
+
+Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9, 2019.
+
+Thomas Schatz. ABX-discriminability measures and applications. PhD thesis, Université Paris 6 (UPMC), 2016.
+
+Martin Schrimpf, Idan Asher Blank, Greta Tuckute, Carina Kauf, Eghbal A Hosseini, Nancy Kanwisher, Joshua B Tenenbaum, and Evelina Fedorenko. The neural architecture of language: Integrative modeling converges on predictive processing. Proceedings of the National Academy of Sciences, 118(45), 2021.
+
+Dan Schwartz, Mariya Toneva, and Leila Wehbe. Inducing brain-relevant bias in natural language processing models. Advances in neural information processing systems, 32, 2019.
+
+Shihab Shamma, Prachi Patel, Shoutik Mukherjee, Guilhem Marion, Bahar Khalighinejad, Cong Han, Jose Herrero, Stephan Bickel, Ashesh Mehta, and Nima Mesgarani. Learning speech production and perception through sensorimotor interactions. Cerebral cortex communications, 2(1): tgaa091, 2021.
+
+Cory Stephenson, Jenelle Feather, Suchismita Padhy, Oguz Elibol, Hanlin Tang, Josh McDermott, and SueYeon Chung. Untangling in invariant speech recognition. Advances in neural information processing systems, 32, 2019.
+
+Jessica AF Thompson, Yoshua Bengio, Elia Formisano, and Marc Schönwiesner. Training neural networks to recognize speech increased their correspondence to the human auditory pathway but did not yield a shared hierarchy of acoustic features. bioRxiv, 2021.
+
+Mariya Toneva and Leila Wehbe. Interpreting and improving natural-language processing (in machines) with natural language-processing (in the brain). arXiv:1905.11833 [cs, q-bio], November 2019. arXiv: 1905.11833.
+
+Aditya R Vaidya, Shailee Jain, and Alexander G Huth. Self-supervised models of audio effectively explain human cortical responses to speech. arXiv preprint arXiv:2205.14252, 2022.
+
+Changhan Wang, Morgane Riviere, Ann Lee, Anne Wu, Chaitanya Talnikar, Daniel Haziza, Mary Williamson, Juan Pino, and Emmanuel Dupoux. Voxpopuli: A large-scale multilingual speech corpus for representation learning, semi-supervised learning and interpretation. arXiv preprint arXiv:2101.00390, 2021.
+
+Lotte Weerts, Stuart Rosen, Claudia Clopath, and Dan FM Goodman. The psychometrics of automatic speech recognition. bioRxiv, 2021.
+
+Qiantong Xu, Alexei Baevski, Tatiana Likhomanenko, Paden Tomasello, Alexis Conneau, Ronan Collobert, Gabriel Synnaeve, and Michael Auli. Self-training and Pre-training are Complementary for Speech Recognition, October 2020. arXiv:2010.11430 [cs, eess].
+
+Daniel LK Yamins and James J DiCarlo. Using goal-driven deep learning models to understand sensory cortex. Nature neuroscience, 19(3):356–365, 2016.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Section 3   
+(b) Did you describe the limitations of your work? [Yes] See Section 4   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] We have, and since we use results of human experiments that were previously made, and that respect these review guidelines, we conform to them.
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] No theoretical results (b) Did you include complete proofs of all theoretical results? [N/A] No theoretical results
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The code will be provided upon request. Otherwise for the rest, see Section 2.2 and 2.1.3   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 2.1.3.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] This is the case for all our figures, see Figure 2, 3 and 4   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] The models we train are not new: the original papers already give that type of information.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See Section 2.2, 2.1.3, and 2.4.   
+(b) Did you mention the license of the assets? [No] They are all under CC0 for the English and French training audio data, the Perceptimatic Benchmark, The Little Prince dataset and the Narratives dataset. Magic data is under CC BY-NC-ND 4.0, Audioset is under CC BY 4.0 for the audio recordings, and CC BY-SA 4.0 for the ontology, Narratives’ preprocessing code is under GNU GENERAL PUBLIC LICENSE and fairseq code is under MIT License.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No] The code will be given upon request   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] See Section 2.2 and 2.4. More details on the way they gave their consent can be seen in the original papers/websites that provided the experimental results.
+
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] The data were anonymized before we used them.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] We did not conduct the experiments ourselves   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] We did not conduct the experiments ourselves   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] We did not conduct the experiments ourselves

parse/dev/gmL46YMpu2J/gmL46YMpu2J.md

@@ -0,0 +1,410 @@
+# PROMPTAGATOR FEW-SHOT DENSE RETRIEVAL FROM 8 EXAMPLES
+
+Zhuyun Dai∗†, Vincent Y. Zhao∗†, Ji $\mathbf { M } \mathbf { a } ^ { * \dagger }$ , Yi Luan∗†, Jianmo Ni, Jing Lu, Anton Bakalov,   
+Kelvin Guu, Keith B. Hall and Ming-Wei Chang†   
+Google Research   
+{zhuyundai, vzhao, maji, luanyi, mingweichang}@google.com   
+∗equal contributions †corresponding authors
+
+# ABSTRACT
+
+Much recent research on information retrieval has focused on how to transfer from one task (typically with abundant supervised data) to various other retrieval tasks where supervision is limited, with the implicit assumption that it is possible to generalize from one task to all the rest. However, this overlooks the fact that there are many diverse and unique retrieval problems, each targeting different search intents, queries, and search domains. In this paper, we suggest to work on Few-shot Dense Retrieval, a setting where each task comes with a short description and a few examples. To address this, we introduce Prompt-based Query Generation for Retrieval (PROMPTAGATOR ): for each task, we feed the few-shot examples to a large language model (LLM) and prompt it to behave as a task-specific query generator. Using this, we can synthetically generate a large number of relevant queries for any document, yielding abundant data for training task-specific retrievers — with no reliance on traditional resources such as Natural Questions (Kwiatkowski et al., 2019) or MS MARCO (Nguyen et al., 2016). Surprisingly, PROMPTAGATOR using only 8 annotated examples enables efficient dual encoder retrievers to outperform computationally more expensive models trained on MS MARCO such as ColBERT v2 (Santhanam et al., 2022) by more than 1.2 points $\mathrm { n D C G } @ 1 0$ on average on 11 retrieval sets. Further training standard-size rerankers using the same generated data yields another 5.0 points $\mathrm { n D C G } @ 1 0$ improvement. Our studies show that synthetic query generation can be far more effective than previously observed, especially when a small amount of task-specific knowledge is given.
+
+# 1 INTRODUCTION
+
+Significant progress has been made on neural retrieval models such as dual encoders, which can search over a large collection of documents containing millions to billions of passages (Yih et al., 2011; Lee et al., 2019; Karpukhin et al., 2020). However, Thakur et al. (2021) recently proposed the BEIR heterogeneous retrieval benchmark, and showed that it is still difficult for neural retrievers to perform well on a wide variety of retrieval tasks that lack dedicated training data. To address this problem, many previous approaches focus on transferring knowledge from high-resource question answering (QA) datasets such as MS MARCO (Nguyen et al., 2016), and propose architectures that possess good inductive biases, such as models that allow fine-grained token-level interaction (e.g., ColBERT (Khattab & Zaharia, 2020; Santhanam et al., 2022) and SPLADE (Formal et al., 2021)) which often come with higher inference cost. Data augmentation via synthetic query generation has previously been explored (Ma et al., 2021; Shakeri et al., 2020), but these question generators are learned from high-resource QA datasets, and often cannot generalize well to new retrieval tasks.
+
+We argue that it is hard to anticipate models based on one or two QA datasets to perform well across all retrieval tasks. First, different retrieval tasks have very different search intents; in other words, different definitions of “relevance”. For example, consider Figure 1(a): both DbpediaEntity (Hasibi et al., 2017) and FEVER (Thorne et al., 2018) are tasks to retrieve documents from Wikipedia. However, Dbpedia-Entity is a task to retrieve entities that are mentioned in the query, while FEVER is a task to find evidence that either supports or refutes a given statement. Which document is relevant to the query can be very different from one task to another task even if they share the same domain. Moreover, different tasks have distinct distributions of queries even when their search intents are similar. For example, queries in HotpotQA (Yang et al., 2018) are long compositional questions, while queries in FiQA (Maia et al., 2018) are short financial questions.
+
+![](images/ee0c43785d29c1f62b49c195278b442952aaf7a8f03a8a4f888a0e81832b1f12.jpg)  
+Figure 1: Few-shot retrieval with PROMPTAGATOR. Left (a): Retrieval tasks from BEIR differ in query distribution, retrieval corpus, and search intents. Middle (b): Most prior work uses supervised setting (2) which trains model on a large QA retrieval datasets and transfer to other retrieval tasks. Right (c): Few-shot PROMPTAGATOR performance. Average nDCG $@ 1 0$ on 11 datasets from BEIR from our PROMPTAGATOR models and previously MS MARCO-supervised models (SPLADE v2).
+
+Motivated by these observations, we advocate to work on the setting of Few-shot Retrieval for diverse retrieval tasks (§2), where each task comes with a short description and a few annotated examples to clearly illustrate the search intent. To address this challenge, we propose Prompt-based Query Generation for Retrieval (PROMPTAGATOR) (§3): for each new retrieval task, we feed the few-shot examples to a large language model (LLM) such as FLAN1 (Wei et al., 2022a) and prompt it to perform doc-to-query generation. Importantly, the few-shot examples ensure that we capture the specific search intent of that task. Using this query generator, we can synthetically generate a large number of relevant queries for any document, yielding abundant data for training any retriever, including highly efficient dual encoder models.
+
+We find that our few-shot LLM query generator can produce good queries without any fine-tuning (§3.1). In fact, as shown in Figure 1(b), our synthetically generated data is strong enough to completely forego using annotated query-document pairs from traditional high-resource datasets such as Natural Questions (Kwiatkowski et al., 2019) or MS MARCO (Nguyen et al., 2016).
+
+While PROMPTAGATOR is not the first application of LLMs for retrieval, prior work did not explore task-specific few-shot adaptation, and often came with high inference cost. Neelakantan et al. (2022) proposes to use GPT-3 (Brown et al., 2020) in dual encoders. However, their embedding dimension is $1 2 \mathrm { k }$ , which makes the search index footprint and inference cost prohibitively high for many applications. Sachan et al. (2022) and Bonifacio et al. (2022) prompt LLMs for question generation, but did not explore the idea of using task-specific few-shot prompts for rapid task adaptation.2 They also focus primarily on models that rerank top retrievals from an existing retriever, rather than directly adapting the underlying retriever which must efficiently search over millions or billions of documents.
+
+To summarize, the contributions of the paper are as follows:
+
+• We highlight previously overlooked differences across retrieval tasks (e.g., search intent and query distribution), and propose a Few-Shot Retrieval evaluation for the BEIR dataset. We propose PROMPTAGATOR, a simple recipe for few-shot retrieval by prompting an LLM to generate synthetic task-specific training data. For the first time, we can train fully neural retrievers and rerankers solely based on a few supervised examples. Our results show that, surprisingly, PROMPTAGATOR with two-to-eight examples produces significantly better retrievers than recent models trained on MS MARCO or NQ that have over 500k human annotated examples (Figure 1(c)) and utilize more expensive architectures: PROMPTAGATOR outperforms ColBERT v2 and SPLADE v2 on 11 retrieval tasks we tested, while reranking boosts results by another 5 points on standard retrieval evaluation metric.
+
+# 2 FEW-SHOT RETRIEVAL TASK
+
+In this section, we first introduce the definition of a few-shot retrieval task and discuss the differences among tasks. We then propose a new Few-Shot Retrieval setting for the BEIR benchmark.
+
+# 2.1 RETRIEVAL TASK
+
+Given a large corpus, a retrieval model is responsible for finding the documents that are most relevant to a provided query $q$ according to a pre-defined notion of relevance. Formally, a retrieval task is:
+
+$$
+T = \{ \mathcal { D } , \mathcal { Q } , \mathcal { Z } \} ,
+$$
+
+where $\mathcal { D } = \{ d _ { 1 } , d _ { 2 } , . . . , d _ { n } \}$ is a large corpus of documents for retrieval, $\mathcal { Q }$ is a query distribution, and $\mathcal { T }$ is the underlying search intent for the task. Depending on the task, $\mathcal { D }$ can be any document collection, such as the web or Wikipedia. $Q$ also varies across tasks, e.g., short keyword search queries, questions, arguments, etc. If $\textstyle { \mathcal { I } } ( q , d ) = 1$ , it means the search intent of $q$ has been satisfied by the document $d$ . For example, in a question answering task $\mathcal { T } _ { \mathrm { Q A } } ( q , d ) = 1$ if $d$ answers $q$ . For the same $( q , d )$ pair, relevance may be either 1 or 0 depending on the search intent. For example, some argument retrieval tasks only seek to retrieve supporting arguments, while others aim to retrieve counterarguments.
+
+In this work, we assume a target retrieval corpus $\mathcal { D } _ { \mathcal { T } }$ is given, but the amount of annotated querydocument pairs for the new task is limited. Most prior research efforts focused on adapting retrievers to a new corpus $\mathcal { D } _ { \mathcal { T } }$ , but didn’t fully account for divergence in queries $\mathcal { Q } _ { T }$ or intents $\mathcal { T } _ { T }$ . Next, we explore how a search intent can be expressed with a short description and very few examples.
+
+# 2.2 FEW-SHOT BEIR SETTING
+
+Intuitively, a person can understand a retrieval task by reading a short prompt and going over a few examples. In this work, we ask if a few (8 or fewer) examples are sufficient to learn a task-specific retriever. To facilitate our study and future research on few-shot retrieval, we define a new few-shot retrieval evaluation built upon the BEIR heterogeneous retrieval benchmark (Thakur et al., 2021).
+
+BEIR has 18 information retrieval datasets across 9 domains, including Bio-Medical, Finance, News, Twitter, Wikipedia, StackExchange, Quora, Scientific, and Misc. These datasets also cover a diverse range of search intents: QA retrieval (question-to-document), duplicate question discovery (questionto-question), fact checking (claim-to-document), etc. Following Santhanam et al. (2022) and Formal et al. (2021), we narrow our focus to the publicly-available datasets in BEIR. The original BEIR evaluation used a zero-shot setup, where no queries or relevant query-document pairs from the evaluation datasets can be used for training.
+
+We relax BEIR to the few-shot setting by randomly taking a few (2 to 8) in-domain relevant querydocument examples as task-specific supervision — in realistic applications, this number of examples is almost always possible to obtain. The examples are sampled from the development set when it is available. For BEIR tasks which only have a test set, we use samples from the test data. To be fair when evaluating our models, we always mark these test-set examples as ‘failed‘: we remove the documents from our retrieved results when computing metrics, even if they are correctly retrieved (the worst possible outcome). The prompts and few-shot examples will be released to the public.
+
+# 3 PROMPTAGATOR
+
+The key idea of PROMPTAGATOR is to transform a few examples into many more examples by prompting an LLM to generate more data, instead of using them to train a retriever directly.
+
+PROMPTAGATOR consists of three components: prompt-based query generation, consistency filtering, and retriever training. During prompt-based query generation, a task-specific prompt will be combined with a large language model to produce relevant queries for all documents in $\mathcal { D } _ { T }$ . Then, a filtering step cleans the generated data based on round-trip consistency. Surprisingly, we found that a retriever trained only on our synthetic data can be used to filter the synthetic data. Finally, a retriever (in this paper, dual encoders) and a cross attention reranker are trained based on the filtered data. Figure 5 in Appendix shows the overall procedure.
+
+# 3.1 PROMPT-BASED QUERY GENERATION
+
+In this first step, we feed our task-specific few-shot examples into a large language model (LLM) and prompt it to perform document-to-query generation. More precisely, let $\{ ( q _ { i } , - d _ { i } ) \bar  \} ^ { k }$ be the $k$ few-shot examples, where each example is a query $( q _ { i } \sim \mathcal { Q } _ { T } )$ and a document relevant to that query $( d _ { i } \in \mathcal { D } _ { T }$ ) according to the target task $T$ $( \mathbb { Z } _ { T } ( q _ { i } , d _ { i } ) = 1 )$ ).
+
+ollowing FLAN (Wei et al., 2022a), we instruction-prompt the LLM with the following string prefix:
+
+$$
+e _ { d o c } ( d _ { i } ) \diamond e _ { q u e r y } ( q _ { 1 } ) \diamond . . . \diamond e _ { d o c } ( d _ { k } ) \diamond e _ { q u e r y } ( q _ { k } ) \diamond e _ { d o c } ( d ) ,
+$$
+
+where $\diamond$ is a separator token, $e _ { d o c } ( d )$ and $e _ { q u e r y } ( q )$ are task-specific document and query descriptions respectively, and $d$ is a new document presented at inference time. Using the ArguAna task as an example, we set $e _ { d o c } ( d ) = \mathrm { \ " { s } ~ } \mathrm { \bar { s } ~ } _ { \mathrm { \bar { d } r g u m e n t : } } \left\{ \mathrm { \Omega } \right.$ $\{ d \}$ ” and $e _ { q u e r y } =$ “Counter Argument: $\{ { q \} } ^ { , }$ to inform the LLM to generate counterarguments 3. The LLM is expected to generate $e _ { q u e r y } ( \hat { q } )$ . We consider it a generation failure if the query description does not precede the actual query; otherwise, we accept $\hat { q }$ and form a synthetic relevant example $( \hat { q } , d )$ .
+
+Running the prompt on all documents from $\mathcal { D } _ { T }$ , we obtain a large set of synthetic $( \hat { q } , d )$ examples, amplifying the information from a few examples into a large synthetic dataset whose query distribution is similar to the true task distribution $\mathcal { Q } _ { T }$ and whose query-document pairs convey the true search intent $\mathcal { I } _ { T }$ . This form of few-shot data extrapolation is similar to Lee et al. (2021).
+
+We use FLAN (Wei et al., 2022a) as our LLM, and refer to our query generator as $p _ { \mathrm { F L A N } } ( q | d )$ . FLAN is trained on a collection of tasks described via instructions and was shown to have good zero/few-shot performance on unseen tasks. We use the 137B parameter checkpoint. During prompt engineering, we use at most 8 examples, and reduce the number if they exceed the input length limit of FLAN. We also manually truncate individual queries and documents in the examples if they are too long. We randomly sample up to 1 million documents from each corpus and generate 8 questions per document using sampling-based decoding with temperature 0.7.
+
+# 3.2 ROUND-TRIP FILTERING GENERATED DATA
+
+We employ round-trip filtering (Alberti et al., 2019; Lewis et al., 2021) to improve the quality of our synthetic data. The main intuition is that for any synthetic query $\hat { q }$ generated from passage $d$ , a good $\hat { q }$ should also retrieve its original passage $d$ . In other words, the original $d$ should have high probability under some retriever, $p ( d | \hat { q } )$ (the reverse direction of query generation). If not, then we filter out $\hat { q }$ .
+
+Round-trip filtering has been very effective for synthetic question generation on QA tasks. However, these techniques typically rely on a question-answering model for the reverse direction filter. Since not all retrieval tasks resemble question-answering, this will not suffice in our setting.
+
+Instead, we train an initial retriever from the unfiltered synthetic data, and then use it to filter the synthetic data. This works surprisingly well over the different search intents observed in BEIR. More precisely, given a synthetic query-document pair $( \hat { q } , d )$ , we use the initial retriever to predict the most relevant passages for $\hat { q }$ . We keep $\hat { q }$ only when $d$ occurs among the Top- $K$ passages returned by the retriever. We show this filter substantially reduces the number of synthetic queries and significantly improves retrieval performance. In Appendix F, we provide more insight into why this can work by viewing our synthetic queries as latent variables.
+
+# 3.3 FEW-SHOT PROMPTAGATOR RETRIEVER
+
+Our synthetically generated data allows training task-specific neutral retrievers for tasks where indomain fine-tuning is challenging due to data scarcity. In this work, we use a standard dual encoder retrieval architecture and we propose a simple pretrain-finetune recipe.
+
+We pretrain our retriever on C4 with the independent cropping task from Contriever (Izacard et al., 2022a), where we treat two random crops from the same document as an artificial positive (query, document) pair and train with a cross-entropy loss over in-batch random negatives. Next, we fine-tune the dual encoder on $( \hat { q } , d )$ pairs from our prompt-based query generation, again with in-batch random negatives. After training for a set number of epochs, we apply round-trip filtering on our synthetic
+
+<table><tr><td rowspan="2"></td><td colspan="2">Training Recipe</td><td colspan="4">Retrieval Architecture</td><td>QGen</td></tr><tr><td>Retrieval Supervision</td><td>Cross-Attn Distillation</td><td>Retriever</td><td>Token-level Retrieval</td><td>#Reranking Doc.</td><td>Serving Model Size</td><td>Model</td></tr><tr><td>Contriever</td><td>NA</td><td></td><td>self</td><td></td><td></td><td>110M</td><td></td></tr><tr><td>GTR-XXL</td><td>MS MARCO (500K)</td><td></td><td>self</td><td></td><td></td><td>6B</td><td></td></tr><tr><td>Splade v2</td><td>MS MARCO (500K)</td><td>//vv</td><td>self</td><td>√</td><td></td><td>110M</td><td></td></tr><tr><td>ColBERT v2</td><td>MS MARCO (500K)</td><td></td><td>self</td><td>√</td><td></td><td>110M</td><td></td></tr><tr><td>GenQ</td><td>MS MARCO (500K)</td><td></td><td>self</td><td></td><td></td><td>110M</td><td>T5 (MS MARCO)</td></tr><tr><td>GPL</td><td>MS MARCO (500K)</td><td></td><td>self</td><td></td><td></td><td>110M</td><td>T5 (MS MARCO)</td></tr><tr><td>MonoT5</td><td>MS MARCO (500K)</td><td></td><td>BM25</td><td>√</td><td>1000</td><td>3B</td><td></td></tr><tr><td>InPars</td><td>Few (3 from MS MARCO)</td><td></td><td>BM25</td><td>√</td><td>1000</td><td>3B</td><td>GPT-3</td></tr><tr><td>UPR</td><td>NA</td><td></td><td>Contriever</td><td></td><td>1000</td><td>110M+3B</td><td>T0*</td></tr><tr><td>PROMPTAGATOR</td><td>Few (0-8)</td><td></td><td>self</td><td></td><td></td><td>110M</td><td>FLAN</td></tr><tr><td>PROMPTAGATOR++</td><td>Few (0-8)</td><td></td><td>PROMPTAGATOR</td><td></td><td>200</td><td>110M+125M</td><td>FLAN</td></tr></table>
+
+Table 1: Comparison of different retrieval frameworks. Our serving models are just a 110M-size dual encoder PROMPTAGATOR and a 125M-size reranker PROMPTAGAT $\mathrm { \Phi _ { O R + + } }$ , as good quality generated data allows simple models/pipeline to achieve strong performance.4 InPars’s few-shot examples are from MS MARCO and is task-independent. See text for more details for UPR’s QGen model5.
+
+data as described in $\ S 3 . 2$ using this initial dual encoder, and then continue to fine-tune the dual encoder on the filtered data.
+
+We also propose PROMPTAGATOR $^ { + + }$ , a reranker trained on the same synthetic data generated from our prompt-based QGen, which refines the retrieved candidates using a slower but more accurate cross-attention model. We train the reranker using a cross-entropy loss with 31 sampled negatives from top 200 passages retrieved by the PROMPTAGATOR retriever, which approximates the inference time distribution (reranking top 200 from the retriever).
+
+Zero-shot PROMPTAGATOR Our prompt-based query generation can run in a zero-shot manner, where we universally apply the following prompt irrespective of the target task: ’{d} Read the passage and generate a query.’. Here {d} denotes the document text. Training retrievers and rerankers on this data leads to zero-shot PROMPTAGATOR and zero-shot PROMPTAGATOR $^ { + + }$ . This recipe serves as a baseline to show the benefits of adapting the few-shot prompt to the target task.
+
+# 3.4 COMPARISON WITH PRIOR METHODS
+
+Table 1 compares the setting of PROMPTAGATOR to some recently proposed approaches. Several dimensions of our recipe are simpler: our dual encoder does not employ hard negative mining, distillation from a cross-attention teacher or token-level retrieval. Also, our 125M parameter reranker is smaller than most other rerankers. We aim to show that even simpler and smaller architectures can achieve excellent results if trained with synthetic data that has been few-shot adapted $( \ S 4 . 3 )$ . Compared to InPars (Bonifacio et al., 2022) and UPR (Sachan et al., 2022), our approach employs task-specific few-shot adaption, while InPars and UPR’s prompts are task-independent and thus bear the same limitation of previous query generation approach (Ma et al., 2021; Wang et al., 2022). Another key difference is that prior works focused on reranking, while we enable few-shot learning for both reranking and retrieval.
+
+# 4 EXPERIMENTS
+
+We evaluate PROMPTAGATOR on the BEIR benchmark. We then dive deeper into the results through ablation studies and qualitative analysis.
+
+# 4.1 IMPLEMENTATION
+
+The original FLAN training set overlapped with 2 datasets in BEIR: NQ and Quora6. Most existing systems use MS MARCO for fully supervised learning and therefore do not report few or zeroshot results on MS MARCO. Therefore we exclude MS MARCO, NQ and Quora from our main evaluations. We report $\mathrm { n D C G } @ 1 0$ , the standard retrieval evaluation metric on BEIR.
+
+Table 2: Main Results. $\mathrm { n D C G } @ 1 0$ on BEIR. Retriever Comparisons (Upper Half): Among the various kind of retrievers, both zero-shot and few-shot PROMPTAGATOR produce strong results. Retriever+Reranker Comparisons (Lower Half): In the scenario where speed is not a concern, reranker is often used. We train PROMPTAGATOR $^ { + + }$ use the same generated data and get significant improvement. See text for more details for Climate-FEVER and Webis-Touché2020.   
+
+<table><tr><td></td><td>arg</td><td>touché</td><td>covid</td><td>nfc</td><td>hotpot</td><td>dbp</td><td>climate</td><td>fever</td><td>scifact</td><td>scidocs</td><td>fiqa</td><td>AVG.</td></tr><tr><td colspan="10"></td><td></td><td></td><td></td><td></td></tr><tr><td>Unsupervised</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>BM25</td><td>31.5</td><td>36.7</td><td>65.6</td><td>32.5</td><td>60.3</td><td>31.3</td><td>21.3</td><td></td><td>75.3</td><td>66.5</td><td>15.8</td><td>23.6</td><td>41.8</td></tr><tr><td>Contriever</td><td>37.9</td><td>19.3</td><td></td><td>27.4</td><td>31.7 48.1</td><td>29.2</td><td></td><td>15.5</td><td>68.2</td><td>64.9</td><td>14.9</td><td>24.5</td><td>34.7</td></tr><tr><td colspan="10">Supervised by MS MARCO</td><td></td><td>16.1</td><td></td><td></td></tr><tr><td>GTR-XXL</td><td>54.0</td><td>25.6</td><td>50.1</td><td>34.2</td><td>59.9</td><td>40.8</td><td>26.7</td><td>74.0</td><td>66.2</td><td></td><td></td><td>46.7</td><td>44.9</td></tr><tr><td>SPLADE v2</td><td>47.9</td><td>27.2</td><td>71.0</td><td>33.4</td><td>68.4</td><td>43.5</td><td>23.5</td><td></td><td>78.6</td><td>69.3</td><td>15.8</td><td>33.6</td><td>46.6</td></tr><tr><td>ColBERTv2</td><td>46.3</td><td>26.3</td><td>73.8</td><td>33.8</td><td>66.7</td><td>44.6</td><td>17.6</td><td></td><td>78.5</td><td>69.3</td><td>15.4</td><td>35.6</td><td>46.2</td></tr><tr><td>GenQ</td><td>49.3</td><td>18.2</td><td>61.9 70.0</td><td>31.9</td><td>53.4</td><td>32.8</td><td></td><td>17.5</td><td>66.9</td><td>64.4</td><td>14.3</td><td>30.8</td><td>40.1</td></tr><tr><td>GPL</td><td>55.7</td><td>25.5</td><td></td><td>34.5</td><td>58.2</td><td>38.4</td><td></td><td>23.5</td><td>75.9</td><td>67.4</td><td>16.9</td><td>34.4</td><td>45.5</td></tr><tr><td colspan="10">PROMPTAGATOR (110M)</td><td></td><td></td><td></td><td></td></tr><tr><td>Zero-shot</td><td>53.8</td><td>26.6</td><td>72.7</td><td>33.4</td><td>60.4</td><td>36.4</td><td>21.4</td><td>76.2</td><td></td><td>62.3</td><td>16.3</td><td>40.4</td><td>45.5</td></tr><tr><td>Few-shot</td><td>59.4</td><td>34.5</td><td>75.6</td><td>33.4</td><td>61.4</td><td>38.0</td><td>16.8 (24.0*)</td><td></td><td>77.0</td><td>65.0</td><td>18.4</td><td>46.2</td><td>47.8</td></tr><tr><td colspan="10">Retriever+ Reranker</td><td></td><td></td><td></td><td></td></tr><tr><td colspan="10">Unsupervised</td><td></td><td></td><td></td><td></td></tr><tr><td>UPR (3B) InPars (3B)</td><td>50.3</td><td>21.3</td><td></td><td>60.4 33.3</td><td>72.2</td><td>33.8</td><td>9.5</td><td></td><td>57.3 1</td><td>69.6</td><td>17.3</td><td>45.0</td><td>42.7</td></tr><tr><td></td><td></td><td>1</td><td>78.4</td><td></td><td></td><td>1</td><td>1</td><td></td><td></td><td></td><td>1</td><td>1</td><td>1</td></tr><tr><td colspan="10">Supervised by MS MARCO</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>monoT5 (220M)</td><td>13.2</td><td>27.7</td><td>77.8</td><td>35.7</td><td>69.5</td><td>41.9</td><td>24.5</td><td></td><td>80.2</td><td>73.6</td><td>16.5</td><td>41.4</td><td>45.6</td></tr><tr><td>monoT5 (3B)</td><td>28.8</td><td>20.0</td><td>79.5</td><td>38.4</td><td>75.9</td><td>47.8</td><td>28.0</td><td></td><td>85.0</td><td>77.7</td><td>19.7</td><td>51.4</td><td>51.1</td></tr><tr><td colspan="10">PROMPTAGATOR++(110M+ 125M)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Zero-shot</td><td>52.1</td><td>27.8</td><td>76.0</td><td>36.0</td><td>71.2</td><td>41.3</td><td>22.6</td><td></td><td>83.8 86.6</td><td>73.2 73.1</td><td>19.1 20.1</td><td>45.9</td><td>49.9</td></tr><tr><td>Few-shot</td><td>63.0</td><td>38.1</td><td>76.2</td><td>37.0</td><td>73.6</td><td>43.4</td><td>20.3 (24.1*)</td><td></td><td></td><td></td><td></td><td>49.4</td><td>52.8</td></tr></table>
+
+For PROMPTAGATOR’s query generation, we sample questions from FLAN with a temperature of 0.7. For round-trip filtering, we find that setting filtering threshold $K$ to 1 gives the best results on MS MARCO and thus use 1 for all BEIR datasets, We implement PROMPTAGATOR’s dual encoders following GTR (Ni et al., 2021).To ensure efficiency, we use the T5-base encoder architecture consisting of 110M parameters. For PROMPTAGATOR $^ { + + }$ reranking models, we initialize from a T5- base version 1.1 encoder checkpoint which has 125M parameters . More details of the reranker implementation can be found in Appendix C. At inference time, we rerank the top 200 candidates retrieved from the PROMPTAGATOR dual encoder.
+
+We mostly follow the hyperparameters used in Ni et al. (2021). By default, we use batch size 6k; however, some of the corpora in BEIR contain only a few thousand documents, making multiple relevant documents appear in the same batch, which interacts negatively with our in-batch softmax loss. To address this issue, we split all datasets into three groups based on corpus size: small $( < 5 0 \mathrm { k } )$ , medium $( 5 0 \mathbf { k } { - } 5 0 0 \mathbf { k } )$ and large $( > 5 0 0 \mathrm { k } )$ . For dual encoder training, we use 128 batch size for small datasets and 6k for others. We finetune for 5k steps for large datasets and 1k for others. For ranking models, we use batch size 64 for all datasets and finetune large datasets for $2 0 \mathrm { k }$ steps, $5 \mathrm { k }$ for others.
+
+# 4.2 MAIN RESULTS
+
+Table 2 shows the experimental results. We first notice that zero-shot PROMPTAGATOR already serves as a strong baseline, comparing favorably to other retrieval baselines trained on $\mathcal { O } ( 1 0 0 \mathrm { K } )$ examples from MS MARCO. Nonetheless, few-shot PROMPTAGATOR markedly improves upon zeroshot PROMPTAGATOR, increasing average nDCG $@ 1 0$ by over 2 points, which highlights the impact of few-shot learning. Few-shot PROMPTAGATOR, despite having a simple training procedure and model architecture, outperforms strong baselines such as GenQ (Thakur et al., 2021) and GPL (Wang et al., 2022) which also use query generation to augment training data, as well as ColBERT v2 (Santhanam et al., 2022) and SPLADE v2 (Formal et al., 2021) which rely on token level interaction architectures and distillation recipes.
+
+![](images/8fd7de3bf06be111ea0a3c358116bf0648d406528f07ae2fda10ca3417e0c05f.jpg)  
+Figure 2: Left (a). Delta in $\mathrm { n D C G } @ 1 0$ between few-shot PROMPTAGATOR with and without filtering. Middle (b): Comparing the effect of the generated data versus the number of supervised data on MS MARCO. PROMPTAGATOR with 8 examples can catch up with 50k labeled examples, when simple dual encoders are used. Right (c): Ablation on query generation model. GenQ is a prior system from Thakur et al. (2021), while NQ-QGen is our in-house NQ-trained T5 query generation model. Other than the generated data, NQ-QGen and PROMPTAGATOR uses the same hyper parameters.
+
+Our reranker PROMPTAGATOR $^ { + + }$ boosts performance by another 5 points on nDCG $@ 1 0$ . It significantly outperforms UPR (Sachan et al., 2022) whose reranker uses T0 (Sanh et al., 2022), an instruction tuned LLM similar to FLAN. It also outperforms monoT5-3B (Nogueira et al., 2020), which achieved previous state-of-the-art reranking performance on BEIR in a recent study (Rosa et al., 2022). Note that most of these reranker approaches use a large 3B parameter model for better generalization, while PROMPTAGATOR $^ { + + }$ uses a standard $1 2 5 \mathbf { M }$ reranker.
+
+Comparing few-shot PROMPTAGATOR to baselines, the biggest improvement is on Webis-Touché2020 (touché), followed by ArguAna (arg) . Webis-Touché2020’s goal is to retrieve documents for a controversial topic, e.g., “should felons who have completed their sentence be allowed to vote?”. ArguAna’s goal is to find the counter-arguments that oppose the input argument, and the input arguments are often several-sentence long. Both tasks are extremely different from traditional QA retrieval data that other models use, which are dominated by factoid questions. On the other hand, few-shot PROMPTAGATOR can successfully adapt to this task with a few examples.
+
+# 4.3 ANALYSIS
+
+Impact of round-trip filtering. In Figure 2(a), we show quality differences between few-shot PROMPTAGATOR with and without filtering. Filtering improves performance on 8 out of 11 datasets and leads to 2.5 points improvement on average, demonstrating the effectiveness of our filtering strategy. Nonetheless, filtering hurts model quality on NFCorpus and SciFact. These are the smallest datasets in terms of generated queries and may indicate overfitting of our retrievers.
+
+We find that the majority of filtered examples are either queries that are too generic that match many documents, or queries that contain additional terms that are irrelevant to the document. Examples are in Fig. 6 in the Appendix. There are also cases where high quality data was incorrectly removed. We suspect that designing dynamic filtering thresholds would help, and leave it to future exploration.
+
+Can generated queries replace human annotated queries? In Figure 2(b), we evaluate 8-shot PROMPTAGATOR on MS MARCO, comparing it against dual encoders trained on MS MARCO’s supervised data. Note that we did not add other components, to make the comparison simple. We chose MS MARCO as there are enough labeled data for this task and neither FLAN nor our models are trained on MS MARCO examples. The results show that eight examples plus an LLM can replace a significant portion of supervised examples.
+
+How does PROMPTAGATOR compare to other query generation approaches? Figure 2(c) compares zero-shot PROMPTAGATOR to two other query generation approaches: GenQ is prior system from Thakur et al. (2021) using a MS MARCO trained T5 query generation model, and NQ-QGen is our in-house T5 QGen model finetuned on NQ. The figure shows the advantages of zero-shot PROMPTAGATOR, outperforming both baselines by large margins. Importantly, NQ-QGen uses the same filtering, dual-encoder training, batch sizes and training steps as PROMPTAGATOR, providing a faircomparison of query generators. This indicates that the main contributing factor to PROMPTAGATOR is better queries from prompting an LLM, not the specific training recipe or hyperparameters.
+
+<table><tr><td></td><td>arg</td><td>touche</td><td>covid</td><td>nfc</td><td>hotpot</td><td>dbp</td><td>climate</td><td>fever</td><td>scifact</td><td>scidocs</td><td>fiqa</td><td>AVG.</td></tr><tr><td>FLAN original</td><td>59.4</td><td>34.5</td><td>75.6</td><td>33.4</td><td>61.4</td><td>38.0</td><td>(24.0*）</td><td>77.0</td><td>65.0</td><td>18.4</td><td>46.2</td><td>48.5</td></tr><tr><td>FLAN w/o NQ and Quora</td><td>58.8</td><td>33.3</td><td>70.2</td><td>33.7</td><td>61.7</td><td>34.4</td><td>(23.5*）</td><td>76.2</td><td>63.8</td><td>18.3</td><td>43.0</td><td>47.0</td></tr></table>
+
+![](images/6b0deda12c0ebe78b5a86065c0e4184c528a916a77041c1ebe8bcd8119ad8c08.jpg)  
+Table 3: Impact of different FLAN versions. This study uses Fever prompt for Climate Fever (§4.3)   
+Figure 3: Top first word distribution on queries generated from different models in the ArguAna dataset. Left (a)(b)(c): Compare gold queries (a) and generated queries (b)(c). Queries generated by few-shot models has closer distribution to the gold queries, while the NQ-QGen queries are mostly questions. Right (d): The few shot FLAN can generate diverse queries even though there are only 4 examples in the prompt. Statistics of more datasets are available in the Appendix (Figure 4).
+
+Does few-shot always improve over zero-shot? As shown in Table 2, few-shot PROMPTAGATOR almost always outperforms zero-shot PROMPTAGATOR except for Climate-FEVER. The original Climate-FEVER dataset uses one of three tags to annotate a query-document pair, namely “supports”, “refutes”, or “not enough info”. However, BEIR treats all these three annotations as relevant, which is problematic. Using query-document pairs annotated “not enough info” in our prompt could be detrimental to generation quality. Therefore, we tried switching to FEVER’s few-shot prompt, as the two datasets share same corpus and similar search intents. With the better annotated examples, few-shot PROMPTAGATOR indeed surpass zero-shot. This result provides some evidence that low quality few-shot examples negatively affect PROMPTAGATOR.
+
+Impact of FLAN Versions FLAN was trained on a collection of datasets which have some overlap with BEIR; specifically, it includes Natural Questions (NQ) and Quora. It was not trained on querydocument pairs from NQ or Quora; however, in order to determine whether the inclusion of this data biased the results on the final retrieval evaluation, we designed an additional ablation experiment. Following the original FLAN recipe (Wei et al., 2022a), we trained an additional LLM excluding both the NQ and Quora datasets. Table 4 shows the results. While the accuracy drops slightly, the overall performance still outperform prior retrievers.
+
+Qualitative Analysis In order to understand the advantages of few-shot PROMPTAGATOR, we analyze the distribution of queries generated by different query generation methods for ArguAna in Figure 3. To easily see the differences, for each distribution we plot the histogram of each query’s first word. Note that the distribution of few-shot PROMPTAGATOR (Fig. 3b) is much closer to the real distribution (Fig. 3a) while the NQ-QGen (Fig. 3c) mostly generated questions even when the queries in this task are generally arguments, not questions. More examples are showcased in Table 6 in the Appendix.
+
+Use few-shot examples directly To study the effect of using the few-shot examples directly, we conduct a study of fine-tuning task-specific models using the GTR dual encoder (110M) (Ni et al., 2021) with the few-shot examples in appendix A. As expected, it does not provide a large amount of impact for GTR base, where the average nDCG decreases from 40.4 to 38.7.
+
+# 5 RELATED WORK
+
+Neural Retrieval and Reranking Models The majority of neural retrievers today employ a dual encoder architecture that encodes queries and documents independently into dense vectors and retrieves documents using maximum inner product search (MIPS). Recent research has primarily focused on the following aspects: developing better pre-training tasks (Lee et al., 2019; Chang et al., 2020; Izacard et al., 2022a; Gao & Callan, 2021; Oguz et al., 2022), improving contrastive negatives (Qu et al., 2021; Xiong et al., 2021; Lu et al., 2021), and improving generalization across different domains (Thakur et al., 2021; Ren et al., 2022).
+
+Although dual encoders enable fast retrieval, their expressivity is limited due to the fact that their score is just a dot-product between a query vector and a document vector. A common solution is to use a cross-attention model to rerank retrieved candidates (Nogueira & Cho, 2019; Nogueira et al., 2020), as cross-attention rerankers can explicitly model the interaction between query and document tokens. Distilling cross-attention models into dual encoders has been effective in closing the gap between the two (Hofstätter et al., 2020; Ren et al., 2021; Reddi et al., 2021; Zhang et al., 2022).
+
+Neural Retrieval with Fine-Grained Interactions An alternative for bridging dense retrievers and cross-attention models is to allow some amount of fine-grained query-document interactions in the retriever. Humeau et al. (2020) and Luan et al. (2021) represent and retrieve queries and documents with multiple vectors instead of a single vector. ColBERT (Khattab & Zaharia, 2020), COIL (Gao et al., 2021a) and SPLADE (Formal et al., 2021) take this further by using token-level interactions between queries and documents. Because these models are not just modeling a dot product, MIPS algorithms cannot be used directly. Hence, these models usually have much higher inference/serving cost compared to dual encoders.
+
+Prompt-based Query Generation The idea of using prompted LLMs for query generation has previously been proposed for improving retrieval reranking. UPR (Sachan et al., 2022) proposed to use prompted LLMs to rerank passages directly. InPars (Bonifacio et al., 2022) is probably the most closely related work to ours. They proposed to use few-shot prompting with GPT-3 to generate synthetic data for training a T5-based reranker. Though InPars was tested on multiple retrieval datasets, they used a task-independent prompt constructed from MS MARCO and did not explore task-specific few-shot learning. They also focused exclusively on reranking, whereas we also address full-scale retrieval.
+
+Few-shot Learning pre-trained LLMs have significantly advanced few-shot learning, thanks to prompting strategies such as in-context learning and instruction-prompting (Brown et al., 2020; Wei et al., 2022b). Some approaches fine-tune LLMs specifically for few-shot learning (Schick & Schütze, 2021a;b;c; Gao et al., 2021b; Logan IV et al., 2022; Izacard et al., 2022b) while others do not (Brown et al., 2020; Bonifacio et al., 2022). We use LLMs as few-shot data generators — this form of data augmentation via few-shot example extrapolation is similar to Lee et al. (2021).
+
+# 6 CONCLUSION AND DISCUSSIONS
+
+In this paper, we present PROMPTAGATOR, a novel approach to few-shot retrieval. We showed that it is possible to create task-specific retrievers and rerankers with only a few annotated examples. The few-shot examples, amplified by prompt-based LLM query generation, simplifies the complexity of training neural retrievers for new tasks and leads to promising performance gains. It hopefully inspires future research towards generalizable retrieval systems that can seamlessly and efficiently adapt to many tasks.
+
+While we demonstrate that LLM-based query generation can be very effective, many questions remain. One of the key issue requiring further investigation is on the generated data efficiency. We have not yet explored exactly how many query-document pairs are needed for each task, or how to use these generated examples more efficiently. Another issue is the sensitivity of the final retriever’s performance with respect to the prompt. Finally, we would like to draw a connection from PROMPTAGATOR to distillation, as the final dual encoders indirectly “learn” from the LLM. Analyzing the headroom and understanding how we can better transfer knowledge from LLMs to retrievers would be a critical topic for the future.
+
+As mentioned in $\ S 6$ , PROMPTAGATOR can be viewed as distilling LLM to standard-sized dual encoders via prompt-based query generation. While the distillation process is computationally expensive, it significantly reduces cost for inference.
+
+# ACKNOWLEDGEMENTS
+
+We thank Kenton Lee, Tom Kwiatkowski, and Daniel Gillick for technical discussion and providing feedback on our manuscript. We thank Alex Salcianu for developing a bulk inference pipeline for large language models.
+
+# REFERENCES
+
+Chris Alberti, Daniel Andor, Emily Pitler, Jacob Devlin, and Michael Collins. Synthetic QA corpora generation with roundtrip consistency. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pp. 6168–6173, Florence, Italy, July 2019. Association for Computational Linguistics. doi: 10.18653/v1/P19-1620. URL https://aclanthology. org/P19-1620.
+
+Luiz Bonifacio, Hugo Abonizio, Marzieh Fadaee, and Rodrigo Nogueira. Inpars: Unsupervised dataset generation for information retrieval. In Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR ’22, pp. 2387–2392, New York, NY, USA, 2022. Association for Computing Machinery. ISBN 9781450387323. doi: 10.1145/3477495.3531863. URL https://doi.org/10.1145/3477495.3531863.
+
+Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language models are few-shot learners. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 1877–1901. Curran Associates, Inc., 2020. URL https://proceedings.neurips. cc/paper/2020/file/1457c0d6bfcb4967418bfb8ac142f64a-Paper.pdf.
+
+Wei-Cheng Chang, Felix X. Yu, Yin-Wen Chang, Yiming Yang, and Sanjiv Kumar. Pre-training tasks for embedding-based large-scale retrieval. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id $=$ rkg-mA4FDr.
+
+A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the em algorithm. JOURNAL OF THE ROYAL STATISTICAL SOCIETY, SERIES B, 39(1):1–38, 1977.
+
+Thibault Formal, Carlos Lassance, Benjamin Piwowarski, and Stéphane Clinchant. SPLADE v2: Sparse lexical and expansion model for information retrieval. CoRR, abs/2109.10086, 2021. URL https://arxiv.org/abs/2109.10086.
+
+Luyu Gao and Jamie Callan. Condenser: a pre-training architecture for dense retrieval. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pp. 981–993, Online and Punta Cana, Dominican Republic, November 2021. Association for Computational Linguistics. doi: 10.18653/v1/2021.emnlp-main.75. URL https://aclanthology.org/ 2021.emnlp-main.75.
+
+Luyu Gao, Zhuyun Dai, and Jamie Callan. COIL: Revisit exact lexical match in information retrieval with contextualized inverted list. In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 3030–3042, Online, June 2021a. Association for Computational Linguistics. doi: 10.18653/v1/ 2021.naacl-main.241. URL https://aclanthology.org/2021.naacl-main.241.
+
+Tianyu Gao, Adam Fisch, and Danqi Chen. Making pre-trained language models better few-shot learners. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pp. 3816–3830, Online, August 2021b. Association for Computational Linguistics. doi: 10.18653/v1/2021.acl-long.295. URL https://aclanthology.org/2021.acl-long. 295.
+
+Faegheh Hasibi, Fedor Nikolaev, Chenyan Xiong, Krisztian Balog, Svein Erik Bratsberg, Alexander Kotov, and Jamie Callan. Dbpedia-entity v2: A test collection for entity search. In Proceedings of the 40th International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR ’17, pp. 1265–1268, New York, NY, USA, 2017. ISBN 9781450350228. URL https://doi.org/10.1145/3077136.3080751.
+
+Sebastian Hofstätter, Sophia Althammer, Michael Schröder, Mete Sertkan, and Allan Hanbury. Improving efficient neural ranking models with cross-architecture knowledge distillation. $A r X i \nu$ , abs/2010.02666, 2020. URL https://arxiv.org/abs/2010.02666.
+
+Samuel Humeau, Kurt Shuster, Marie-Anne Lachaux, and Jason Weston. Poly-encoders: Architectures and pre-training strategies for fast and accurate multi-sentence scoring. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net, 2020. URL https://openreview.net/forum?id $=$ SkxgnnNFvH.
+
+Gautier Izacard, Mathilde Caron, Lucas Hosseini, Sebastian Riedel, Piotr Bojanowski, Armand Joulin, and Edouard Grave. Unsupervised dense information retrieval with contrastive learning. Transactions on Machine Learning Research, 2022a. URL https://openreview.net/ forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ jKN1pXi7b0.
+
+Gautier Izacard, Patrick Lewis, Maria Lomeli, Lucas Hosseini, Fabio Petroni, Timo Schick, Jane Dwivedi-Yu, Armand Joulin, Sebastian Riedel, and Edouard Grave. Few-shot learning with retrieval augmented language models, 2022b. URL https://arxiv.org/abs/2208.03299.
+
+Vladimir Karpukhin, Barlas Oguz, Sewon Min, Patrick Lewis, Ledell Wu, Sergey Edunov, Danqi Chen, and Wen-tau Yih. Dense passage retrieval for open-domain question answering. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 6769–6781, Online, November 2020. Association for Computational Linguistics. doi: 10.18653/v1/ 2020.emnlp-main.550. URL https://aclanthology.org/2020.emnlp-main.550.
+
+Omar Khattab and Matei Zaharia. ColBERT: Efficient and effective passage search via contextualized late interaction over BERT. In Jimmy Huang, Yi Chang, Xueqi Cheng, Jaap Kamps, Vanessa Murdock, Ji-Rong Wen, and Yiqun Liu (eds.), Proceedings of the 43rd International ACM SIGIR conference on research and development in Information Retrieval, SIGIR 2020, Virtual Event, China, July 25-30, 2020, pp. 39–48. ACM, 2020. doi: 10.1145/3397271.3401075. URL https: //doi.org/10.1145/3397271.3401075.
+
+Tom Kwiatkowski, Jennimaria Palomaki, Olivia Redfield, Michael Collins, Ankur Parikh, Chris Alberti, Danielle Epstein, Illia Polosukhin, Jacob Devlin, Kenton Lee, Kristina Toutanova, Llion Jones, Matthew Kelcey, Ming-Wei Chang, Andrew M. Dai, Jakob Uszkoreit, Quoc Le, and Slav Petrov. Natural questions: A benchmark for question answering research. Transactions of the Association for Computational Linguistics, 7:452–466, March 2019. doi: 10.1162/tacl_a_00276. URL https://aclanthology.org/Q19-1026.
+
+Kenton Lee, Ming-Wei Chang, and Kristina Toutanova. Latent retrieval for weakly supervised open domain question answering. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pp. 6086–6096, Florence, Italy, July 2019. Association for Computational Linguistics. doi: 10.18653/v1/P19-1612. URL https://aclanthology.org/P19-1612.
+
+Kenton Lee, Kelvin Guu, Luheng He, Tim Dozat, and Hyung Won Chung. Neural data augmentation via example extrapolation. CoRR, abs/2102.01335, 2021. URL https://arxiv.org/abs/ 2102.01335.
+
+Patrick Lewis, Yuxiang Wu, Linqing Liu, Pasquale Minervini, Heinrich Küttler, Aleksandra Piktus, Pontus Stenetorp, and Sebastian Riedel. PAQ: 65 million probably-asked questions and what you can do with them. Transactions of the Association for Computational Linguistics, 9:1098–1115, 2021. doi: 10.1162/tacl_a_00415. URL https://aclanthology.org/2021.tacl-1. 65.
+
+Robert Logan IV, Ivana Balazevic, Eric Wallace, Fabio Petroni, Sameer Singh, and Sebastian Riedel. Cutting down on prompts and parameters: Simple few-shot learning with language models. In Findings of the Association for Computational Linguistics: ACL 2022, pp. 2824–2835, Dublin,
+
+Ireland, May 2022. Association for Computational Linguistics. doi: 10.18653/v1/2022.findings-acl.   
+222. URL https://aclanthology.org/2022.findings-acl.222.
+
+Jing Lu, Gustavo Hernandez Abrego, Ji Ma, Jianmo Ni, and Yinfei Yang. Multi-stage training with improved negative contrast for neural passage retrieval. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pp. 6091–6103, Online and Punta Cana, Dominican Republic, November 2021. Association for Computational Linguistics. doi: 10.18653/v1/ 2021.emnlp-main.492. URL https://aclanthology.org/2021.emnlp-main.492.
+
+Yi Luan, Jacob Eisenstein, Kristina Toutanova, and Michael Collins. Sparse, Dense, and Attentional Representations for Text Retrieval. Transactions of the Association for Computational Linguistics, 9:329–345, 04 2021. doi: 10.1162/tacl_a_00369. URL https://doi.org/10.1162/ tacl_a_00369.
+
+Ji Ma, Ivan Korotkov, Yinfei Yang, Keith Hall, and Ryan McDonald. Zero-shot neural passage retrieval via domain-targeted synthetic question generation. In Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume, pp. 1075–1088, Online, April 2021. Association for Computational Linguistics. doi: 10.18653/v1/ 2021.eacl-main.92. URL https://aclanthology.org/2021.eacl-main.92.
+
+Macedo Maia, Siegfried Handschuh, André Freitas, Brian Davis, Ross McDermott, Manel Zarrouk, and Alexandra Balahur. Www’18 open challenge: financial opinion mining and question answering. In Companion proceedings of the the web conference 2018, pp. 1941–1942, 2018. URL https: //doi.org/10.1145/3184558.3192301.
+
+Arvind Neelakantan, Tao Xu, Raul Puri, Alec Radford, Jesse Michael Han, Jerry Tworek, Qiming Yuan, Nikolas A. Tezak, Jong Wook Kim, Chris Hallacy, Johannes Heidecke, Pranav Shyam, Boris Power, Tyna Eloundou Nekoul, Girish Sastry, Gretchen Krueger, David P. Schnurr, Felipe Petroski Such, Kenny Sai-Kin Hsu, Madeleine Thompson, Tabarak Khan, Toki Sherbakov, Joanne Jang, Peter Welinder, and Lilian Weng. Text and code embeddings by contrastive pre-training. ArXiv, abs/2201.10005, 2022. URL https://arxiv.org/abs/2201.10005.
+
+Tri Nguyen, Mir Rosenberg, Xia Song, Jianfeng Gao, Saurabh Tiwary, Rangan Majumder, and Li Deng. Ms marco: A human generated machine reading comprehension dataset. In CoCo@NIPS, 2016. URL http://ceur-ws.org/Vol-1773/CoCoNIPS_2016_paper9.pdf.
+
+Jianmo Ni, Chen Qu, Jing Lu, Zhuyun Dai, Gustavo Hernández Ábrego, Ji Ma, Vincent Y. Zhao, Yi Luan, Keith B. Hall, Ming-Wei Chang, and Yinfei Yang. Large dual encoders are generalizable retrievers. CoRR, abs/2112.07899, 2021. URL https://arxiv.org/abs/2112.07899.
+
+Rodrigo Nogueira and Kyunghyun Cho. Passage re-ranking with bert. arXiv, 2019. URL https: //arxiv.org/abs/1901.04085.
+
+Rodrigo Nogueira, Zhiying Jiang, Ronak Pradeep, and Jimmy Lin. Document ranking with a pretrained sequence-to-sequence model. In Trevor Cohn, Yulan He, and Yang Liu (eds.), Findings of the Association for Computational Linguistics: EMNLP 2020, Online Event, 16-20 November 2020, volume EMNLP 2020 of Findings of ACL, pp. 708–718. Association for Computational Linguistics, 2020. doi: 10.18653/v1/2020.findings-emnlp.63. URL https://doi.org/10. 18653/v1/2020.findings-emnlp.63.
+
+Barlas Oguz, Kushal Lakhotia, Anchit Gupta, Patrick Lewis, Vladimir Karpukhin, Aleksandra Piktus, Xilun Chen, Sebastian Riedel, Scott Yih, Sonal Gupta, and Yashar Mehdad. Domain-matched pre-training tasks for dense retrieval. In Findings of the Association for Computational Linguistics: NAACL 2022, pp. 1524–1534, Seattle, United States, July 2022. Association for Computational Linguistics. doi: 10.18653/v1/2022.findings-naacl.114. URL https://aclanthology. org/2022.findings-naacl.114.
+
+Yingqi Qu, Yuchen Ding, Jing Liu, Kai Liu, Ruiyang Ren, Wayne Xin Zhao, Daxiang Dong, Hua Wu, and Haifeng Wang. RocketQA: An optimized training approach to dense passage retrieval for open-domain question answering. In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 5835–5847, June 2021. doi: 10.18653/v1/2021.naacl-main.466. URL https: //aclanthology.org/2021.naacl-main.466.
+
+Sashank J. Reddi, Rama Kumar Pasumarthi, Aditya Krishna Menon, Ankit Singh Rawat, Felix X. Yu, Seungyeon Kim, Andreas Veit, and Sanjiv Kumar. Rankdistil: Knowledge distillation for ranking. In AISTATS, pp. 2368–2376, 2021. URL http://proceedings.mlr.press/ v130/reddi21a.html.
+
+Ruiyang Ren, Shangwen Lv, Yingqi Qu, Jing Liu, Wayne Xin Zhao, QiaoQiao She, Hua Wu, Haifeng Wang, and Ji-Rong Wen. PAIR: Leveraging passage-centric similarity relation for improving dense passage retrieval. In Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, pp. 2173–2183, Online, August 2021. Association for Computational Linguistics. doi: 10.18653/ v1/2021.findings-acl.191. URL https://aclanthology.org/2021.findings-acl. 191.
+
+Ruiyang Ren, Yingqi Qu, Jing Liu, Wayne Xin Zhao, Qifei Wu, Yuchen Ding, Hua Wu, Haifeng Wang, and Ji-Rong Wen. A thorough examination on zero-shot dense retrieval, 2022. URL https://arxiv.org/abs/2204.12755.
+
+Guilherme Moraes Rosa, Luiz Henrique Bonifacio, Vitor Jeronymo, Hugo Abonizio, Marzieh Fadaee, Roberto de Alencar Lotufo, and Rodrigo Nogueira. No parameter left behind: How distillation and model size affect zero-shot retrieval. CoRR, abs/2206.02873, 2022. doi: 10.48550/arXiv.2206. 02873. URL https://doi.org/10.48550/arXiv.2206.02873.
+
+Devendra Singh Sachan, Mike Lewis, Mandar Joshi, Armen Aghajanyan, Wen-tau Yih, Joelle Pineau, and Luke Zettlemoyer. Improving passage retrieval with zero-shot question generation. arXiv, 2022. URL https://arxiv.org/abs/2204.07496.
+
+Victor Sanh, Albert Webson, Colin Raffel, Stephen Bach, Lintang Sutawika, Zaid Alyafeai, Antoine Chaffin, Arnaud Stiegler, Arun Raja, Manan Dey, M Saiful Bari, Canwen Xu, Urmish Thakker, Shanya Sharma Sharma, Eliza Szczechla, Taewoon Kim, Gunjan Chhablani, Nihal V. Nayak, Debajyoti Datta, Jonathan Chang, Mike Tian-Jian Jiang, Han Wang, Matteo Manica, Sheng Shen, Zheng Xin Yong, Harshit Pandey, Rachel Bawden, Thomas Wang, Trishala Neeraj, Jos Rozen, Abheesht Sharma, Andrea Santilli, Thibault Févry, Jason Alan Fries, Ryan Teehan, Teven Le Scao, Stella Biderman, Leo Gao, Thomas Wolf, and Alexander M. Rush. Multitask prompted training enables zero-shot task generalization. In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022. OpenReview.net, 2022. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ 9Vrb9D0WI4.
+
+Keshav Santhanam, Omar Khattab, Jon Saad-Falcon, Christopher Potts, and Matei Zaharia. ColBERTv2: Effective and efficient retrieval via lightweight late interaction. In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 3715–3734, July 2022. URL https://aclanthology.org/2022.naacl-main.272.
+
+Timo Schick and Hinrich Schütze. Exploiting cloze-questions for few-shot text classification and natural language inference. In Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume, pp. 255–269, Online, April 2021a. Association for Computational Linguistics. doi: 10.18653/v1/2021.eacl-main.20. URL https://aclanthology.org/2021.eacl-main.20.
+
+Timo Schick and Hinrich Schütze. It’s not just size that matters: Small language models are also few-shot learners. In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 2339–2352, Online, June 2021b. Association for Computational Linguistics. doi: 10.18653/v1/2021.naacl-main. 185. URL https://aclanthology.org/2021.naacl-main.185.
+
+Timo Schick and Hinrich Schütze. Few-shot text generation with natural language instructions. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pp. 390–402, Online and Punta Cana, Dominican Republic, November 2021c. Association for Computational Linguistics. doi: 10.18653/v1/2021.emnlp-main.32. URL https: //aclanthology.org/2021.emnlp-main.32.
+
+Siamak Shakeri, Cicero Nogueira dos Santos, Henghui Zhu, Patrick Ng, Feng Nan, Zhiguo Wang, Ramesh Nallapati, and Bing Xiang. End-to-end synthetic data generation for domain adaptation of question answering systems. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 5445–5460. Association for Computational Linguistics, 2020. URL https://aclanthology.org/2020.emnlp-main.439.
+
+Nandan Thakur, Nils Reimers, Andreas Rücklé, Abhishek Srivastava, and Iryna Gurevych. BEIR: A heterogeneous benchmark for zero-shot evaluation of information retrieval models. In Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2), 2021. URL https://openreview.net/forum?id ${ . } = { }$ wCu6T5xFjeJ.
+
+Romal Thoppilan, Daniel De Freitas, Jamie Hall, Noam Shazeer, Apoorv Kulshreshtha, Heng-Tze Cheng, Alicia Jin, Taylor Bos, Leslie Baker, Yu Du, YaGuang Li, Hongrae Lee, Huaixiu Steven Zheng, Amin Ghafouri, Marcelo Menegali, Yanping Huang, Maxim Krikun, Dmitry Lepikhin, James Qin, Dehao Chen, Yuanzhong Xu, Zhifeng Chen, Adam Roberts, Maarten Bosma, Yanqi Zhou, Chung-Ching Chang, Igor Krivokon, Will Rusch, Marc Pickett, Kathleen S. Meier-Hellstern, Meredith Ringel Morris, Tulsee Doshi, Renelito Delos Santos, Toju Duke, Johnny Soraker, Ben Zevenbergen, Vinodkumar Prabhakaran, Mark Diaz, Ben Hutchinson, Kristen Olson, Alejandra Molina, Erin Hoffman-John, Josh Lee, Lora Aroyo, Ravi Rajakumar, Alena Butryna, Matthew Lamm, Viktoriya Kuzmina, Joe Fenton, Aaron Cohen, Rachel Bernstein, Ray Kurzweil, Blaise Aguera-Arcas, Claire Cui, Marian Croak, Ed H. Chi, and Quoc Le. Lamda: Language models for dialog applications. CoRR, abs/2201.08239, 2022. URL https://arxiv.org/abs/2201. 08239.
+
+James Thorne, Andreas Vlachos, Christos Christodoulopoulos, and Arpit Mittal. FEVER: a large-scale dataset for fact extraction and VERification. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), pp. 809–819, New Orleans, Louisiana, June 2018. Association for Computational Linguistics. doi: 10.18653/v1/N18-1074. URL https://aclanthology.org/N18-1074.
+
+Kexin Wang, Nandan Thakur, Nils Reimers, and Iryna Gurevych. GPL: Generative pseudo labeling for unsupervised domain adaptation of dense retrieval. In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 2345–2360, Seattle, United States, July 2022. Association for Computational Linguistics. doi: 10.18653/v1/2022.naacl-main.168. URL https://aclanthology.org/ 2022.naacl-main.168.
+
+Jason Wei, Maarten Paul Bosma, Vincent Zhao, Kelvin Guu, Adams Wei Yu, Brian Lester, Nan Du, Andrew Mingbo Dai, and Quoc V. Le. Finetuned language models are zero-shot learners. In International Conference on Learning Representations, 2022a. URL https://openreview. net/forum?id ${ . } = { }$ gEZrGCozdqR.
+
+Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yogatama, Maarten Bosma, Denny Zhou, Donald Metzler, Ed H. Chi, Tatsunori Hashimoto, Oriol Vinyals, Percy Liang, Jeff Dean, and William Fedus. Emergent abilities of large language models. Transactions on Machine Learning Research, 2022b. URL https://openreview.net/forum? id $=$ yzkSU5zdwD.
+
+Lee Xiong, Chenyan Xiong, Ye Li, Kwok-Fung Tang, Jialin Liu, Paul N. Bennett, Junaid Ahmed, and Arnold Overwijk. Approximate nearest neighbor negative contrastive learning for dense text retrieval. In International Conference on Learning Representations, 2021. URL https: //openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ zeFrfgyZln.
+
+Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Bengio, William Cohen, Ruslan Salakhutdinov, and Christopher D. Manning. HotpotQA: A dataset for diverse, explainable multi-hop question answering. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pp. 2369–2380, Brussels, Belgium, October-November 2018. Association for Computational Linguistics. doi: 10.18653/v1/D18-1259. URL https://aclanthology.org/D18-1259.
+
+Wen-tau Yih, Kristina Toutanova, John C. Platt, and Christopher Meek. Learning discriminative projections for text similarity measures. In Proceedings of the Fifteenth Conference on Computational Natural Language Learning, pp. 247–256, Portland, Oregon, USA, June 2011. Association for Computational Linguistics. URL https://aclanthology.org/W11-0329.
+
+Hang Zhang, Yeyun Gong, Yelong Shen, Jiancheng Lv, Nan Duan, and Weizhu Chen. Adversarial retriever-ranker for dense text retrieval. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id $=$ MR7XubKUFB.
+
+<table><tr><td></td><td>arg</td><td>touché</td><td>covid</td><td>nfc</td><td>hotpot</td><td>dbp</td><td>climate</td><td>fever</td><td>scifact</td><td>scidocs</td><td>fiqa</td><td>AVG.</td></tr><tr><td>GTR-base</td><td>51.1</td><td>20.5</td><td>53.9</td><td>30.8</td><td>53.5</td><td>34.7</td><td>(24.1*)</td><td>66.0</td><td>60.0</td><td>14.9</td><td>34.9</td><td>40.4</td></tr><tr><td>GTR-base with 8 examples</td><td>51.9</td><td>24.1</td><td>56.2</td><td>30.5</td><td>23.4</td><td>35.4</td><td>(25.4*）</td><td>68.2</td><td>60.1</td><td>14.8</td><td>35.3</td><td>38.7</td></tr></table>
+
+Table 4: Directly using eight examples might not improve the results. To showcase this, we fine-tuned a GTR-based model with the same few-shot examples used for tuning.
+
+# COMPUTE USAGE AND ENVIRONMENTAL IMPACT
+
+We used the 137B FLAN, which is based on LaMDA (Thoppilan et al., 2022). LaMDA was pretrained on a large corpus consisting of 1.56T words, costing 451 MWh energy and $2 5 . 2 \mathrm { t C O } 2 \mathrm { e }$ carbon footprint. In PROMPTAGATOR, we generated 29.23M queries $^ { * } 2$ prompts $= 5 8 . 4 6 \mathrm { M }$ queries, for a total of 610M words.
+
+# A FINE-TUNING GTR WITH FEW-SHOT EXAMPLES
+
+We study the effect of directly adding few-shot (no more than 8 examples) on top of a 110M dual encoder, GTR-base Ni et al. (2021). To maximum the utilities of the 8 examples, we associated each positive example with 32 random negative examples from the target corpus to construct a batch, and ran 50 steps of fine-tuning for each task. As expected, the 8 examples did not provide a large amount of impact for GTR base, where the average ndcg go from 40.4 to 38.7 after fine-tuning using few-shot examples.
+
+# B ANALYSIS ON PROMPTS
+
+Table 5 shows the list of prompt templates on different BEIR datasets. In order to further analysis the difference between zero-shot and few-shot prompts, we compare the few-shot and zero-shot generated queries given the same paragraph, randomly sampled from three datasets in Table 6. We observe that in general, the few-shot generated queries are closer to the original queries, while zero-shot queries are mostly questions. For example, in the ArguAna dataset, the few-shot queries are in general longer and more claim-like. In contrary, the zero-shot queries are most short question-like queries. Interestingly, for the HotpotQA dataset, even though both few-shot and zero-shot queries are generating questions-like queries, few-shot queries sometimes generate multi-hop questions, while zero-shot mostly generates single-hop questions. We further conduct first word distribution across different generation models for all datasets in Figure 4.
+
+Table 5: Prompt template for each dataset.   
+
+<table><tr><td>Dataset</td><td colspan="8">Prompt</td></tr><tr><td>ArguAna</td><td></td><td>0 Argument: passage X 1 Counter argument: query X</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FiQA</td><td></td><td>0 passage X 1 query X</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>HotpotQA</td><td></td><td>l0 Evidence: passage X 1 Vexed question: query X</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DBPedia-Entity</td><td></td><td>0 entity: passage X 1 query: query X</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>NFCorpus</td><td></td><td>l0 Article: passage X 1 Query: query X</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Toucheé-2020</td><td></td><td>|0 passage X 1 Debate: query X</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>TREC-Covid</td><td></td><td>l0 passage X 1 Question: query X</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SciFact</td><td></td><td>l0 passage X 1 Finding: query X</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SCIDOCS</td><td></td><td>l0 passage X 1 The passage is about query X</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FEVER</td><td></td><td>|0 passage X 1 Is it true that query X</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 6: Few-shot and zero-shot generated queries randomly sampled from ArguAna, FiQA and HotpotQA dataset.   
+
+<table><tr><td>Paragraph</td><td>Few-shot</td><td>Zero-shot</td><td>Analysis In ArguAna dataset,</td></tr><tr><td>ArguAna: Banning skin whiten- ing on such a basis also requires the acceptance of the racial over- tones.Some form of tan is popular in many societies of people of Eu- ropean ethnic origins- that is not a racial matter, it is more based on economic social perceptions (that of holidays to warmer climates). Ascribing a racial element to ev- erything to do with skin tone is at best a lazy analysis.Irrespective of issues of race and perceptions of ethnic origins,and its intersec- tion with beauty standards,some people will be given advantages in life due to their appearance.Ban- ninga certain form of cosmetic, even if it can have some racial and ethnic undertones,won&#x27;t change that.</td><td>Query 1:1 Banning skin whitening on such a ba- sis also requires the accep- tance of the racial overtones. However,the racial over- tones are simply a side ef- fect of the product being used incorrectly in order to attain a lighter skin tone than they are naturally born with． The product itself is not racist, but rather the peo- ple who use it. For exam- ple,a person of Asian de- scent may use a skin whiten- ing cream to attain a lighter skin tone so that they may fit into a more white-dominant society. The cream itself is not racist,but rather the fact that the person has used it to attain a lighter skin tone so that they may fit into a more</td><td>Query 1: What did they try to ban？ Query 2: Is skin whitening racist? Query 3:Banning skin whiten- ing on such a basis also re- quires the acceptance of the racial overtones.Some form of tan is popular in many so- cieties of people of European ethnic origins- that is not a racial matter, it is more based on economic social perceptions (that of holidays to warmer cli- mates). Query 4: What are some of the racial overtones in skin- whitening? Query 5: Are tans a sign of af- fluence?</td><td>the few-shot ex- amplesaremore statement-likeand are longer than zero-shot.</td></tr><tr><td>white-dominant society. Touché-2020:Psychologically, diet-wiseandacademically. Housewives can be both educated and uneducated but the fact that they spend more time with their children means that they put in a greater effort and since they are not preoccupied with their job(read:cellphone)； they_pay the close attention needed to keep their children out of trouble. Think of housewives as inspectors in your home...</td><td>Query 1: Housewives are better parents Query 2: Should house- wives be considered better parents?</td><td>Query 1: diet-wise and aca- demically. Query 2: People want to be relaxed all the time(At home AND at work) that is why all the research on work-life bal- ance. Query 3: has shown that home- makers are more relaxed, less stressed,healthier,happier - the list goes on.</td><td>The few-shot gener- atesargument-like queries that are more controversial, while zero-shot generate randomstatements that sometimes even have grammar problems.</td></tr><tr><td>HotpotQA: The Yung See San Fong House,or Young&#x27;s Home in the Heart of the Hills, is a house in Los Gatos in the U.S.state of California.It is a combination of Asian decorative motifs and pagoda roofs together with West- ern massing and layout.It was completed in 1917 forRuth Com- fort Mitchell Young,a writer,and her husband, Sanborn Young,a gentleman farmer,conservationist and later California State Senator.</td><td>Query 1: Which author was the husband of Ruth Com- fort Mitchell Young? Query 2: Which is known as the Yount-Lee House Query 3:Who is the archi- tect of the Yung See San Fong House? Query 4:The Yung See San Fong House,or Young&#x27;s Home in the Heart of the Hills;isa house in which U.S. state?</td><td>Query 1: Yung See San Fong House was completed in 1917 for Ruth Comfort Mitchell Young,a writer,and her hus- band, Sanborn Young. Query 2: How old is the Yung See San Fong House? Query 3: Yung See San Fong House was completed in 1917 for Ruth Comfort Mitchell Young,a writer,and her hus- band, Sanborn Young.</td><td>Few-shot examples sometimes will create multihop questions indicated in blue, whichrarelyhap- pensin zero-shot examples.</td></tr></table>
+
+# C DETAILED IMPLEMENTATION
+
+Figure 5 shows the overall process of PROMPTAGATOR $^ { + + }$ , the details of which are in Section 3.
+
+Our cross attention reranker is a listwise model based on T5. Specifically, it takes a list of documents given a query as the input. We represent each query-document pair as “Query: {q} Document: $\{ \mathrm { d } \} ^ { \prime \prime }$ and feed it into the encoder of a T5 model. We then apply a projection layer on the output encodings of the first token and use the output as the ranking score. We optimize the model using softmax cross entropy loss over over a ranking list consisting of a positive $( q , \bar { d } ^ { + } )$ pair and 31 sampled negative $( q , d ^ { - } )$ pairs. Unlike monoT5 (Nogueira et al., 2020), which is pointwise reranker that uses an encoder-decoder model and is trained to generate a relevance label, our model is a listwise reranker and is directly optimizes for ranking performance.
+
+# D QUERY GENERATION STATISTICS
+
+In Table 7, we analyze the length of the generated questions by different query generation systems. Note that NQ-QGen always generates short queries due to the query generation models being fine-tuned on the NQ dataset, and all of the generated questions have similar length to those questions of NQ. Interestingly, zero-shot PROMPTAGATOR already obtains more variance in terms of length compared to NQ-QGen. Finally, few-shot PROMPTAGATOR offers significantly more variance in terms of the length of generated queries.
+
+Table 7: Average query length.   
+
+<table><tr><td></td><td>Few-shot</td><td>Zero-shot</td><td>NQ QGen</td></tr><tr><td>ArguAna</td><td>98.2</td><td>26.0</td><td>9.7</td></tr><tr><td>Touche-2020</td><td>7.8</td><td>13.4</td><td>9.8</td></tr><tr><td>TREC-Covid</td><td>10.8</td><td>11.4</td><td>10.2</td></tr><tr><td>NFCorpus</td><td>8.3</td><td>11.5</td><td>10.3</td></tr><tr><td>HotpotQA</td><td>11.2</td><td>12.2</td><td>8.8</td></tr><tr><td>DBPedia-Entity</td><td>8.2</td><td>13.8</td><td>8.8</td></tr><tr><td>Fever</td><td>12.1</td><td>10.7</td><td>8.8</td></tr><tr><td>Climate-Fever</td><td>12.9</td><td>10.7</td><td>8.8</td></tr><tr><td>SciFact</td><td>12.6</td><td>12.4</td><td>10.0</td></tr><tr><td>SCIDOCS</td><td>7.4</td><td>15.7</td><td>10.7</td></tr><tr><td>FiQA-2018</td><td>12.5</td><td>10.1</td><td>9.5</td></tr><tr><td>AVG.</td><td>17.8</td><td>13.5</td><td>9.6</td></tr></table>
+
+# E ROUND-TRIP FILTERING EXAMPLES
+
+Figure 6 shows examples of queries from few-shot PROMPTAGATOR that were removed by round-trip filtering.
+
+# F ROUND-TRIP FILTERING AS LATENT VARIABLE MODELING
+
+This section aims to share some insight into why our round-trip filtering method is effective, by viewing queries as latent variables that we estimate. First, consider a hypothetical graphical model in which each query $q$ is a latent variable and the documents retrieved for that query are observed variables following some distribution, $p ( d | q , \theta ^ { * } )$ , where $\theta ^ { * }$ represents the parameters of a hypothetical “optimal” retriever that always selects the “best” documents for any query. For synthetic data generation, we make it our goal to sample queries from the posterior, $p ( q | d , \theta ^ { * } )$ , which according to Bayes rule is:
+
+$$
+p ( q | d , \theta ^ { * } ) = \frac { p ( d | q , \theta ^ { * } ) p ( q | \theta ^ { * } ) } { \sum _ { q ^ { \prime } } p ( d | q ^ { \prime } , \theta ^ { * } ) p ( q ^ { \prime } | \theta ^ { * } ) }
+$$
+
+where $p ( q | \theta ^ { * } )$ is a prior over queries. We will assume it is an “uninformative prior”that is uniform over all $q$ and later write it as just $p ( q )$ . If we knew $\theta ^ { * }$ (the parameters of an optimal retriever), we could just directly compute the above expression. But in practice we do not, so we will estimate $\theta$ using expectation maximization (EM) (Dempster et al., 1977), which will turn out to mirror our round-trip filtering algorithm. EM and other latent variable learning methods have long been used to impute missing data with great success.
+
+In EM, we estimate $\theta$ by approximately maximizing the marginal likelihood of the observed variables:
+
+$$
+\hat { \theta } = \arg \operatorname* { m a x } _ { \theta } \prod _ { d \in \mathcal { D } _ { T } } p ( d ) = \prod _ { d \in \mathcal { D } _ { T } } \sum _ { q } p ( d | q , \theta ) p ( q )
+$$
+
+The first step in EM is to make an initial estimate of $p ( q | d )$ for every document $d$ . We use our FLAN query generator as our initial estimate: $p _ { \mathrm { F L A N } } ( q | d )$ . We then proceed to the M-step of EM, which computes:
+
+$$
+\hat { \theta } = \arg \operatorname* { m a x } _ { \theta } \sum _ { d \in \mathcal { D } _ { T } } \sum _ { q } p _ { \mathrm { F L A N } } ( q | d ) p ( d | q , \theta )
+$$
+
+This is equivalent to training an initial retriever $p ( d | q , \hat { \theta } )$ on documents from $\mathcal { D } _ { T }$ that have been paired with our FLAN-generated queries (what we do). Finally, we proceed to the E-step of EM, which estimates $p ( q | d , \hat { \theta } )$ for every document $d$ :
+
+$$
+p ( q | d , \hat { \theta } ) \propto p ( d | q , \hat { \theta } ) p ( q )
+$$
+
+From this, we see that the highest probability queries under $p ( q | d , \hat { \theta } )$ are the ones with the highest probability of retrieving $d$ under our initial retriever (since $p ( q )$ is an uninformative prior that is uniform over all $q$ ).
+
+To sample from $p ( q | d , \hat { \theta } )$ , we could employ importance sampling. In importance sampling, we first sample $q$ from any proposal distribution that we choose, ${ \bar { p } } _ { \mathrm { p r o p } } { \bar { ( } } q | d )$ . We would then weight that sample by the ratio $p ( q | d , \hat { \theta } ) / p _ { \mathrm { p r o p } } ( q | d )$ . We choose $p _ { \mathrm { F L A N } } ( q | d )$ as our proposal distribution. Then, instead of actually importance-weighting each sample, our filtering procedure just discards samples with a low value of $p ( d | q , \hat { \theta } )$ , which is proportional to the numerator in the importance weight. This reveals both the differences and connections between our method and EM.
+
+Although we could repeat this EM-like procedure until convergence, we found that a single round of filtering yielded sufficient quality gains. It is also worth noting that the training of our final retriever could be viewed as another M-step.
+
+![](images/f2f805a5569b3d0696d3404b4ca626a70ebadd14470da677cc8df0ac5633e8f4.jpg)
+
+![](images/c557b60c369f7c8ad18b9f3ad02e06ce7bb6d6e40b24a596c8fcb9c6da9a6602.jpg)  
+Figure 4: Top first word distribution on queries generated from different models in all other BEIR datasets.
+
+![](images/9e9f4dad11d0e8ff0ddf2ffc474843fddec973536abf8c874edfde444b984b73.jpg)  
+Figure 5: PROMPTAGATOR $^ { + + }$ Training pipeline.
+
+Passage: As the COVID-19 pandemic sweeps the globe, evolving containment measures have created an unprecedented need for rapid and effective science communication that is able to engage the public in behavioural change on a mass scale. Public health bodies, governments, and media outlets have turned to comics in this time of need and found a natural and capable medium for responding to the challenge...
+
+Query: What is the authors’ purpose for writing this article?
+
+Remarks: The query lacks context. This query is also generic and can be used to all articles
+
+Passage: Big Bad Love is a 2001 film directed by Arliss Howard , who co-wrote the script with his brother , James Howard , based on a collection of short stories of the same name by Larry Brown . The story recounts an episode in the life of an alcoholic Vietnam veteran and struggling writer named Leon Barlow , who is played by Arliss Howard , and his wife , played by Howard ’s wife Debra Winger . The soundtrack includes music by Tom Verlaine , the Kronos Quartet , and R. L. Burnside .
+
+Query: music artist is from the United States
+
+Remarks: The query lacks context.
+
+Passage: Aminopeptidase N (APN) is the major cell surface receptor for group 1 coronaviruses. In this study, we have isolated and characterized a feline APN cDNA and shown that the transfection of human embryonic kidney cells with this cDNA renders them susceptible to infection with the feline coronavirus feline infectious peritonitis virus, the human coronavirus (HCV) 229E and the porcine coronavirus porcine transmissible gastroenteritis virus. ...
+
+Query: What is the function of Aminopeptidase N?
+
+Remarks: The query is too general and can match many documents that mentions APN.
+
+Passage: Selena Danielle Coppa (born February 25, 1983) was a military intelligence Sergeant in the United States Army. She is primarily notable for her organizing and activism against the US Occupation of Iraq while serving as an active duty military member, including serving on the Executive Board of Iraq Veterans Against the War. In 2009 it was announced that she was heading a committee responsible for gaining and training more active duty anti-war soldiers.
+
+Query: who was the first president of the senate?
+
+Remarks: Hallucination. The passage didn’t mention president of the senate.
+
+Passage: Porcine deltacoronavirus (PDCoV) is an emerging swine coronavirus that causes severe diarrhea, resulting in high mortality in neonatal piglets. Despite widespread outbreaks in many countries, no effective PDCoV vaccines are currently available. Here, we generated, for the first time, a full-length infectious cDNA clone of PDCoV. We further manipulated the infectious clone by replacing the NS6 gene with a green fluorescent protein (GFP) to generate rPDCoV-NS6-GFP...
+
+Query: What virus is the first one that we’ve been able to create a vaccine for?
+
+Remarks: Hallucination. The passage didn’t mention that Porcine deltacoronavirus is the first virus one can create a vaccine.
+
+Passage: Nacoleia rectistrialis is a moth in the Crambidae family. It was described by Hampson in 1912. It is found on the Key Islands.   
+Query: Who composed the music for the movie the ring?   
+Remarks: Hallucination. The passage is about moth and is unrelated to “music for the movie the ring”.
+
+Figure 6: Examples of queries from few-shot PROMPTAGATOR’s prompt-based query generation that were removed by round-trip filtering.

parse/dev/ksVGCOlOEba/ksVGCOlOEba_content_list.json

@@ -0,0 +1,1201 @@
+[
+    {
+        "type": "text",
+        "text": "Optimal Brain Compression: A Framework for Accurate Post-Training Quantization and Pruning ",
+        "text_level": 1,
+        "bbox": [
+            196,
+            122,
+            803,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Elias Frantar ∗ IST Austria elias.frantar@ist.ac.at ",
+        "bbox": [
+            186,
+            226,
+            383,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sidak Pal Singh ETH Zurich sidak.singh@inf.ethz.ch ",
+        "bbox": [
+            403,
+            226,
+            601,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dan Alistarh IST Austria & Neural Magic dan.alistarh@ist.ac.at ",
+        "bbox": [
+            620,
+            226,
+            812,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            304,
+            535,
+            320
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We consider the problem of model compression for deep neural networks (DNNs) in the challenging one-shot/post-training setting, in which we are given an accurate trained model, and must compress it without any retraining, based only on a small amount of calibration input data. This problem has become popular in view of the emerging software and hardware support for executing models compressed via pruning and/or quantization with speedup, and well-performing solutions have been proposed independently for both compression approaches. In this paper, we introduce a new compression framework which covers both weight pruning and quantization in a unified setting, is time- and space-efficient, and considerably improves upon the practical performance of existing post-training methods. At the technical level, our approach is based on an exact and efficient realization of the classical Optimal Brain Surgeon (OBS) framework of [LeCun, Denker, and Solla, 1990] extended to also cover weight quantization at the scale of modern DNNs. From the practical perspective, our experimental results show that it can improve significantly upon the compression-accuracy trade-offs of existing post-training methods, and that it can enable the accurate compound application of both pruning and quantization in a post-training setting. ",
+        "bbox": [
+            233,
+            338,
+            766,
+            574
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            604,
+            310,
+            622
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The impressive recent progress of deep learning for solving challenging tasks across several domains has been accompanied by a significant increase in parameter counts and computational costs for executing such models. A natural consequence has been a growing effort to reduce such costs via model compression, and the two most popular approaches for model compression are pruning— removing neural network weights by setting them to zero—and quantization, reducing the precision at which neural network weights and activations are stored and manipulated. Hundreds of such pruning and quantization approaches have been proposed and analyzed [17, 11], with the general goal of obtaining efficient variants of deep neural nets (DNNs) which would preserve accuracy while maximizing compression. Despite impressive progress, compression is still a laborious process: the pruning and quantization stages are often done independently, and recovering model accuracy after compression often requires partial or even full retraining of the compressed model. ",
+        "bbox": [
+            174,
+            638,
+            825,
+            790
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "An alternative but challenging scenario is the post-training compression setup [31, 24, 19, 25], in which we are given a trained but uncompressed model, together with a small amount of calibration data, and must produce an accurate compressed model in one shot, i.e. a single compression step, without retraining, and with limited computational costs. This is motivated by practical scenarios such as the MLPerf Inference Benchmark [37], and is the setting we focus on in this paper. ",
+        "bbox": [
+            174,
+            797,
+            825,
+            867
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Compression via weight pruning started with seminal work by LeCun et al. [23], complemented by Hassibi and Stork [13], who proposed a mathematical framework called the Optimal Brain Surgeon (OBS), for choosing the set of weights to remove from a trained neural network, by leveraging secondorder information. (We describe their approach in Section 3.) Recent advances, e.g. [6, 40, 39, 10] showed that OBS can lead to state-of-the-art compression at DNN scale, by introducing numerical methods which can approximate the second-order information needed by OBS at the massive parameter counts of modern models. However, these approaches do not apply to the post-training setting, as they require gradual pruning, as well as significant retraining, to recover good accuracy. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            203
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "An alternative approach, which is standard in the context of post-training compression, has been to break the compression task into layer-wise sub-problems, identifying a compressed weight approximation for each layer, given a sub-sample of the layer’s inputs and outputs based on calibration data. This line of work, e.g. [42, 31, 19], introduced elegant solvers for the resulting layer-wise weight quantization problem, which achieve state-of-the-art results for post-training quantization. Recently, AdaPrune [18] showed that this approach can also be effective for post-training weight pruning. ",
+        "bbox": [
+            174,
+            208,
+            825,
+            291
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this context, a natural question is whether existing approaches for pruning and quantization can be unified in order to cover both types of compression in the post-training setting, thus making DNN compression simpler and, hopefully, more accurate. This question is also of practical importance, since both GPU and CPU platforms now jointly support sparse and quantized formats [30, 35], and, as we illustrate experimentally, the resulting models could be executed with compound speedups. ",
+        "bbox": [
+            174,
+            297,
+            825,
+            367
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contribution. In this paper, we provide a mathematical framework for compression via pruning or quantization, which leads to state-of-the-art accuracy-versus-compression trade-offs in the challenging post-training compression setup. Our framework starts from the layer-wise compression problem described above, by which the global compression task, defined either for pruning or quantization, is first split into layer-wise sub-problems, based on the layer behavior on the calibration data. Specifically, given a layer $\\ell$ defined by weights $\\mathbf { W } _ { \\ell }$ , and layer inputs $\\mathbf { X } _ { \\ell }$ , we aim to find a compressed version of the weights $\\widehat { \\mathbf { W } } _ { \\ell }$ which minimizes the output difference relative to the uncompressed layer, measured via the squared error between the original and compressed layer, acting on the sample input $\\| { \\mathbf { W } } _ { \\ell } { \\mathbf { X } } _ { \\ell } - \\widehat { { \\mathbf { W } } } _ { \\ell } { \\mathbf { X } } _ { \\ell } \\| _ { 2 } ^ { 2 }$ , under a fixed compression constraint on $\\widehat { \\mathbf { W } } _ { \\ell }$ . ",
+        "bbox": [
+            174,
+            373,
+            826,
+            505
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Although solving this problem optimally for sparsity or quantization constraints is NP-hard [2, 31], it is a key step in all state-of-the-art post-training compression methods, both for pruning [18, 9] and for quantization [31, 18, 24]. Once this is solved per layer, a solution to the global problem can be obtained by combining layer-wise solutions, which is handy especially for non-uniform compression, e.g. [15, 9]. Thus, several approximations for this problem have been proposed [31, 19, 18]. ",
+        "bbox": [
+            174,
+            511,
+            825,
+            580
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We show that there still is significant room for improvement when solving the layer-wise compression problem. Roughly, our approach is to specialize the OBS framework to the squared error formulation above: in this case, the framework can in theory produce an exact greedy solution, but a direct implementation would have infeasible $\\Theta ( d ^ { 4 } )$ computational cost, where $d$ is the layer dimension. Our main technical contribution is a series of algorithms which reduce this computational cost, without any approximations, to $O ( d \\cdot d _ { c o l } ^ { 2 } )$ where $d _ { c o l }$ is the column dimension of the weight matrix. In practice, these improvements are significant enough to allow us to implement the exact OBS greedy solution, which prunes one weight at a time, and updates all remaining weights after each step, at the scale of modern DNNs with tens of millions of parameters, within reasonable time, on a single GPU. We provide efficient implementations of our methods at https://github.com/IST-DASLab/OBC. ",
+        "bbox": [
+            174,
+            585,
+            825,
+            724
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In turn, this algorithmic development allows us to apply the OBS approach to quantization. The resulting algorithm, called the Optimal Brain Quantizer (OBQ), quantizes weights iteratively one-ata-time, depending on their impact on the loss increase, after which it applies a closed-form update to the remaining unquantized weights, further reducing the loss. This solves the two problems efficiently, and in a unified manner—we call the unified framework the Optimal Brain Compressor (OBC). ",
+        "bbox": [
+            174,
+            731,
+            825,
+            800
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Experimental Results. We apply OBC to standard tasks and models covering image classification, object detection, and language modelling applications. We first show that our framework yields significantly better solutions for the layer-wise compression problem, which leads to higher-accuracy end-to-end compressed models for both pruning and quantization, relative to the corresponding stateof-the-art techniques, often by significant margins. Second, we show that our pruning and quantization approaches can be compounded, with surprisingly strong results: we obtain a $1 2 \\times$ reduction in theoretical operations with a $2 \\%$ accuracy drop for GPU-supported compound compression [30], and a $4 \\times$ speedup in actual runtime with only $1 \\%$ accuracy drop for a CPU-based sparsity-aware runtime [35]. Together, these results suggest for the first time that post-training compression can be competitive with full retraining. ",
+        "bbox": [
+            174,
+            806,
+            825,
+            904
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            823,
+            133
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            159,
+            321,
+            175
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Optimal Brain Surgeon (OBS). The classic OBS framework [23, 13] was originally applied to networks with hundreds of weights; more recently, methods such as WoodFisher [39] rendered the approach computationally feasible for DNNs by using a block-diagonal Fisher approximation of the Hessian, while follow-up methods introduced more efficient and general algorithms for handling the inverse Fisher matrix, or customize this approximation to specific model families [21]. Earlier work called Layer-wise OBS (L-OBS) [6] was inspired by the K-FAC approximation [29, 12]: L-OBS approximates the OBS framework not for the global objective, but for a quadratic per-layer loss, while also pruning all weights based on a single Hessian computation. At a high level, our approach is similar, in that we apply OBS layer-wise; however, we apply OBS exactly, that is, pruning one weight at a time, and exactly recomputing the Hessian after every pruning step. This is made computationally tractable by several new algorithmic ideas, and yields significantly improved results relative to L-OBS. This prior work on pruning considered settings with extensive finetuning. By contrast, we will focus on the post-training setting, where only a small amount of calibration data is available. ",
+        "bbox": [
+            174,
+            194,
+            825,
+            375
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Post-Training Quantization. This setting has been primarily considered for quantization, and most state-of-the-art methods work by performing layer-wise compression. Specifically, BitSplit [6] optimizes the quantized weights bit by bit, while AdaRound [31] finds a weight rounding policy through gradient based optimization with an annealed penalty term that encourages weights to move towards points on the quantization grid. AdaQuant [19] relaxes the AdaRound constraint, allowing weights to change during quantization-aware optimization, via straight-through estimation [33]. BRECQ [24] suggested that accuracy can be improved further by integrating second-order information into the layer-wise losses and by jointly optimizing hand-crafted blocks of related layers. ",
+        "bbox": [
+            174,
+            380,
+            825,
+            491
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A key step of AdaRound, AdaQuant and BRECQ is to quantize layers incrementally, in sequential order, so that errors accumulated in earlier layers can be compensated by weight adjustments in later ones. This significantly improves performance, but reduces flexibility, as the entire process may need to be re-done whenever we wish to change compression parameters of one layer. We instead target independent compression of each layer, allowing the end model to be simply “stitched” together from layer-wise results. Despite operating independently on each layer, we find that, after correcting basic statistics such as batchnorm, our method performs on par to sequential ones for uniform quantization. ",
+        "bbox": [
+            174,
+            497,
+            825,
+            594
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Post-Training Sparsification. The layer-wise approach was shown to also be effective for posttraining pruning by AdaPrune [18], which pruned weights to the GPU-supported N:M pattern [45]. AdaPrune first drops parameters according to their magnitude [46] and then reoptimizes the remaining weights to reconstruct the pre-compression calibration set output. This is similar to [16, 8] which also perform layer-wise reoptimization of the remaining weights. Follow-up work [10] noted that the results of AdaPrune can be improved further by performing more frequent pruning/optimization steps. Our algorithm pushes this idea to the limit, performing full reoptimization after every single pruned weight, while remaining computationally tractable. We further use a more sophisticated weight selection metric which incorporates second-order information. Finally, [10] also introduces global AdaPrune, a more expensive global optimization step applied on top of the layer-wise AdaPrune results, which can bring additional accuracy gains. This can also be applied to our pruned models. ",
+        "bbox": [
+            173,
+            599,
+            825,
+            752
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Non-Uniform Compression. An orthogonal practical question is how to compress different layers to maximize accuracy under a given resource constraint, such as latency or energy consumption. Existing methods can be roughly categorized into search-based and solver-based approaches. The former, e.g. AMC [15] or HAQ [41], search for a layer-wise compression policy directly via, for example, reinforcement learning or genetic programming [43], whereas the latter, e.g. HAWQv3 [44] or AdaQuant [19], construct a relaxed version of the overall problem that is then solved exactly. We focus here on solver-based approaches, as they can rapidly adapt to different scenarios when combined with accurate independent layer-wise compression schemes; however, our techniques could be of interest for search-based methods as well. Concretely, we use the problem formulation of AdaQuant [19] to which we apply the DP algorithm of SPDY [10] to achieve fast solving times even with a large number of possible choices per layer. ",
+        "bbox": [
+            174,
+            758,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Problem Definition and Background ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            88,
+            509,
+            107
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Layerwise Compression Problem. Following prior work on post-training compression, e.g. [31, 19], we define the problem as follows. Mathematically, we model a layer $\\ell$ as a function $f _ { \\ell } ( X _ { \\ell } , W _ { \\ell } )$ acting on inputs $X _ { \\ell }$ , parametrized by weights $W _ { \\ell }$ . The goal of layer-wise compression is to find a “compressed” version of $W _ { \\ell }$ that performs as similarly as possible to the original weights. More formally, the compressed weights $\\widehat { W } _ { \\ell }$ should minimize the expected layer output change as measured by some loss $\\mathcal { L }$ while at the same time satisfying a generic compression constraint, which we denote by ${ \\mathcal { C } } ( \\widehat { W } _ { \\ell } ) > C$ , which will be customized depending on the compression type: ",
+        "bbox": [
+            173,
+            119,
+            825,
+            224
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b5ec35afc69c7d5eb05a923e8c298d22d8dbc3999ed11be11fd0d795b08a27f6.jpg",
+        "text": "$$\n\\mathrm { a r g m i n } _ { \\widehat { W } _ { \\ell } } \\quad \\mathbb { E } _ { X _ { \\ell } } \\mathcal { L } \\big ( f _ { \\ell } ( X _ { \\ell } , W _ { \\ell } ) , f _ { \\ell } ( X _ { \\ell } , \\widehat { W } _ { \\ell } ) \\big ) \\quad \\mathrm { s u b j e c t ~ t o } \\quad \\mathcal { C } ( \\widehat { W } _ { \\ell } ) > C .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            228,
+            736,
+            251
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The expectation over the layer inputs $X _ { \\ell }$ is usually approximated by taking the mean over a small set of $N$ input samples. This low-data setting is one of the primary applications of layer-wise compression. Further, most works [42, 31, 19] focus on compressing linear and convolutional layers, which can be unfolded into linear ones, as these are prevalent in practice, and use the squared loss to measure the approximation error. This definition of the loss can be motivated, via a sequence of approximations, from second-order information: please see [31] for a precise derivation. Furthermore, this approximation approach has been shown to work well in many applications [31, 19, 9]. ",
+        "bbox": [
+            173,
+            256,
+            826,
+            354
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We follow these conventions as well, and work with the specific layer-wise compression problem stated formally below, where the weights $\\mathbf { W } _ { \\ell }$ are a $d _ { \\mathrm { r o w } } \\times d _ { \\mathrm { c o l } }$ matrix (for conv-layers $d _ { \\mathrm { c o l } }$ corresponds to the total number of weights in a single filter), and the input $\\mathbf { X } _ { \\ell }$ has dimensions $d _ { \\mathrm { c o l } } \\times N$ . ",
+        "bbox": [
+            174,
+            361,
+            825,
+            404
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/77d00f817a312063833e0fcae5c63b78c6d18fee24e5c00da45ff20cecaa7a70.jpg",
+        "text": "$$\n\\operatorname { a r g m i n } _ { \\widehat { \\mathbf { W } } _ { \\ell } } \\quad | | \\mathbf { W } _ { \\ell } \\mathbf { X } _ { \\ell } - \\widehat { \\mathbf { W } } _ { \\ell } \\mathbf { X } _ { \\ell } | | _ { 2 } ^ { 2 } \\quad \\mathrm { s . t . } \\quad \\mathcal { C } ( \\widehat { \\mathbf { W } } _ { \\ell } ) > C .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            407,
+            679,
+            431
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Optimal Brain Surgeon (OBS) Framework. The OBS framework [23, 13] considers the problem of accurately pruning a trained dense neural network. It starts from the Taylor approximation at the given point (assumed to have negligible gradient), and provides explicit formulas for the optimal single weight to remove, as well as the optimal update of the remaining weights which would compensate for the removal. More precisely, let $\\mathbf { H }$ denote the Hessian matrix of the loss at the given (dense) model. Then the weight to prune $w _ { p }$ which incurs the minimal increase in loss and the corresponding update of the remaining weights $\\delta _ { p }$ can be calculated as follows: ",
+        "bbox": [
+            173,
+            443,
+            825,
+            541
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4872538c6eae395f33fb403b94cf06785c300565b8715bebcb1f9a3355baf028.jpg",
+        "text": "$$\nw _ { p } = \\operatorname * { a r g m i n } _ { w _ { p } } \\frac { w _ { p } ^ { 2 } } { [ \\mathbf { H } ^ { - 1 } ] _ { p p } } , \\quad \\delta _ { p } = - \\frac { w _ { p } } { [ \\mathbf { H } ^ { - 1 } ] _ { p p } } \\cdot \\mathbf { H } _ { : , p } ^ { - 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            547,
+            673,
+            584
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $[ \\mathbf { H } ^ { - 1 } ] _ { p p }$ denotes the $p$ th diagonal entry of the inverse Hessian, and $\\mathbf { H } _ { : , p } ^ { - 1 }$ is its $p$ th column. ",
+        "bbox": [
+            168,
+            590,
+            803,
+            607
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "OBS for Layer-Wise Pruning. We will now instantiate this framework for the layer-wise pruning problem, defined above. First, the loss in equation (2) is quadratic and since our starting point is given by the dense weights achieving the minimal loss of 0, the assumptions of the OBS framework are fully met, meaning that its formulas are exact for this specific problem formulation. Thus, iterating the OBS framework to remove one weight at a time would yield an exact greedy solution for the layer-wise pruning problem, as it takes the (locally) optimal decision at each step. While this greedy approach does not guarantee convergence to a global optimum, such approaches can be very effective for dealing with problem instances that are too large to be handled by exact methods. ",
+        "bbox": [
+            173,
+            611,
+            825,
+            723
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 An Optimal Greedy Solver for Sparsity ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            741,
+            532,
+            758
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The obvious challenge is that applying the OBS framework in its true form, i.e. pruning a single weight at a time using the exact formulas in (3), is computationally very demanding. The Hessian $\\mathbf { H }$ is a $d { \\times } d$ matrix where $d = d _ { \\mathrm { r o w } } \\cdot d _ { \\mathrm { c o l } }$ , which is already expensive to store and compute with. Additionally, this matrix needs to be updated and inverted at each of the $O ( d )$ steps with a computational complexity of $\\Theta ( d ^ { 3 } )$ . Clearly, an $O ( d ^ { 4 } )$ total runtime is too inefficient for pruning most layers of modern neural networks, as $d$ is usually $\\geq 1 0 ^ { 5 }$ or even $\\geq 1 0 ^ { 6 }$ for several layers. However, as we will now show, it is actually possible to reduce the overall costs of this process to $O ( d _ { \\mathrm { r o w } } \\cdot d _ { \\mathrm { c o l } } ^ { 3 } )$ time and $\\Theta ( d _ { \\mathrm { c o l } } ^ { 2 } )$ memory, making it efficient enough to prune e.g. all layers of a medium-sized model such as ResNet50 in a bit more than one hour on a single NVIDIA RTX 3090 GPU. We emphasize that the techniques we introduce are exact; unlike prior work [6, 39], we do not rely on any approximations. ",
+        "bbox": [
+            173,
+            772,
+            825,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The ExactOBS Algorithm. In the following, we introduce our efficient instantiation of the OBS framework, for the layer-wise compression problem, which we call ExactOBS, in step-by-step fashion. We start by rewriting the matrix squared error in (2) as the sum of the squared errors for each row in the weight matrix. As we are always dealing with a fixed layer $\\ell .$ , we drop the subscript $\\ell$ to simplify notation. The objective is then equivalent to $\\begin{array} { r } { \\sum _ { i = 1 } ^ { d _ { \\mathrm { r o w } } } | | \\mathbf { W } _ { \\mathbf { i } , : } \\mathbf { \\widehat X } - \\widehat { \\mathbf { W } } _ { \\mathbf { i } , : } \\mathbf { X } | | _ { 2 } ^ { 2 } } \\end{array}$ . ",
+        "bbox": [
+            173,
+            90,
+            825,
+            165
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This way of writing the error makes it clear that removing a single weight $[ \\mathbf { W } ] _ { i j }$ only affects the error of the corresponding output row $\\mathbf { Y _ { i , : } } = \\mathbf { W _ { i , : } } \\mathbf { X }$ . Hence, there is no Hessian interaction between different rows and so it suffices to work only with the individual $d _ { \\mathrm { c o l } } \\times d _ { \\mathrm { c o l } }$ Hessians corresponding to each of the $d _ { \\mathrm { r o w } }$ rows. Further, as the dense layer output $\\mathbf { Y } = \\mathbf { W } \\mathbf { X }$ is fixed, the objective for each row has standard least squares form and its Hessian is given by $\\mathbf { H } = \\mathbf { 2 } \\mathbf { X } \\mathbf { X } ^ { \\top }$ . ",
+        "bbox": [
+            173,
+            170,
+            825,
+            239
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Although this observation already reduces computational complexity, two key challenges remain: (a) applying OBS to each row still costs $O ( d _ { \\mathrm { c o l } } \\cdot \\bar { d } _ { \\mathrm { c o l } } ^ { 3 } )$ time, which is too slow for large layers, and (b) we need fast access to the Hessian inverses of all $d _ { \\mathrm { r o w } }$ rows, since we want to prune the minimum score weight of the whole matrix rather than just per row in each step. In particular, (b) requires $O ( d _ { \\mathrm { r o w } } \\cdot \\bar { d _ { \\mathrm { c o l } } ^ { 2 } } )$ GPU memory, which is likely to be infeasible. ",
+        "bbox": [
+            174,
+            244,
+            825,
+            315
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Step 1: Handling a Single Row. We first describe how to efficiently prune weights from a single row with $d _ { \\mathrm { c o l } }$ parameters. For simplicity, we denote such a row by w with corresponding Hessian $\\mathbf { H }$ . The full algorithm for this procedure is given in Algorithm 1; in the following, we provide a detailed description. The key idea is to avoid having to do the full $\\Theta ( N \\cdot d _ { \\mathrm { c o l } } ^ { 2 } )$ calculation and $\\Theta ( d _ { \\mathrm { c o l } } ^ { 3 } )$ inversion of $\\mathbf { H }$ in each step. The former is easy, as the weights themselves do not enter the calculation of $\\mathbf { H } = 2 \\mathbf { X } \\mathbf { X } ^ { \\top }$ , and the Hessian for the weights with pruning mask $M$ denoted by ${ \\bf { H } } _ { M }$ is thus simply comprised of the corresponding rows and columns in the fully dense version $\\mathbf { H }$ . Hence, we only have to compute $\\mathbf { H }$ (which is actually the same for all rows) once, from which we can then extract the rows and columns corresponding to $M$ as needed. ",
+        "bbox": [
+            173,
+            320,
+            825,
+            446
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Critically, this trick is not applicable to the inverse, as $( \\mathbf { H } _ { M } ) ^ { - 1 } \\neq ( \\mathbf { H } ^ { - 1 } ) _ { M }$ . However, using the fact that the removal of one parameter $p$ simply drops the corresponding row and column from $\\mathbf { H }$ , we can actually update the inverse to remove parameter $p$ directly using a single step of Gaussian elimination, with cost $\\Theta ( d _ { c o l } ^ { 2 } )$ . The following result, whose proof is in the Appendix, formalizes this. ",
+        "bbox": [
+            173,
+            452,
+            826,
+            508
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1 (Row & Column Removal). Given an invertible $d _ { c o l } \\times d _ { c o l }$ matrix $\\mathbf { H }$ and its inverse ${ \\bf H } ^ { - 1 }$ , we want to efficiently compute the inverse of $\\mathbf { H }$ with row and column $p$ removed, which we denote by $\\mathbf { H } _ { - p }$ . This can be accomplished through the following formula: ",
+        "bbox": [
+            173,
+            512,
+            825,
+            555
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/00e11a0cb14a37ae0072ea6b62c8b09e13c71149cbe6a0faf513389250635f2d.jpg",
+        "text": "$$\n{ \\bf H } _ { - p } ^ { - 1 } = \\Big ( { \\bf H } ^ { - 1 } - \\frac { 1 } { [ { \\bf H } ^ { - 1 } ] _ { p p } } { \\bf H } _ { : , p } ^ { - 1 } { \\bf H } _ { p , : } ^ { - 1 } \\Big ) _ { - p } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            560,
+            635,
+            593
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which corresponds to performing Gaussian elimination of row and column $p$ in ${ \\bf H } ^ { - 1 }$ followed by dropping them completely. This has $\\Theta ( d _ { c o l } ^ { 2 } )$ time complexity. ",
+        "bbox": [
+            173,
+            598,
+            825,
+            628
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The resulting pseudocode is shown in Algorithm 1, where we avoid constantly resizing ${ \\bf H } ^ { - 1 }$ (and correspondingly changing indices) by utilizing the fact that row and column $p$ have no effect on any future calculations after they have been eliminated by Lemma 1 as they are 0 (and the non-zero diagonal element is never accessed again). One can check that this algorithm applies OBS to a single row of W with a per-step cost of $\\Theta ( \\bar { d } _ { \\mathrm { c o l } } ^ { 2 } )$ , and thus $\\Theta ( k \\cdot d _ { \\mathrm { c o l } } ^ { 2 } )$ overall time for pruning $k$ weights. ",
+        "bbox": [
+            173,
+            636,
+            825,
+            708
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/76d0a9fd79490f2e170b0c7f5a206e398d70df3802bd8ec03201cb54faf783ee.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Algorithm1Prune k ≤ dcol weights from row w with inverse Hessian H-1 = (2XXT)-1 according to OBS in O(k · d²ol) time.</td><td rowspan=\"5\"></td></tr><tr><td>M = {1,...,dco1} fori=1,...,k do</td><td></td></tr><tr><td>p ← argminp∈M 1 [H-1]</td><td></td></tr><tr><td>pp w←w-H-1 1 .Wp</td><td></td></tr><tr><td>：，p [H-1] pp H-1←H-1 1 H-11</td><td>H-1 p，</td></tr></table>",
+        "bbox": [
+            173,
+            729,
+            485,
+            893
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/420513c26a2b97b398b281b3e7ef3eb773ec0bd92c2b0cc71eac615d9769ed42.jpg",
+        "image_caption": [
+            "Figure 1: Efficient global OBS using the rowwise results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            516,
+            728,
+            821,
+            864
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Step 2: Jointly Considering All Rows. Applying the OBS framework to the full weight matrix W rather than just to each row independently requires fast access to all $d _ { \\mathrm { r o w } }$ row-wise inverse Hessians, in order to select the weight with the smallest overall pruning score in each step. However, storing $d _ { \\mathrm { r o w } }$ matrices of size $d _ { \\mathrm { c o l } } \\times d _ { \\mathrm { c o l } }$ each in GPU memory can be too expensive; while it would be possible to offload some Hessians to main memory, this could result in a large number of expensive memory transfers. However, since there is no Hessian interaction between rows, the final compressed weights of each row only depend on the total number of parameters that were pruned in it. Similarly, the change in loss incurred by pruning some weight only depends on the previously pruned weights in the same row, which also means that the order in which weights are pruned in each row is fixed. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            215
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The consequence of these insights is that we can process each row independently, pruning all weights in order while always recording the corresponding change in loss $\\delta \\bar { \\mathcal { L } _ { p } } = w _ { p } ^ { 2 } \\big / [ \\bar { \\mathbf { H } } ^ { - 1 } ] _ { p p }$ . At the end, we know $\\delta { \\mathcal { L } } _ { p }$ for all $d$ weights and can then simply determine the global mask that would be chosen by OBS on the full matrix by selecting the weights with the lowest values in order, requiring only $\\Theta ( d )$ extra memory. We note that once the per-row masks $M _ { i }$ are known, we can directly solve for the optimal update of the remaining weights via the corresponding group OBS formula [21] $\\delta _ { M _ { i } } = \\bar { \\mathbf { H } } _ { : , M _ { i } } ^ { - 1 } ( ( \\bar { \\mathbf { H } } ^ { - 1 } ) _ { M _ { i } } ) ^ { - 1 } \\mathbf { w } _ { M _ { i } }$ . This will be considerably faster in practice than simply rerunning the iterative pruning process in Algorithm 1. Alternatively, if enough CPU memory is available, one can keep the full pruning trace of each row, that is, the full weight vector after every individual pruning step, in CPU memory and ultimately simply reload the entries corresponding to the global mask. This requires $O ( d _ { \\mathrm { r o w } } \\cdot d _ { \\mathrm { c o l } } ^ { 2 } )$ extra CPU memory but avoids a second computation pass to reconstruct the not pruned weights and will therefore be faster. Figure 1 visualizes both options just discussed. ",
+        "bbox": [
+            173,
+            222,
+            826,
+            404
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Implementation Details. In practice, the matrix $\\mathbf { H }$ might not always be invertible for reasons such as using too few data samples or dead / linearly dependent inputs. The former can usually be addressed by extending the calibration dataset with augmentations (additional augmented samples only need to be accumulated into the Hessian once and are thus very cheap to include) and the latter can be prevented by adding a small diagonal dampening term to the Hessian before inverting it. Second, a direct GPU implementation of Algorithm 1 will perform a large number of small CUDA calls, which can be expensive. This overhead can be removed by using batch operations to process multiple matrix rows simultaneously—for more details please see our sample implementation. Finally, when applied to an already sparse weight matrix, the complexity of our algorithm can scale cubicly with the row-density by working with a dense version of the weights / Hessians consisting only of the non-zero elements and mapping the pruning result back at the end. ",
+        "bbox": [
+            173,
+            410,
+            825,
+            563
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "N:M Sparsity. Our method can be easily extended to various forms of semi-structured sparsity. This includes, for example, the N:M sparsity pattern [45], which enforces exactly $N$ non-zero values in each block of $M$ consecutive weights, and is becoming popular due to support on newer NVIDIA hardware [30]. Adapting our algorithm to this pattern requires only one simple change: instead of selecting the weight with the smallest change in loss, we select the weight with the smallest change in loss that is in a block with $< N$ pruned weights. We note that all rows have exactly the same sparsity $1 - N / M$ in the N:M pattern and so we can terminate per-row pruning as soon as this target sparsity value is reached. For the same reason, there is no need for the global mask selection step described earlier. Thus, our method will be even more efficient in this case. ",
+        "bbox": [
+            173,
+            569,
+            825,
+            694
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Block-Sparsity. Another practically relevant pruning pattern, particularly in the context of CPU acceleration [7, 22], is block-pruning, where zeros appear only in consecutive blocks of size $c$ , which is typically a small number like 4 or 8. We follow recent work [21] that extends the OBS framework to pruning small groups of connected weights in order to account for the correlation between them, using the following formulas for the target block and weight update, respectively: ",
+        "bbox": [
+            173,
+            700,
+            825,
+            770
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4507ef5b02463e9006aa063bfa9f9fac61e3c18601aa2f12c7443c7e3cfe825f.jpg",
+        "text": "$$\n\\begin{array} { r } { { \\bf w } _ { P } = \\operatorname * { a r g m i n } _ { { \\bf w } _ { P } } { \\bf w } _ { P } ^ { \\top } ( ( { \\bf H } ^ { - 1 } ) _ { P } ) ^ { - 1 } { \\bf w } _ { P } , \\quad \\delta _ { P } = - { \\bf H } _ { : , P } ^ { - 1 } ( ( { \\bf H } ^ { - 1 } ) _ { P } ) ^ { - 1 } { \\bf w } _ { P } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            787,
+            738,
+            809
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $P$ denotes the set of indices corresponding to one block. Algorithm 1 can easily be adapted to operate on blocks using the above equations and applying the update of ${ \\bf H } ^ { - 1 }$ via Lemma 1 successively for all $p \\in P$ . Although there are now only $d _ { c o l } / c$ steps per row, each update of ${ \\bf H } ^ { - 1 }$ also takes ${ \\dot { O } } ( c \\cdot d _ { \\mathrm { c o l } } ^ { 2 } )$ time and so the overall asymptotic runtime stays the same. Additional practical overhead only comes from the extra $O ( c ^ { 2 } \\cdot d _ { \\mathrm { c o l } } ^ { 2 } )$ terms that are the result of computing and multiplying with the $c \\times c$ matrices $\\big ( ( \\mathbf { H } ^ { - 1 } \\big ) _ { P } \\big ) ^ { - 1 }$ . ",
+        "bbox": [
+            174,
+            827,
+            825,
+            912
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 The Optimal Brain Quantizer (OBQ) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            88,
+            516,
+            107
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Although the classical OBS framework [23, 13] has inspired a long line of work on pruning methods for DNNs [39, 10, 27], so far it has not been used for quantization. We now show that our results from the previous section can in fact be extended to quantization in an effective and accurate way, via a method which we call the Optimal Brain Quantizer (OBQ), in the spirit of [23, 13]. ",
+        "bbox": [
+            173,
+            119,
+            825,
+            176
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The Quantization Order and Update Derivations. Under the standard assumption that the gradient at the current point w is negligible, the OBS formulas for the optimal weight to be pruned $w _ { p }$ and the corresponding update $\\delta _ { p }$ can be derived by writing the locally quadratic problem under the constraint that element $p$ of $\\delta _ { p }$ is equal to $- w _ { p }$ , which means that $w _ { p }$ is zero after applying the update to w. This problem has the following Lagrangian: ",
+        "bbox": [
+            173,
+            181,
+            825,
+            252
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b1ddf2b04acbaddba1cc4ab66797416b202efc71b03d2439d52b95c2a15b140d.jpg",
+        "text": "$$\n\\begin{array} { r } { L ( \\delta _ { p } , \\lambda ) = \\delta _ { \\mathbf { p } } ^ { \\top } \\mathbf H \\delta _ { p } + \\lambda ( \\mathbf e _ { \\mathbf { p } } ^ { \\top } \\delta _ { p } - ( - w _ { p } ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            256,
+            638,
+            276
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { H }$ denotes the Hessian at w and $\\mathbf { e _ { p } }$ is the pth canonical basis vector. The optimal solution is then derived by first finding the optimal solution to $\\delta _ { p }$ via setting the derivative $\\partial L / \\partial \\delta _ { p }$ to zero and then substituting this solution back into $L$ and solving for $\\lambda$ ; please see e.g. [13, 39] for examples. ",
+        "bbox": [
+            173,
+            279,
+            825,
+            321
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assume a setting in which we are looking to quantize the weights in a layer on a fixed grid of width $\\Delta$ while minimizing the loss. To map OBS to a quantized projection, we can set the target of the Lagrangian constraint in (6) to ( ${ \\mathfrak { q u a n t } } ( w _ { p } ) - w _ { p } )$ , where quant $( w _ { p } )$ is the weight rounding given by quantization; then $w _ { p } = { \\tt q u a n t } ( w _ { p } )$ after the update. ",
+        "bbox": [
+            173,
+            327,
+            825,
+            383
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assuming we wish to quantize weights iteratively, one-at-a-time, we can derive formulas for the “optimal” weight to quantize at a step, in terms of minimizing the loss increase, and for the corresponding optimal update to the unquantized weights, in similar fashion as discussed above: ",
+        "bbox": [
+            174,
+            388,
+            825,
+            431
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/05d05fbee8acd57b2de51180f84cce0c0741b60b646c5bf61b3056501552dbc0.jpg",
+        "text": "$$\n\\begin{array} { r } { w _ { p } = \\operatorname * { a r g m i n } _ { w _ { p } } \\frac { ( \\mathrm { q u a n t } ( w _ { p } ) - w _ { p } ) ^ { 2 } } { [ \\mathbf { H } ^ { - 1 } ] _ { p p } } , \\quad \\delta _ { p } = - \\frac { w _ { p } - \\mathrm { q u a n t } ( w _ { p } ) } { [ \\mathbf { H } ^ { - 1 } ] _ { p p } } \\cdot \\mathbf { H } _ { : , p } ^ { - 1 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            435,
+            736,
+            470
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In fact, since $- w _ { p }$ is a constant during all derivations, we can just substitute it with $( \\mathrm { q u a n t } ( w _ { p } ) - w _ { p } )$ in the final result. We note that the resulting formulas are a generalization of standard OBS for pruning, if quant $( \\cdot )$ always “quantizes” a weight to 0, then we recover the original form. ",
+        "bbox": [
+            174,
+            474,
+            826,
+            517
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Quantizing Full Layers. At first glance, OBQ might appear curious since one usually quantizes all weights in a layer, leaving no more weights to update. At the same time, the weight selection metric influences only the quantization order, but not the quantization value. However, this view changes when considering OBQ in the context of our efficient one-weight-at-a-time pruning algorithm described in the previous section. Specifically, using OBQ, we can greedily quantize the currently “easiest” weight by the above metric, and then adjust all the remaining unquantized weights to compensate for this loss of precision, thus changing their value. We then choose the next weight to quantize, and so on. This can result in quantization assignments that are different from the ones that would have been chosen by rounding initially, and in better overall quantization results. Concretely, to realize this, we can plug (7) into Algorithm 1 in order to iteratively quantize weights for a given layer, leading to the similar Algorithm in the Appendix, thus essentially unifying pruning and quantization. ",
+        "bbox": [
+            173,
+            522,
+            825,
+            676
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Quantization Outliers. One practical issue with this greedy scheme can occur especially when applied to quantization grids that permit some outliers in order to achieve a lower error on the majority of weights, which are currently standard [4, 34]. Since these outliers can have high quantization error, they will usually be quantized last, when there are only few other unquantized weights available that may be adjusted to compensate for the large error incurred by quantizing the outliers. This effect can become worse when some weights are pushed even further outside the grid by intermediate updates. We prevent this with a simple but effective heuristic: we quantize outliers, e.g. weights with a quantization error $> \\Delta / 2$ where $\\Delta$ is the distance between quantized values, as soon as they appear (which typically happens only a few times per layer). With this heuristic, OBQ yields a highly effective layer-wise quantization scheme, as our experiments in the next section demonstrate. Finally, we note that the OBQ version of the techniques discussed in Section 4 has all the same runtime and memory characteristics (barring the global step in Figure 1, which is unnecessary for quantization). ",
+        "bbox": [
+            173,
+            681,
+            825,
+            848
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            866,
+            312,
+            883
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Objectives, Models & Datasets. To demonstrate the effectiveness and flexibility of our method, we consider several different standard post-training compression scenarios [31, 19, 18]. We begin with settings where only a single type of compression is applied: concretely, we consider unstructured pruning for given FLOP targets, global 2:4 and 4:8 pruning, as well as uniform weight quantization. Additionally, we also study two practical tasks that feature joint pruning and quantization: a GPU scenario where quantization and N:M pruning are combined, as well as a CPU scenario combining quantization and block pruning. We work with variants of the following models and tasks: ResNet [14] for image classification on Imagenet [38], YOLOv5 [20] for object detection on COCO [26] and BERT [5] for question answering on SQuAD [36]. Our smaller BERT models denoted by BERT3 and BERT6 correspond to the smaller 3 and 6 layer variants of BERT-base, respectively, trained by [21]. The Appendix contains additional experiments as well as runtime information of our algorithms. ",
+        "bbox": [
+            174,
+            897,
+            821,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            825,
+            229
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Experimental Setup. All of our calibration datasets consist of 1024 random training samples. For ImageNet, where we use roughly $0 . 1 \\%$ of the training data, we additionally apply standard flipping and cropping augmentations to artificially increase the size of this dataset by $1 0 \\times$ ; other tasks do not use any augmentations. While the effect of augmentations is typically minor, they are very cheap to include for our method. For ResNet models, batchnorm statistics are reset using 100 batches of 128 samples from the calibration set with standard augmentations. For other models, we apply mean and variance correction [32, 1] after all normalization layers (so that the correction parameters can be easily merged and incur no extra cost) on a single batch of samples of size 128 (for YOLO) and 512 (for BERT). We found this to be more effective than batchnorm tuning for YOLO, and the BERT models have no batchnorm layers. ",
+        "bbox": [
+            174,
+            236,
+            825,
+            375
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "When compressing to a given FLOP or timing constraint, we need to solve the problem of identifying per-layer compression targets, which match the constraint, while maximizing accuracy. To identify these non-uniform targets, we follow the approach of [10]: we first collect a “model database” containing for each compression level (e.g. bit-width or sparsity setting) the corresponding (independently) compressed version of each layer. For building a joint sparse and quantized database we simply sparsify layers first and then apply quantization to the remaining weights. Next, similarly to [19], we compute the layer-wise calibration losses (without augmentations) for all compression levels, corresponding to the models with exactly one layer compressed to a certain level. Then, given layer-wise FLOP or timing information, we set up a constrained layer-wise compression problem of the form described in AdaQuant [19] and solve it with the dynamic programming algorithm of SPDY [10]. This returns an optimal per-layer assignment of compression levels, for which we can then easily produce the corresponding model, via a two-step process: we first stitch together layers at the corresponding compression levels from the database, and then perform the discussed statistics correction to recover some extra accuracy [19]. ",
+        "bbox": [
+            174,
+            381,
+            825,
+            574
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Unstructured Sparsity. We begin our experiments with unstructured sparsity, comparing against global magnitude pruning (GMP) [46], the approximate layer-wise OBS method L-OBS [6], and the post-training pruning state-of-the-art method AdaPrune [18]. As a sanity check, we examine in Figure 2 whether our method provides better results in terms of layer-wise squared error, pruning the first layer of a ResNet18 (RN18) model to several sparsities. In this metric, ExactOBS performs best by a wide margin ahead of AdaPrune, which significantly outperforms the other two methods. ",
+        "bbox": [
+            174,
+            580,
+            825,
+            664
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Next, in Table 1, we turn our attention to the practical problem of pruning various models to achieve a given FLOP reduction of $2 \\times - 4 \\times$ , applying the per-layer target sparsity optimization technique described above. Our ExactOBS generally performs best (except for YOLOv5l $2 \\times$ where all methods perform similarly in terms of mAP $@ 0 . 5$ ) and at $4 \\times$ FLOP reduction even with a $> 1 \\%$ gap to the next best method. Interestingly, on the hard-to-prune BERT model, ExactOBS appears to be the only method which still produces reasonable results at higher reduction targets. For BERT $3 \\times$ and $4 \\times$ , where the performance drop of all methods is $> 2 \\%$ , we additionally assess the compatibility of our results with the more powerful (but also more expensive) post processing method global AdaPrune [10]. While this global optimization technique is able to recover lost accuracy, the ExactOBS models still maintain $a > 0 . { \\bar { 5 } } \\%$ and $> 2 \\%$ F1 advantage, respectively (see Table 5). ",
+        "bbox": [
+            174,
+            670,
+            825,
+            808
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "N:M Sparsity. Next, we study the performance of our method for semi-structured sparsity via the N:M pattern. Specifically, we compare against the 4:8 results of AdaPrune with batchnorm tuning [18] on ResNet models (see Table 2) in addition to a 2:4 comparison on BERT models (see Table 3). We highlight that ExactOBS matches or even slightly exceeds the 4:8 results of AdaPrune with the considerably more stringent 2:4 pattern, which is already well supported on NVIDIA hardware. Furthermore, in a 2:4 comparison on BERT models, ExactOBS achieves $1 { - } 2 \\%$ higher F1 scores. ",
+        "bbox": [
+            174,
+            814,
+            825,
+            897
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e63c4464a79a22fecd310d811360589f3a30affa49cccf71cee521d9e2137d65.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            84,
+            349,
+            186
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/67146397ce577e9b78f15267fb42c76359ad133a9421a92f245b6d8cc33515ef.jpg",
+        "table_caption": [
+            "Table 1: Unstructured pruning for different FLOP reduction targets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">ResNet50 -76.13</td><td colspan=\"3\">YOLOv5l-66.97</td><td colspan=\"3\">BERT-88.53</td></tr><tr><td>2×</td><td>3×</td><td>4×</td><td>2×</td><td>3×</td><td>4×</td><td>2×</td><td>3×</td><td>4×</td></tr><tr><td>GMP</td><td>74.86</td><td>71.44</td><td>64.84</td><td>65.83</td><td>62.30</td><td>55.09</td><td>65.64</td><td>12.52</td><td>09.23</td></tr><tr><td>L-OBS</td><td>75.48</td><td>73.73</td><td>71.24</td><td>66.21</td><td>64.47</td><td>61.15</td><td>77.67</td><td>3.62</td><td>6.63</td></tr><tr><td>AdaPrune</td><td>75.53</td><td>74.47</td><td>72.39</td><td>66.00</td><td>64.88</td><td>62.71</td><td>87.12</td><td>70.32</td><td>18.75</td></tr><tr><td>ExactOBS</td><td>75.64</td><td>75.01</td><td>74.05</td><td>66.14</td><td>65.35</td><td>64.05</td><td>87.81</td><td>85.87</td><td>82.10</td></tr></table>",
+        "bbox": [
+            361,
+            93,
+            812,
+            172
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/aba7d382aa6cf2d1d84bc67276eb1ed4175b3d757bd6786d47122e0917c30d80.jpg",
+        "table_caption": [
+            "Figure 2: RN18 squared error. ",
+            "Table 2: Semi-structured N:M pruning ( $^ +$ batchnorm tuning) of all layers except the first and the last. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Dense</td><td>AdaPrune 4:8</td><td colspan=\"2\">ExactOBS 2:4 4:8</td></tr><tr><td>ResNet18</td><td>69.76</td><td>68.63</td><td>68.81</td><td>69.18</td></tr><tr><td>ResNet34</td><td>73.31</td><td>72.36</td><td>72.66</td><td>72.95</td></tr><tr><td>ResNet50</td><td>76.13</td><td>74.75</td><td>74.71</td><td>75.20</td></tr></table>",
+        "bbox": [
+            214,
+            212,
+            475,
+            276
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/aa2c2f50aeded4c8e5833b5299c958d886a71bfa86c3608ad06d304ff6c92b82.jpg",
+        "table_caption": [
+            "Table 3: Semi-structured 2:4 pruning of all layers except the embeddings. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Dense</td><td>AdaPrune</td><td>ExactOBS</td></tr><tr><td>BERT3</td><td>84.66</td><td>82.75</td><td>83.54</td></tr><tr><td>BERT6</td><td>88.33</td><td>85.02</td><td>86.97</td></tr><tr><td>BERT</td><td>88.53</td><td>85.24</td><td>86.77</td></tr></table>",
+        "bbox": [
+            565,
+            217,
+            795,
+            270
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Quantization. Additionally, we compare OBQ’s independent performance (after batchnorm tuning) with the state-of-the-art sequential post-training methods AdaQuant [19], AdaRound [31] and BRECQ [24]. We perform standard asymmetric per-channel quantization of all weights, using the authors’ implementations. We rerun all methods on Torchvision [28] ResNets to ensure a uniform baseline. The quantization grids for OBQ as well as AdaRound are determined with the same LAPQ [34] procedure that is used by BRECQ. Surprisingly, we find that, despite optimizing layers independently, OBQ achieves very similar (sometimes even slightly better) accuracies as existing non-independent methods for 4 and 3 bits. This suggests that it should be well-suited for mixed precision applications where one needs to quickly generate many non-uniform models optimized for different constraints. (However, we note that ExactOBS can also be applied sequentially; see Appendix.) ",
+        "bbox": [
+            173,
+            343,
+            825,
+            497
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/52e52494e4f913579129f612ca95556d0752e874d9872653613255304971e56e.jpg",
+        "table_caption": [
+            "Table 5: Further improving results in Table 1 with $5 \\ 3 \\%$ performance drops through more expensive post-processing via global AdaPrune (gAP). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td colspan=\"2\">BERT 3× 4×</td></tr><tr><td>gAP + AdaPrune</td><td>86.99</td><td>84.10</td></tr><tr><td>gAP +ExactOBS</td><td>87.57</td><td>86.42</td></tr></table>",
+        "bbox": [
+            617,
+            512,
+            807,
+            566
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/19670099f45b4ac03a1141fd9d3839c596e458ac05bd4819f6293a3f464ff0b6.jpg",
+        "table_caption": [
+            "Table 4: Comparison with state-of-the-art post-training methods for asymmetric per-channel weight quantization of all layers. We mark whether methods are Layer-wise (Lw.) or Independent (Ind.). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Lw.</td><td rowspan=\"2\">Ind.</td><td colspan=\"3\">ResNet18-69.76</td><td colspan=\"3\">ResNet50-76.13</td></tr><tr><td>4bit</td><td>3bit</td><td>2bit</td><td>4bit</td><td>3bit</td><td>2bit</td></tr><tr><td rowspan=\"3\">AdaRound AdaQuant BRECQ</td><td>yes</td><td>no</td><td>69.34</td><td>68.37</td><td>63.37</td><td>75.84</td><td>75.14</td><td>71.58</td></tr><tr><td>yes</td><td>no</td><td>68.12</td><td>59.21</td><td>00.10</td><td>74.68</td><td>64.98</td><td>00.10</td></tr><tr><td>no</td><td>no</td><td>69.37</td><td>68.47</td><td>64.70</td><td>75.88</td><td>75.32</td><td>72.41</td></tr><tr><td rowspan=\"2\">OBQ(ours)</td><td>yes</td><td>yes</td><td>69.56</td><td>68.69</td><td>64.04</td><td>75.72</td><td>75.24</td><td>70.71</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
+            184,
+            512,
+            583,
+            592
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "BOP-Constrained Mixed GPU Compression. We now consider a practical setting where we are given a trained model together with some calibration data and want to compress this model for efficient inference on an NVIDIA GPU which supports 8-bit and 4-bit arithmetic, also in combination with 2:4 sparsity. Thus, there are 4 possible compression choices per layer: 8bit weights $+ ~ 8 \\mathrm { b i t }$ activations (8w8a), 4w4a, $8 \\mathrm { w } 8 \\mathrm { a } + 2 { : } 4$ and $4 \\mathrm { w } 4 \\mathrm { a } + 2 { : } 4$ . Unlike in the previous section, we do symmetric per-channel quantization of the weights as it has better hardware support; activations are quantized asymmetrically per-tensor. We then generate mixed precision configurations for various BOP (number of bits times FLOPs) reduction targets and visualize the resulting compression-accuracy trade-off curves in Figure 3. In summary, at the cost of a $\\approx 2 . 5 \\%$ relative performance drop, we can achieve a $1 2 - 1 4 \\times$ BOP reduction for ResNets and a $7 - 8 \\times$ reduction for the more challenging YOLO and BERT models (relative to the compute in compressible layers). To the best of our knowledge, we are the first to consider joint N:M pruning and quantization in a post-training setting. Recent work [3] also studies joint $4 \\mathrm { w } 4 \\mathrm { a } + 2 { : } 4$ compression for ResNet18 but with 90 epochs of (sparse) Quantization-Aware Training (QAT) on the full dataset and report $6 7 . 3 3 \\%$ accuracy. Although not perfectly comparable (we keep the first layer dense and their dense baseline has $0 . 9 4 \\%$ higher accuracy and uses 4:8 sparse activations), we achieve similar $6 7 . 2 0 \\%$ accuracy for $4 \\mathrm { w } 4 \\mathrm { a } + 2 { : } 4$ post training, which emphasizes the effectiveness of our methods for joint sparsification and quantization. ",
+        "bbox": [
+            173,
+            641,
+            825,
+            876
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Time-Constrained CPU Compression. Lastly, we explore a similar scenario, but targeting actual CPU inference speedup on a 12-core Intel Xeon Silver 4214 CPU using the DeepSparse inference engine [35, 22], which provides acceleration for joint 8-bit quantization and block-sparsity with blocksize 4. In this case, we work with real layer-wise timing data (for batchsize 64), as in [9]. There are 30 available block-sparsity targets per-layer, in steps of pruning $10 \\%$ of the remaining weights, all of which are further quantized to 8 bits. The base acceleration of the dense 8 bit model is $\\approx 2 . 7 \\times$ on top of which sparsity speedup acts roughly multiplicatively. Figure 2d shows results for ResNet50 and several (real-time) speedup targets—we achieve $4 \\times$ and $5 \\times$ (actual) speedup with $1 \\%$ and $2 \\%$ accuracy loss, respectively. These are the first full post-training results in this setting (the authors of [10] only performed 4-block pruning post-training, followed by 5 epochs of QAT on the entire ImageNet dataset), and they show very encouraging accuracy-speedup trade-offs. ",
+        "bbox": [
+            174,
+            883,
+            821,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/48b7ae4ee49d35553186d141fe1a71bab7edce99ae5239859f187d9fe93726c1.jpg",
+        "image_caption": [
+            "Mixed quantization pruning ",
+            "Mixed quantization pruning ",
+            "Figure 3: (a) to (c) Mixed quantization and 2:4 pruning for various BOP reduction targets. (d) Joint block-pruning and quantization for CPU inference time speedups. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            223,
+            95,
+            771,
+            450
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            515,
+            825,
+            640
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusions & Future Work ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            660,
+            441,
+            678
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We have presented a new efficient and accurate approach for solving the layer-wise compression problem, and built on it to obtain state-of-the-art post-training compression solutions for both pruning and quantization. Our framework should be naturally extensible to structured pruning, which in fact should allow for further optimizations, and should also be compatible with further compression via unstructured pruning and quantization. Our results suggest that post-training compression may be able to reach comparable accuracies to much more expensive retraining methods. We plan to investigate this in future work, in particular in the context of more resource-intensive models, such as very large-scale language models. ",
+        "bbox": [
+            173,
+            693,
+            825,
+            803
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "8 Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            823,
+            367,
+            840
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We gratefully acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 programme (grant agreement No 805223 ScaleML), as well as computational support from AWS EC2. We thank Eldar Kurtic for providing us BERT code and pretrained models, and the Neural Magic Team, notably Michael Goin and Mark Kurtz, for support with their software. ",
+        "bbox": [
+            176,
+            856,
+            825,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Ron Banner, Yury Nahshan, and Daniel Soudry. Post training 4-bit quantization of convolutional networks for rapid-deployment. In Conference on Neural Information Processing Systems (NeurIPS), 2019.   \n[2] Thomas Blumensath and Mike E Davies. Iterative thresholding for sparse approximations. Journal of Fourier Analysis and Applications, 14(5-6):629–654, 2008.   \n[3] Brian Chmiel, Itay Hubara, Ron Banner, and Daniel Soudry. Optimal fine-grained N:M sparsity for activations and neural gradients. arXiv preprint arXiv:2203.10991, 2022.   \n[4] Yoni Choukroun, Eli Kravchik, Fan Yang, and Pavel Kisilev. Low-bit quantization of neural networks for efficient inference. In International Conference on Computer Vision Workshop (ICCVW), 2019.   \n[5] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In North American Chapter of the Association for Computational Linguistics (NAACL), 2019.   \n[6] Xin Dong, Shangyu Chen, and Sinno Jialin Pan. Learning to prune deep neural networks via layer-wise optimal brain surgeon. In Conference on Neural Information Processing Systems (NeurIPS), 2017.   \n[7] Erich Elsen, Marat Dukhan, Trevor Gale, and Karen Simonyan. Fast sparse convnets. In Conference on Computer Vision and Pattern Recognition (CVPR), 2020.   \n[8] Utku Evci, Nicolas Le Roux, Pablo Castro, and Leon Bottou. Mean replacement pruning. 2018.   \n[9] Elias Frantar and Dan Alistarh. SPDY: Accurate pruning with speedup guarantees. arXiv preprint arXiv:2201.13096, 2022.   \n[10] Elias Frantar, Eldar Kurtic, and Dan Alistarh. M-FAC: Efficient matrix-free approximations of second-order information. In Conference on Neural Information Processing Systems (NeurIPS), 2021.   \n[11] Amir Gholami, Sehoon Kim, Zhen Dong, Zhewei Yao, Michael W Mahoney, and Kurt Keutzer. A survey of quantization methods for efficient neural network inference. arXiv preprint arXiv:2103.13630, 2021.   \n[12] Roger Grosse and James Martens. A Kronecker-factored approximate Fisher matrix for convolution layers. In International Conference on Machine Learning (ICML), 2016.   \n[13] Babak Hassibi, David G Stork, and Gregory J Wolff. Optimal brain surgeon and general network pruning. In IEEE International Conference on Neural Networks, 1993.   \n[14] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Conference on Computer Vision and Pattern Recognition (CVPR), 2016.   \n[15] Yihui He, Ji Lin, Zhijian Liu, Hanrui Wang, Li-Jia Li, and Song Han. AMC: AutoML for model compression and acceleration on mobile devices. In European Conference on Computer Vision (ECCV), 2018.   \n[16] Yihui He, Xiangyu Zhang, and Jian Sun. Channel pruning for accelerating very deep neural networks. In International Conference on Computer Vision (ICCV), 2017.   \n[17] Torsten Hoefler, Dan Alistarh, Tal Ben-Nun, Nikoli Dryden, and Alexandra Peste. Sparsity in deep learning: Pruning and growth for efficient inference and training in neural networks. arXiv preprint arXiv:2102.00554, 2021.   \n[18] Itay Hubara, Brian Chmiel, Moshe Island, Ron Banner, Seffi Naor, and Daniel Soudry. Accelerated sparse neural training: A provable and efficient method to find N:M transposable masks. In Conference on Neural Information Processing Systems (NeurIPS), 2021. ",
+        "bbox": [
+            173,
+            75,
+            828,
+            920
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[19] Itay Hubara, Yury Nahshan, Yair Hanani, Ron Banner, and Daniel Soudry. Accurate post training quantization with small calibration sets. In International Conference on Machine Learning (ICML), 2021. ",
+        "bbox": [
+            173,
+            90,
+            821,
+            133
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[20] Glenn Jocher. YOLOv5. https://github.com/ultralytics/yolov5, 2022.   \n[21] Eldar Kurtic, Daniel Campos, Tuan Nguyen, Elias Frantar, Mark Kurtz, Benjamin Fineran, Michael Goin, and Dan Alistarh. The Optimal BERT Surgeon: Scalable and accurate secondorder pruning for large language models. arXiv preprint arXiv:2203.07259, 2022.   \n[22] Mark Kurtz, Justin Kopinsky, Rati Gelashvili, Alexander Matveev, John Carr, Michael Goin, William Leiserson, Sage Moore, Bill Nell, Nir Shavit, and Dan Alistarh. Inducing and exploiting activation sparsity for fast inference on deep neural networks. In International Conference on Machine Learning (ICML), 2020.   \n[23] Yann LeCun, John S Denker, and Sara A Solla. Optimal brain damage. In Conference on Neural Information Processing Systems (NeurIPS), 1990.   \n[24] Yuhang Li, Ruihao Gong, Xu Tan, Yang Yang, Peng Hu, Qi Zhang, Fengwei Yu, Wei Wang, and Shi Gu. BRECQ: Pushing the limit of post-training quantization by block reconstruction. In International Conference on Learning Representations (ICLR), 2021.   \n[25] Tailin Liang, John Glossner, Lei Wang, Shaobo Shi, and Xiaotong Zhang. Pruning and quantization for deep neural network acceleration: A survey. Neurocomputing, 461:370–403, 2021.   \n[26] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft COCO: Common objects in context. In European Conference on Computer Vision (ECCV), 2014.   \n[27] Liyang Liu, Shilong Zhang, Zhanghui Kuang, Aojun Zhou, Jing-Hao Xue, Xinjiang Wang, Yimin Chen, Wenming Yang, Qingmin Liao, and Wayne Zhang. Group fisher pruning for practical network compression. In International Conference on Machine Learning (ICML), 2021.   \n[28] Sébastien Marcel and Yann Rodriguez. Torchvision the machine-vision package of torch. In ACM International Conference on Multimedia, 2010.   \n[29] James Martens and Roger Grosse. Optimizing neural networks with kronecker-factored approximate curvature. In International Conference on Machine Learning (ICML), 2015.   \n[30] Asit Mishra, Jorge Albericio Latorre, Jeff Pool, Darko Stosic, Dusan Stosic, Ganesh Venkatesh, Chong Yu, and Paulius Micikevicius. Accelerating sparse deep neural networks. arXiv preprint arXiv:2104.08378, 2021.   \n[31] Markus Nagel, Rana Ali Amjad, Mart Van Baalen, Christos Louizos, and Tijmen Blankevoort. Up or down? Adaptive rounding for post-training quantization. In International Conference on Machine Learning (ICML), 2020.   \n[32] Markus Nagel, Mart van Baalen, Tijmen Blankevoort, and Max Welling. Data-free quantization through weight equalization and bias correction. In International Conference on Computer Vision (ICCV), 2019.   \n[33] Markus Nagel, Marios Fournarakis, Rana Ali Amjad, Yelysei Bondarenko, Mart van Baalen, and Tijmen Blankevoort. A white paper on neural network quantization. arXiv preprint arXiv:2106.08295, 2021.   \n[34] Yury Nahshan, Brian Chmiel, Chaim Baskin, Evgenii Zheltonozhskii, Ron Banner, Alex M Bronstein, and Avi Mendelson. Loss aware post-training quantization. Machine Learning, 110(11):3245–3262, 2021.   \n[35] NeuralMagic. The DeepSparse Inference Engine. https://github.com/neuralmagic/deepsparse, 2022.   \n[36] Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. SQuAD: $^ { 1 0 0 , 0 0 0 + }$ questions for machine comprehension of text. In Conference on Empirical Methods in Natural Language Processing (EMNLP), 2016.   \n[37] Vijay Janapa Reddi, Christine Cheng, David Kanter, Peter Mattson, Guenther Schmuelling, Carole-Jean Wu, Brian Anderson, Maximilien Breughe, Mark Charlebois, William Chou, et al. Mlperf inference benchmark. In 2020 ACM/IEEE 47th Annual International Symposium on Computer Architecture (ISCA), pages 446–459. IEEE, 2020.   \n[38] Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International Journal of Computer Vision, 115(3):211–252, 2015.   \n[39] Sidak Pal Singh and Dan Alistarh. WoodFisher: Efficient second-order approximation for neural network compression. In Conference on Neural Information Processing Systems (NeurIPS), 2020.   \n[40] Chaoqi Wang, Roger Grosse, Sanja Fidler, and Guodong Zhang. Eigendamage: Structured pruning in the Kronecker-factored eigenbasis. In International Conference on Machine Learning (ICML), 2019.   \n[41] Kuan Wang, Zhijian Liu, Yujun Lin, Ji Lin, and Song Han. HAQ: Hardware-aware automated quantization with mixed precision. In Conference on Computer Vision and Pattern Recognition (CVPR), 2019.   \n[42] Peisong Wang, Qiang Chen, Xiangyu He, and Jian Cheng. Towards accurate post-training network quantization via bit-split and stitching. In International Conference on Machine Learning (ICML), 2020.   \n[43] Haichuan Yang, Shupeng Gui, Yuhao Zhu, and Ji Liu. Automatic neural network compression by sparsity-quantization joint learning: A constrained optimization-based approach. In Conference on Computer Vision and Pattern Recognition (CVPR), 2020.   \n[44] Zhewei Yao, Zhen Dong, Zhangcheng Zheng, Amir Gholami, Jiali Yu, Eric Tan, Leyuan Wang, Qijing Huang, Yida Wang, Michael Mahoney, et al. HAWQ-v3: Dyadic neural network quantization. In International Conference on Machine Learning (ICML), 2021.   \n[45] Aojun Zhou, Yukun Ma, Junnan Zhu, Jianbo Liu, Zhijie Zhang, Kun Yuan, Wenxiu Sun, and Hongsheng Li. Learning N:M fine-grained structured sparse neural networks from scratch. In International Conference on Learning Representations (ICLR), 2021.   \n[46] Michael Zhu and Suyog Gupta. To prune, or not to prune: exploring the efficacy of pruning for model compression. arXiv preprint arXiv:1710.01878, 2017. ",
+        "bbox": [
+            169,
+            103,
+            828,
+            911
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            84,
+            826,
+            650
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Checklist ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            254,
+            106
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            116,
+            339,
+            131
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes]   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            135,
+            825,
+            227
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            214,
+            231,
+            493,
+            244
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] ",
+        "bbox": [
+            236,
+            248,
+            735,
+            280
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            212,
+            285,
+            393,
+            299
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ",
+        "bbox": [
+            238,
+            303,
+            826,
+            421
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            215,
+            426,
+            823,
+            440
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes]   \n(b) Did you mention the license of the assets? [N/A]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [No] Assets are open-source with standard permissive licenses.   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            238,
+            444,
+            825,
+            565
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            214,
+            569,
+            705,
+            584
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            588,
+            825,
+            678
+        ],
+        "page_idx": 13
+    }
+]

parse/dev/tHK5ntjp-5K/tHK5ntjp-5K.md

@@ -0,0 +1,445 @@
+# LION: Latent Point Diffusion Models for 3D Shape Generation
+
+Xiaohui Zeng1,2,3,∗ Arash Vahdat1 Francis Williams1
+
+Zan Gojcic1 Or Litany1 Sanja Fidler1,2,3 Karsten Kreis1
+
+1NVIDIA 2University of Toronto 3Vector Institute {xzeng,avahdat,fwilliams,zgojcic,olitany,sfidler,kkreis}@nvidia.com
+
+# Abstract
+
+Denoising diffusion models (DDMs) have shown promising results in 3D point cloud synthesis. To advance 3D DDMs and make them useful for digital artists, we require (i) high generation quality, (ii) flexibility for manipulation and applications such as conditional synthesis and shape interpolation, and (iii) the ability to output smooth surfaces or meshes. To this end, we introduce the hierarchical Latent Point Diffusion Model (LION) for 3D shape generation. LION is set up as a variational autoencoder (VAE) with a hierarchical latent space that combines a global shape latent representation with a point-structured latent space. For generation, we train two hierarchical DDMs in these latent spaces. The hierarchical VAE approach boosts performance compared to DDMs that operate on point clouds directly, while the point-structured latents are still ideally suited for DDM-based modeling. Experimentally, LION achieves state-of-the-art generation performance on multiple ShapeNet benchmarks. Furthermore, our VAE framework allows us to easily use LION for different relevant tasks: LION excels at multimodal shape denoising and voxel-conditioned synthesis, and it can be adapted for text- and image-driven 3D generation. We also demonstrate shape autoencoding and latent shape interpolation, and we augment LION with modern surface reconstruction techniques to generate smooth 3D meshes. We hope that LION provides a powerful tool for artists working with 3D shapes due to its high-quality generation, flexibility, and surface reconstruction. Project page and code: https://nv-tlabs.github.io/LION.
+
+# 1 Introduction
+
+Generative modeling of 3D shapes has extensive applications in 3D content creation and has become an active area of research [1–52]. However, to be useful as a tool for digital artists, generative models of 3D shapes have to fulfill several criteria: (i) Generated shapes need to be realistic and of highquality without artifacts. (ii) The model should enable flexible and interactive use and refinement: For example, a user may want to refine a generated shape and synthesize versions with varying details. Or an artist may provide a coarse or noisy input shape, thereby guiding the model to produce multiple realistic high-quality outputs. Similarly, a user may want to interpolate different shapes. (iii) The model should output smooth meshes, which are the standard representation in most graphics software.
+
+Existing 3D generative models build on various frameworks, including generative adversarial networks (GANs) [1–23], variational autoencoders (VAEs) [24–30], normalizing flows [31–34], autoregressive models [35–38], and more [39–44]. Most recently, denoising diffusion models (DDMs)
+
+![](images/a5ecbde7333db061775c6917bb95434b89bc4661fe3aa6fbf5e867c83ff8aba7.jpg)  
+Figure 1: LION is set up as a hierarchical point cloud VAE with denoising diffusion models over the shape latent and latent point distributions. PointVoxel CNNs (PVCNN) with adaptive Group Normalization (Ada. GN) are used as neural networks. The latent points can be interpreted as a smoothed version of the input point cloud. Shape As Points (SAP) is optionally used for mesh reconstruction.
+
+have emerged as powerful generative models, achieving outstanding results not only on image synthesis [53–64] but also for point cloud-based 3D shape generation [45–47]. In DDMs, the data is gradually perturbed by a diffusion process, while a deep neural network is trained to denoise. This network can then be used to synthesize novel data in an iterative fashion when initialized from random noise [53, 65–67]. However, existing DDMs for 3D shape synthesis struggle with simultaneously satisfying all criteria discussed above for practically useful 3D generative models.
+
+Here, we aim to develop a DDM-based generative model of 3D shapes overcoming these limitations. We introduce the Latent Point Diffusion Model (LION) for 3D shape generation (see Fig. 1). Similar to previous 3D DDMs, LION operates on point clouds, but it is constructed as a VAE with DDMs in latent space. LION comprises a hierarchical latent space with a vector-valued global shape latent and another point-structured latent space. The latent representations are predicted with point cloud processing encoders, and two latent DDMs are trained in these latent spaces. Synthesis in LION proceeds by drawing novel latent samples from the hierarchical latent DDMs and decoding back to the original point cloud space. Importantly, we also demonstrate how to augment LION with modern surface reconstruction methods [68] to synthesize smooth shapes as desired by artists. LION has multiple advantages:
+
+Expressivity: By mapping point clouds into regularized latent spaces, the DDMs in latent space are effectively tasked with learning a smoothed distribution. This is easier than training on potentially complex point clouds directly [58], thereby improving expressivity. However, point clouds are, in principle, an ideal representation for DDMs. Because of that, we use latent points, this is, we keep a point cloud structure for our main latent representation. Augmenting the model with an additional global shape latent variable in a hierarchical manner further boosts expressivity. We validate LION on several popular ShapeNet benchmarks and achieve state-of-the-art synthesis performance.
+
+Varying Output Types: Extending LION with Shape As Points (SAP) [68] geometry reconstruction allows us to also output smooth meshes. Fine-tuning SAP on data generated by LION’s autoencoder reduces synthesis noise and enables us to generate high-quality geometry. LION combines (latent) point cloud-based modeling, ideal for DDMs, with surface reconstruction, desired by artists.
+
+Flexibility: Since LION is set up as a VAE, it can be easily adapted for different tasks without retraining the latent DDMs: We can efficiently fine-tune LION’s encoders on voxelized or noisy inputs, which a user can provide for guidance. This enables multimodal voxel-guided synthesis and shape denoising. We also leverage LION’s latent spaces for shape interpolation and autoencoding. Optionally training the DDMs conditioned on CLIP embeddings enables image- and text-driven 3D generation.
+
+In summary, we make the following contributions: (i) We introduce LION, a novel generative model for 3D shape synthesis, which operates on point clouds and is built on a hierarchical VAE framework with two latent DDMs. (ii) We validate LION’s high synthesis quality by reaching state-of-the-art performance on widely used ShapeNet benchmarks. (iii) We achieve high-quality and diverse 3D shape synthesis with LION even when trained jointly over many classes without conditioning. (iv) We propose to combine LION with SAP-based surface reconstruction. (v) We demonstrate the flexibility of our framework by adapting it to relevant tasks such as multimodal voxel-guided synthesis.
+
+# 2 Background
+
+Traditionally, DDMs were introduced in a discrete-step fashion: Given samples $\mathbf { x } _ { 0 } \sim q ( \mathbf { x } _ { 0 } )$ from a data distribution, DDMs use a Markovian fixed forward diffusion process defined as [65, 53]
+
+![](images/50a69b4fae41081a6f71ae4fa30d4db630414574be115671a763f0682d5ca458.jpg)  
+Figure 2: Generated meshes with LION. Right: Synthesizing different details by diffuse-denoise (see Sec. 3.1) in latent space, while preserving overall shapes.
+
+$$
+q ( \mathbf { x } _ { 1 : T } | \mathbf { x } _ { 0 } ) : = \prod _ { t = 1 } ^ { T } q ( \mathbf { x } _ { t } | \mathbf { x } _ { t - 1 } ) , \qquad q ( \mathbf { x } _ { t } | \mathbf { x } _ { t - 1 } ) : = \mathcal { N } ( \mathbf { x } _ { t } ; \sqrt { 1 - \beta _ { t } } \mathbf { x } _ { t - 1 } , \beta _ { t } I ) ,
+$$
+
+where $T$ denotes the number of steps and $q \big ( \mathbf { x } _ { t } | \mathbf { x } _ { t - 1 } \big )$ is a Gaussian transition kernel, which gradually adds noise to the input with a variance schedule $\beta _ { 1 } , . . . , \beta _ { T }$ . The $\beta _ { t }$ are chosen such that the chain approximately converges to a standard Gaussian distribution after $T$ steps, $q ( \mathbf { x } _ { T } ) { \approx } { \mathcal { N } } ( \mathbf { x } _ { T } ; \mathbf { 0 } , I )$ . DDMs learn a parametrized reverse process (model parameters $\pmb \theta$ ) that inverts the forward diffusion:
+
+$$
+p _ { \theta } ( \mathbf { x } _ { 0 : T } ) : = p ( \mathbf { x } _ { T } ) \prod _ { t = 1 } ^ { T } p _ { \theta } ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } ) , \qquad p _ { \theta } ( \mathbf { x } _ { t - 1 } | \mathbf { x } _ { t } ) : = \mathcal { N } ( \mathbf { x } _ { t - 1 } ; \mu _ { \theta } ( \mathbf { x } _ { t } , t ) , \rho _ { t } ^ { 2 } I ) .
+$$
+
+This generative reverse process is also Markovian with Gaussian transition kernels, which use fixed variances $\rho _ { t } ^ { 2 }$ . DDMs can be interpreted as latent variable models, where $\mathbf { x } _ { 1 } , . . . , \mathbf { x } _ { T }$ are latents, and the forward process $q \big ( \mathbf { x } _ { 1 : T } | \mathbf { x } _ { 0 } \big )$ acts as a fixed approximate posterior, to which the generative $p _ { \pmb { \theta } } ( \mathbf { x } _ { 0 : T } )$ is fit. DDMs are trained by minimizing the variational upper bound on the negative log-likelihood of the data $\mathbf { x } _ { \mathrm { 0 } }$ under $p _ { \theta } ( \mathbf { x } _ { 0 : T } )$ . Up to irrelevant constant terms, this objective can be expressed as [53]
+
+$$
+\operatorname* { m i n } _ { \theta } \mathbb { E } _ { t \sim U \left\{ 1 , T \right\} , \mathbf { x } _ { 0 } \sim p \left( \mathbf { x } _ { 0 } \right) , \epsilon \sim \mathcal { N } ( \mathbf { 0 } , I ) } \left[ w ( t ) | | \epsilon - \epsilon _ { \theta } ( \alpha _ { t } \mathbf { x } _ { 0 } + \sigma _ { t } \epsilon , t ) | | _ { 2 } ^ { 2 } \right] , w ( t ) = \frac { \beta _ { t } ^ { 2 } } { 2 \rho _ { t } ^ { 2 } ( 1 - \beta _ { t } ) ( 1 - \alpha _ { t } ^ { 2 } ) } ,
+$$
+
+where $\alpha _ { t } = \sqrt { \prod _ { s = 1 } ^ { t } ( 1 - \beta _ { s } ) }$ and $\sigma _ { t } ~ = ~ \sqrt { 1 - \alpha _ { t } ^ { 2 } }$ are the parameters of the tractable diffused distribution after $t$ steps $q ( \mathbf { x } _ { t } | \mathbf { x } _ { 0 } ) = \mathcal { N } ( \mathbf { x } _ { t } ; \alpha _ { t } \mathbf { x } _ { 0 } , \sigma _ { t } ^ { 2 } I )$ . Furthermore, Eq. (3) employs the widely used parametrization µθ(xt, t) := √ 11−βt $\begin{array} { r } { \mu _ { \theta } ( \mathbf { x } _ { t } , t ) : = \frac { 1 } { \sqrt { 1 - \beta _ { t } } } \left( \mathbf { x } _ { t } - \frac { \beta _ { t } } { \sqrt { 1 - \alpha _ { t } ^ { 2 } } } \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) \right) } \end{array}$ . It is common practice to set $w ( t ) = 1$ , instead of the one in Eq. (3), which often promotes perceptual quality of the generated output. In the objective of Eq. (3), the model $\epsilon _ { \theta }$ is, for all possible steps $t$ along the diffusion process, effectively trained to predict the noise vector $\epsilon$ that is necessary to denoise an observed diffused sample $\mathbf { x } _ { t }$ . After training, the DDM can be sampled with ancestral sampling in an iterative fashion:
+
+$$
+\begin{array} { r } { \mathbf { x } _ { t - 1 } = \frac { 1 } { \sqrt { 1 - \beta _ { t } } } ( \mathbf { x } _ { t } - \frac { \beta _ { t } } { \sqrt { 1 - \alpha _ { t } ^ { 2 } } } \pmb { \epsilon } _ { \theta } ( \mathbf { x } _ { t } , t ) ) + \rho _ { t } \pmb { \eta } , } \end{array}
+$$
+
+where $\eta \sim \mathcal { N } ( \eta ; 0 , I )$ . This sampling chain is initialized from a random sample $\mathbf { x } _ { T } \sim \mathcal { N } ( \mathbf { x } _ { T } ; \mathbf { 0 } , I )$ .   
+Furthermore, the noise injection in Eq. 4 is usually omitted in the last sampling step.
+
+DDMs can also be expressed with a continuous-time framework [67, 69]. In this formulation, the diffusion and reverse generative processes are described by differential equations. This approach allows for deterministic sampling and encoding schemes based on ordinary differential equations (ODEs). We make use of this framework in Sec. 3.1 and we review this approach in more detail in App. B.
+
+# 3 Hierarchical Latent Point Diffusion Models
+
+We first formally introduce LION, then discuss various applications and extensions in Sec. 3.1, and finally recapitulate its unique advantages in Sec. 3.2. See Fig. 1 for a visualization of LION.
+
+We are modeling point clouds $\mathbf { x } \in \mathbb { R } ^ { 3 \times N }$ , consisting of $N$ points with xyz-coordinates in $\mathbb { R } ^ { 3 }$ . LION is set up as a hierarchical VAE with DDMs in latent space. It uses a vector-valued global shape latent $\mathbf { z } _ { 0 } \in \mathbb { R } ^ { D _ { \mathbf { z } } }$ and a point cloud-structured latent $\mathbf { h } _ { 0 } \in \mathbf { \bar { \mathbb { R } } } ^ { ( 3 + D _ { \mathbf { h } } ) \times N }$ . Specifically, $\mathbf { h } _ { 0 }$ is a latent point cloud consisting of $N$ points with $x y z$ -coordinates in $\mathbb { R } ^ { 3 }$ . In addition, each latent point can carry additional $D _ { \mathbf { h } }$ latent features. Training of LION is then performed in two stages—first, we train it as a regular VAE with standard Gaussian priors; then, we train the latent DDMs on the latent encodings.
+
+First Stage Training. Initially, LION is trained by maximizing a modified variational lower bound on the data log-likelihood (ELBO) with respect to the encoder and decoder parameters $\phi$ and $\boldsymbol { \xi }$ [70, 71]:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { E L B O } } ( \boldsymbol { \phi } , \boldsymbol { \xi } ) = \mathbb { E } _ { p ( \mathbf { x } ) , q _ { \phi } ( \mathbf { z } _ { 0 } | \mathbf { x } ) , q _ { \phi } ( \mathbf { h } _ { 0 } | \mathbf { x } , \mathbf { z } _ { 0 } ) } [ \log p _ { \xi } ( \mathbf { x } | \mathbf { h } _ { 0 } , \mathbf { z } _ { 0 } )  } \\ & {  \qquad -  \lambda _ { \mathbf { z } } D _ { \mathrm { K L } } ( q _ { \phi } ( \mathbf { z } _ { 0 } | \mathbf { x } ) \vert p ( \mathbf { z } _ { 0 } ) ) - \lambda _ { \mathbf { h } } D _ { \mathrm { K L } } ( q _ { \phi } ( \mathbf { h } _ { 0 } | \mathbf { x } , \mathbf { z } _ { 0 } ) \vert p ( \mathbf { h } _ { 0 } ) ) ] . } \end{array}
+$$
+
+![](images/794b4b6104af7191905670a7a7a67c74af466190e3278dba537a59617fa285aa.jpg)  
+Figure 3: Generated shapes (top: point clouds, bottom: corresponding meshes) from LION trained jointly over 13 classes of ShapeNet-vol without conditioning (Sec. 5.2).
+
+Here, the global shape latent $\mathbf { z } _ { 0 }$ is sampled from the posterior distribution $q _ { \phi } ( \mathbf { z } _ { 0 } | \mathbf { x } )$ , which is parametrized by factorial Gaussians, whose means and variances are predicted via an encoder network. The point cloud latent $\mathbf { h } _ { 0 }$ is sampled from a similarly parametrized posterior $q _ { \phi } ( \mathbf { h } _ { 0 } | \mathbf { x } , \mathbf { z } _ { 0 } )$ , while also conditioning on $\mathbf { z } _ { 0 }$ ( $\cdot \phi$ denotes the parameters of both encoders). Furthermore, $p _ { \pmb { \xi } } ( \mathbf { x } | \mathbf { h } _ { 0 } , \mathbf { z } _ { 0 } )$ denotes the decoder, parametrized as a factorial Laplace distribution with predicted means and fixed unit scale parameter (corresponding to an $L _ { 1 }$ reconstruction loss). $\lambda _ { \mathbf { z } }$ and $\lambda _ { \mathbf { h } }$ are hyperparameters balancing reconstruction accuracy and Kullback-Leibler regularization (note that only for $\lambda _ { \mathbf { z } } = \lambda _ { \mathbf { h } } = 1$ we are optimizing a rigorous ELBO). The priors $p ( \mathbf { z } _ { 0 } )$ and $p ( \mathbf { h } _ { 0 } )$ are $\mathcal { N } ( \mathbf { 0 } , \pmb { I } )$ . Also see Fig. 1 again.
+
+Second Stage Training. In principle, we could use the VAE’s priors to sample encodings and generate new shapes. However, the simple Gaussian priors will not accurately match the encoding distribution from the training data and therefore produce poor samples (prior hole problem [58, 72–79]). This motivates training highly expressive latent DDMs. In particular, in the second stage we freeze the VAE’s encoder and decoder networks and train two latent DDMs on the encodings $\mathbf { z } _ { 0 }$ and $\mathbf { h } _ { 0 }$ sampled from $q _ { \phi } ( \mathbf { z } _ { 0 } | \mathbf { x } )$ and $q _ { \phi } ( \mathbf { h } _ { 0 } | \mathbf { x } , \mathbf { z } _ { 0 } )$ , minimizing score matching (SM) objectives similar to Eq. (2):
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { S M } ^ { \mathbf { z } } } ( \pmb { \theta } ) = \mathbb { E } _ { t \sim U \{ 1 , T \} , p ( \mathbf { x } ) , q _ { \phi } ( \mathbf { z } _ { 0 } | \mathbf { x } ) , \epsilon \sim \mathcal { N } ( \mathbf { 0 } , I ) } | | \epsilon - \epsilon _ { \theta } ( \mathbf { z } _ { t } , t ) | | _ { 2 } ^ { 2 } , } \\ & { \mathcal { L } _ { \mathrm { S M } ^ { \mathbf { h } } } ( \pmb { \psi } ) = \mathbb { E } _ { t \sim U \{ 1 , T \} , p ( \mathbf { x } ) , q _ { \phi } ( \mathbf { z } _ { 0 } | \mathbf { x } ) , q _ { \phi } ( \mathbf { h } _ { 0 } | \mathbf { x } , \mathbf { z } _ { 0 } ) , \epsilon \sim \mathcal { N } ( \mathbf { 0 } , I ) } | | \epsilon - \epsilon _ { \psi } ( \mathbf { h } _ { t } , \mathbf { z } _ { 0 } , t ) | | _ { 2 } ^ { 2 } , } \end{array}
+$$
+
+where ${ \bf z } _ { t } = \alpha _ { t } { \bf z } _ { 0 } + \sigma _ { t } \epsilon$ and $\mathbf { h } _ { t } = \alpha _ { t } \mathbf { h } _ { 0 } + \sigma _ { t } \mathbf { \epsilon } \epsilon$ are the diffused latent encodings. Furthermore, $\pmb \theta$ denotes the parameters of the global shape latent DDM $\boldsymbol { \epsilon } _ { \boldsymbol { \theta } } ( \mathbf { z } _ { t } , t )$ , and $\psi$ refers to the parameters of the conditional DDM $\mathbf { \epsilon } _ { \psi } ( \mathbf { h } _ { t } , \bar { \mathbf { z } _ { 0 } } , t )$ trained over the latent point cloud (note the conditioning on $\mathbf { z } _ { 0 }$ ).
+
+Generation. With the latent DDMs, we can formally define a hierarchical generative model $\begin{array} { r } { p _ { \xi , \psi , \theta } ( \mathbf { x } , \mathbf { h } _ { 0 } , \mathbf { z } _ { 0 } ) = p _ { \xi } ( \mathbf { x } | \mathbf { h } _ { 0 } , \mathbf { z } _ { 0 } ) p _ { \psi } ( \mathbf { h } _ { 0 } | \mathbf { z } _ { 0 } ) p _ { \theta } ( \mathbf { z } _ { 0 } ) } \end{array}$ , where $p _ { \pmb { \theta } } ( \mathbf { z } _ { 0 } )$ denotes the distribution of the global shape latent DDM, $p _ { \psi } ( \mathbf { h } _ { 0 } | \mathbf { z } _ { 0 } )$ refers to the DDM modeling the point cloud-structured latents, and $p _ { \pmb { \xi } } ( \mathbf { x } | \mathbf { h } _ { 0 } , \mathbf { z } _ { 0 } )$ is LION’s decoder. We can hierarchically sample the latent DDMs following Eq. (4) and then translate the latent points back to the original point cloud space with the decoder.
+
+Network Architectures and DDM Parametrization. Let us briefly summarize key implementation choices. The encoder networks, as well as the decoder and the latent point DDM, operating on point clouds $\mathbf { x }$ , are all implemented based on Point-Voxel CNNs (PVCNNs) [80], following Zhou et al. [46]. PVCNNs efficiently combine the point-based processing of PointNets [81, 82] with the strong spatial inductive bias of convolutions. The DDM modeling the global shape latent uses a ResNet [83] structure with fully-connected layers (implemented as $1 \times 1$ -convolutions). All conditionings on the global shape latent are implemented via adaptive Group Normalization [84] in the PVCNN layers. Furthermore, following Vahdat et al. [58] we use a mixed score parametrization in both latent DDMs. This means that the score models are parametrized to predict a residual correction to an analytic standard Gaussian score. This is beneficial since the latent encodings are regularized towards a standard Gaussian distribution during the first training stage (see App. D for all details).
+
+# 3.1 Applications and Extensions
+
+Here, we discuss how LION can be used and extended for different relevant applications.
+
+Multimodal Generation. We can synthesize different variations of a given shape, enabling multimodal generation in a controlled manner: Given a shape, i.e., its point cloud $\mathbf { x }$ , we encode it into latent space. Then, we diffuse its encodings $\mathbf { z } _ { 0 }$ and $\mathbf { h } _ { 0 }$ for a small number of steps $\tau < T$ towards intermediate ${ \bf z } _ { \tau }$ and ${ \bf h } _ { \tau }$ along the diffusion process such that only local details are destroyed. Running the reverse generation process from this intermediate $\tau$ , starting at ${ \bf z } _ { \tau }$ and ${ \bf h } _ { \tau }$ , leads to variations of the original shape with different details (see, for instance, Fig. 2). We refer to this procedure as diffuse-denoise (details in App. C.1). Similar techniques have been used for image editing [85].
+
+Encoder Fine-tuning for Voxel-Conditioned Synthesis and Denoising. In practice, an artist using a 3D generative model may have a rough idea of the desired shape. For instance, they may be able to quickly construct a coarse voxelized shape, to which the generative model then adds realistic details.
+
+![](images/85429d590b634ce75ac7892abdbde7713e22da5e9b4c3ea48ef5c5e6e54822dc.jpg)  
+Figure 4: Voxelguided synthesis with LION. We run diffuse-denoise in latent space (see Sec. 3.1) to generate diverse plausible clean shapes.
+
+In LION, we can support such applications: using a similar ELBO as in Eq. (5), but with a frozen decoder, we can fine-tune LION’s encoder networks to take voxelized shapes as input (we simply place points at the voxelized shape’s surface) and map them to the corresponding latent encodings $\mathbf { z } _ { 0 }$ and $\mathbf { h } _ { 0 }$ that reconstruct the original non-voxelized point cloud. Now, a user can utilize the fine-tuned encoders to encode voxelized shapes and generate plausible detailed shapes. Importantly, this can be naturally combined with the diffuse-denoise procedure to clean up imperfect encodings and to generate different possible detailed shapes (see Fig. 4).
+
+Furthermore, this approach is general. Instead of voxel-conditioned synthesis, we can also fine-tune the encoder networks on noisy shapes to perform multimodal shape denoising, also potentially combined with diffuse-denoise. LION supports these applications easily without re-training the latent DDMs due to its VAE framework with additional encoders and decoders, in contrast to previous works that train DDMs on point clouds directly [46, 47]. See App. C.2 for technical details.
+
+Shape Interpolation. LION also enables shape interpolation: We can encode different point clouds into LION’s hierarchical latent space and use the probability flow ODE (see App. B) to further encode into the latent DDMs’ Gaussian priors, where we can safely perform spherical interpolation and expect valid shapes along the interpolation path. We can use the intermediate encodings to generate the interpolated shapes (see Fig. 7; details in App. C.3).
+
+Surface Reconstruction. While point clouds are an ideal 3D representation for DDMs, artists may prefer meshed outputs. Hence, we propose to combine LION with modern geometry reconstruction methods (see Figs. 2, 4 and 5). We use Shape As Points (SAP) [68], which is based on differentiable Poisson surface reconstruction and can be trained to extract smooth meshes from noisy point clouds. Moreover, we fine-tune SAP on training data generated by LION’s autoencoder to better adjust SAP to the noise distribution in point clouds generated by LION. Specifically, we take clean shapes, encode them into latent space, run a few steps of diffuse-denoise that only slightly modify some details, and decode back. The diffuse-denoise in latent space results in noise in the generated point ated points. clouds similar to what is observed during unconditional synthesis (details in App. C.4).
+
+![](images/e8246239ce1cb83b69f6bfa0190e0a475971dd591c1a606190b19903eaf14fce.jpg)  
+Figure 5: Reconstructing a mesh from LION’s gener
+
+# 3.2 LION’s Advantages
+
+We now recapitulate LION’s unique advantages. LION’s structure as a hierarchical VAE with latent DDMs is inspired by latent DDMs on images [57, 58, 77]. This framework has key benefits:
+
+(i) Expressivity: First training a VAE that regularizes the latent encodings to approximately fall under standard Gaussian distributions, which are also the DDMs’ equilibrium distributions towards which the diffusion processes converge, results in an easier modeling task for the DDMs: They have to model only the remaining mismatch between the actual encoding distributions and their own Gaussian priors [58]. This translates into improved expressivity, which is further enhanced by the additional decoder network. However, point clouds are, in principle, an ideal representation for the DDM framework, because they can be diffused and denoised easily and powerful point cloud processing architectures exist. Therefore, LION uses point cloud latents that combine the advantages of both latent DDMs and 3D point clouds. Our point cloud latents can be interpreted as smoothed versions of the original point clouds that are easier to model (see Fig. 1). Moreover, the hierarchical VAE setup with an additional global shape latent increases LION’s expressivity even further and results in natural disentanglement between overall shape and local details captured by the shape latents and latent points (Sec. 5.2).
+
+(ii) Flexibility: Another advantage of LION’s VAE framework is that its encoders can be fine-tuned for various relevant tasks, as discussed previously, and it also enables easy shape interpolation. Other 3D point cloud DDMs operating on point clouds directly [47, 46] do not offer simultaneously as much flexibility and expressivity out-of-the-box (see quantitative comparisons in Secs. 5.1 and 5.4).
+
+(iii) Mesh Reconstruction: As discussed, while point clouds are ideal for DDMs, artists likely prefer meshed outputs. As explained above, we propose to use LION together with modern surface reconstruction techniques [68], again combining the best of both worlds—a point cloud-based VAE backbone ideal for DDMs, and smooth geometry reconstruction methods operating on the synthesized point clouds to generate practically useful smooth surfaces, which can be easily transformed into meshes.
+
+![](images/3a7faca45d0a42a04bfe777486d20f82a5448e7fd0bfdf16d03a15e9ff8b2a17.jpg)  
+Figure 6: Unconditional shape generation with 2,048 points for airplane, car and chair classes (class-specific models trained on PointFlow’s ShapeNet data with global normalization).
+
+# 4 Related Work
+
+We are building on DDMs [53, 65–67], which have been used most prominently for image [53–63] and speech synthesis [86–91]. We train DDMs in latent space, an idea that has been explored for image [57, 58, 77] and music [92] generation, too. However, these works did not train separate conditional DDMs. Hierarchical DDM training has been used for generative image upsampling [54], text-to-image generation [63, 64], and semantic image modeling [60]. Most relevant among these works is Preechakul et al. [60], which extracts a high-level semantic representation of an image with an auxiliary encoder and then trains a DDM that adds details directly in image space. We are the first to explore related concepts for 3D shape synthesis and we also train both DDMs in latent space. Furthermore, DDMs and VAEs have also been combined in such a way that the DDM improves the output of the VAE [93].
+
+Most related to LION are “Point-Voxel Diffusion” (PVD) [46] and “Diffusion Probabilistic Models for 3D Point Cloud Generation” (DPM) [47]. PVD trains a DDM directly on point clouds, and our decision to use PVCNNs is inspired by this work. DPM, like LION, uses a shape latent variable, but models its distribution with Normalizing Flows [94, 95], and then trains a weaker point-wise conditional DDM directly on the point cloud data (this allows DPM to learn useful representations in its latent variable, but sacrifices generation quality). As we show below, neither PVD nor DPM easily enables applications such as multimodal voxel-conditioned synthesis and denoising. Furthermore, LION achieves significantly stronger generation performance. Finally, neither PVD nor DPM reconstructs meshes from the generated point clouds. Point cloud and 3D shape generation have also been explored with other generative models: PointFlow [31], DPF-Net [33] and SoftFlow [32] rely on Normalizing Flows [94–97]. SetVAE [29] treats point cloud synthesis as set generation and uses VAEs. ShapeGF [45] learns distributions over gradient fields that model shape surfaces. Both IM-GAN [7], which models shapes as neural fields, and l-GAN [2] train GANs over latent variables that encode the shapes, similar to other works [3], while r-GAN [2] generates point clouds directly. PDGN [52] proposes progressive deconvolutional networks within a point cloud GAN. SP-GAN [19] uses a spherical point cloud prior. Other progressive [22, 37] and graph-based architectures [4, 6] have been used, too. Also generative cellular automata (GCAs) can be employed for voxel-based 3D shape generation [43]. In orthogonal work, point cloud DDMs have been used for generative shape completion [46, 98].
+
+Recently, image-driven [8–16, 44] training of 3D generative models as well as text-driven 3D generation [34, 49–51] have received much attention. These are complementary directions to ours; in fact, augmenting LION with additional image-based training or including text-guidance are promising future directions. Finally, we are relying on SAP [68] for mesh generation. Strong alternative approaches for reconstructing smooth surfaces from point clouds exist [99–103].
+
+![](images/4415796a634b4f9ff0b2056e1bed79ad61f4bef0f5dc3c549987eef0c051b295.jpg)  
+Figure 7: Interpolating different shapes by interpolating their encodings in the standard Gaussian priors of LION’s latent DDMs (details in App. C.3).
+
+Table 1: Generation metrics (1-NNA↓) on airplane, chair, car categories from ShapeNet dataset from PointFlow [31]. Training and test data normalized globally into [-1, 1].   
+
+<table><tr><td rowspan="2"></td><td>Airplane</td><td></td><td>Chair</td><td>Car</td></tr><tr><td>CD EMD</td><td>CD</td><td>EMD</td><td>CD</td><td>EMD</td></tr><tr><td>r-GAN[2]</td><td>98.40 96.79</td><td>83.69</td><td>99.70</td><td>94.46</td><td>599.01</td></tr><tr><td>1-GAN (CD) [2]</td><td>87.30 ）93.95</td><td>68.58</td><td>83.84</td><td>66.49</td><td>88.78</td></tr><tr><td>1-GAN (EMD)[2]</td><td>89.49 76.91</td><td>71.90</td><td>64.65</td><td>71.16</td><td>66.19</td></tr><tr><td>PointFlow [31]</td><td>75.68 370.74</td><td>62.84</td><td>60.57</td><td>58.10</td><td>56.25</td></tr><tr><td>SoftFlow [32]</td><td>76.05 65.80</td><td>59.21</td><td>60.05</td><td>64.77</td><td>60.09</td></tr><tr><td>SetVAE [29]</td><td>76.54 67.65</td><td>58.84</td><td>60.57</td><td>59.94</td><td>59.94</td></tr><tr><td>DPF-Net [33]</td><td>75.18 65.55</td><td>62.00</td><td>58.53</td><td>62.35</td><td>54.48</td></tr><tr><td>DPM[47]</td><td>76.42 86.91</td><td>60.05</td><td>74.77</td><td>68.89</td><td>79.97</td></tr><tr><td>PVD [46]</td><td>73.82 64.81</td><td>56.26</td><td>53.32</td><td>54.55</td><td>53.83</td></tr><tr><td>LION (ours)</td><td>67.41 61.23</td><td>53.70</td><td>52.34</td><td></td><td>53.41 51.14</td></tr></table>
+
+Table 2: Generation results (1-NNA↓) on ShapeNet dataset from PointFlow [31]. All data normalized individually into [-1, 1].   
+
+<table><tr><td rowspan="2"></td><td>Airplane</td><td></td><td>Chair</td><td>Car</td></tr><tr><td>CD</td><td>EMD</td><td>CD EMD</td><td>CD EMD</td></tr><tr><td>TreeGAN[ [6]</td><td></td><td>97.53 99.88 88.37</td><td>96.37 89.77</td><td>94.89</td></tr><tr><td>ShapeGF[ [45]</td><td>81.23 80.86</td><td>58.01</td><td>61.25 61.79</td><td>57.24</td></tr><tr><td>SP-GAN[19]</td><td>94.69 93.95</td><td>72.58</td><td>83.69</td><td>87.36 85.94</td></tr><tr><td>PDGN [52]</td><td>94.94 91.73</td><td>71.83</td><td></td><td>79.00 89.35 87.22</td></tr><tr><td>GCA [43]</td><td>88.15 85.93</td><td>64.27</td><td>64.50 70.45</td><td>64.20</td></tr><tr><td>LION(ours)</td><td></td><td></td><td>76.30 67.04 56.50 53.85 59.52 49.29</td><td></td></tr></table>
+
+Table 3: Results (1-NNA↓) on ShapeNet-vol.   
+
+<table><tr><td rowspan="2"></td><td>Airplane</td><td>Chair</td><td>Car</td></tr><tr><td>CD EMD</td><td>CD EMD</td><td>CD EMD</td></tr><tr><td>IM-GAN[7]</td><td>79.70 77.85 57.09</td><td>58.20 88.92</td><td>84.58</td></tr><tr><td>DPM[47] PVD [46]</td><td>83.04 96.04 61.96 66.46 56.06 61.89</td><td>74.96 77.30 57.90 64.49</td><td>87.12 55.74</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td colspan="2">LION (ours) 53.47 53.84 52.07</td><td>48.67</td><td>54.81 50.53</td></tr></table>
+
+# 5 Experiments
+
+We provide an overview of our most interesting experimental results in the main paper. All experiment details and extensive additional experiments can be found in App. E and App. F, respectively.
+
+# 5.1 Single-Class 3D Shape Generation
+
+Datasets. To compare LION against existing methods, we use ShapeNet [104], the most widely used dataset to benchmark 3D shape generative models. Following previous works [31, 46, 47], we train on three categories: airplane, chair, car. Also like previous methods, we primarily rely on PointFlow’s [31] dataset splits and preprocssing. It normalizes the data globally across the whole dataset. However, some baselines require per-shape normalization [19, 43, 45, 52]; hence, we also train on such data. Furthermore, training SAP requires signed distance fields (SDFs) for volumetric supervision, which the PointFlow data does not offer. Hence, for simplicity we follow Peng et al. [68, 101] and also use their data splits and preprocessing, which includes SDFs.We train LION, DPM, PVD, and IM-GAN (which synthesizes shapes as SDFs) also on as ShapeNet-vol here). This data is also per-shape normalized.
+
+Evaluation. Model evaluation follows previous works [31, 46]. Various metrics to evaluate point cloud generative models exist, with different advantages and disadvantages, discussed in detail by Yang et al. [31]. Following recent works [31, 46], we use 1-NNA (with both Chamfer distance (CD) and earth mover distance (EMD)) as our main metric. It quantifies the distributional similarity between generated shapes and validation set and measures both quality and diversity [31]. For fair comparisons, all metrics are computed on point clouds, not meshed outputs (App. E.2 discusses different metrics; further results on coverage (COV) and minimum matching distance (MMD) in App. F.2).
+
+![](images/5b55e07dfb5406b0da8472c26a7620d4b2d2f093be889f901eda2c6ce9df0988.jpg)  
+Figure 8: Samples from our unconditional 13-class model: In each column, we use the same global shape latent $\mathbf { z } _ { 0 }$ .
+
+this dataset version (denoted Dataset details in App. E.1.
+
+Table 4: Generation results (1- NNA↓) of LION trained jointly on 13 classes of ShapeNet-vol.   
+
+<table><tr><td>Model</td><td>CD</td><td>EMD</td></tr><tr><td>TreeGAN [6]</td><td>96.80</td><td>96.60</td></tr><tr><td>PointFlow [31]</td><td>63.25</td><td>66.05</td></tr><tr><td>ShapeGF[45]</td><td>55.65</td><td>59.00</td></tr><tr><td>SetVAE [29]</td><td>79.25</td><td>95.25</td></tr><tr><td>PDGN [52]</td><td>71.05</td><td>86.00</td></tr><tr><td>DPF-Net [33]</td><td>67.10</td><td>64.75</td></tr><tr><td>DPM[47]</td><td>62.30</td><td>86.50</td></tr><tr><td>PVD [46]</td><td>58.65</td><td>57.85</td></tr><tr><td>LION (ours)</td><td>51.85</td><td>48.95</td></tr></table>
+
+Results. Samples from LION are shown in Fig. 6 and quantitative results in Tabs. 1-3 (see Sec. 4 for details about baselines—to reduce the number of baselines to train, we are focusing on the most recent and competitive ones). LION outperforms all baselines and achieves state-of-the-art performance on all classes and dataset versions. Importantly, we outperform both PVD and DPM, which also leverage DDMs, by large margins. Our samples are diverse and appear visually pleasing.
+
+![](images/9b9618423aeeac129aa5e6e358feb49f5e1ebb18eaf986c614bfbfb4c5e35683.jpg)  
+Figure 9: Generated point clouds from LION trained jointly over 55 classes of ShapeNet-vol (no conditioning).
+
+![](images/7e583fae1e920eca77cfbcfa30b8e62210c56760f2035b8269f5be44b2ae0c30.jpg)  
+Figure 10: Samples from LION trained on ShapeNet’s Mug and Bottle classes, and on Turbosquid animals.
+
+Mesh Reconstruction. As explained in Sec. 3.1, we combine LION with mesh reconstruction, to directly synthesize practically useful meshes. We show generated meshes in Fig. 2, which look smooth and of high quality. In Fig. 2, we also visually demonstrate how we can vary the local details of synthesized shapes while preserving the overall shape with our diffuse-denoise technique (Sec. 3.1). Details about the number of diffusion steps for all diffuse-denoise experiments are in App. E.
+
+Shape Interpolation. As discussed in Sec. 3.1, LION also enables shape interpolation, potentially useful for shape editing applications. We show this in Fig. 7, combined with mesh reconstruction. The generated shapes are clean and semantically plausible along the entire interpolation path. In App. F.12.1, we also show interpolations from PVD [46] and DPM [47] for comparison.
+
+# 5.2 Many-class Unconditional 3D Shape Generation
+
+13-Class LION Model. We train a LION model jointly without any class conditioning on 13 different categories (airplane, chair, car, lamp, table, sofa, cabinet, bench, telephone, loudspeaker, display, watercraft, rifle) from ShapeNet (ShapeNet-vol version). Training a single model without conditioning over such diverse shapes is challenging, as the data distribution is highly complex and multimodal. We show LION’s generated samples in Fig. 3, including meshes: LION synthesizes high-quality and diverse plausible shapes even when trained on such complex data. We report the model’s quantitative generation performance in Tab. 4, and we also trained various strong baseline methods under the same setting for comparison. We find that LION significantly outperforms all baselines by a large margin. We further observe that the hierarchical VAE architecture of LION becomes crucial: The shape latent variable $\mathbf { z } _ { 0 }$ captures global shape, while the latent points $\mathbf { h } _ { 0 }$ model details. This can be seen in Fig. 8: we show samples when fixing the global shape latent $\mathbf { z } _ { 0 }$ and only sample $\mathbf { h } _ { 0 }$ (details in App. F.3).
+
+55-Class LION Model. Encouraged by these results, we also trained a LION model again jointly without any class conditioning on all 55 different categories from ShapeNet. Note that we did on purpose not use class-conditioning in these experiments to create a difficult 3D generation task and thereby explore LION’s scalability to highly complex and multimodal datasets. We show generated point cloud samples in Fig. 9 (we did not train an SAP model on the 55 classes data): LION synthesizes high-quality and diverse shapes. It can even generate samples from the cap class, which contributes with only 39 training data samples, indicating that LION has an excellent mode coverage that even includes the very rare classes. To the best of our knowledge no previous 3D shape generative models have demonstrated satisfactory generation performance for such diverse and multimodal 3D data without relying on conditioning information (details in App. F.4). In conclusion, we observe that LION out-of-the-box easily scales to highly complex multi-category shape generation.
+
+# 5.3 Training LION on Small Datasets
+
+Next, we explore whether LION can also be trained successfully on very small datasets. To this end, we train models on the Mug and Bottle ShapeNet classes. The number of training samples is 149 and 340, respectively, which is much smaller than the common classes like chair, car and airplane. Furthermore, we also train LION on 553 animal assets from the TurboSquid data repository. Generated shapes from the three models are shown in Fig. 10. LION is able to generate correct mugs and bottles as well as diverse and high-quality animal shapes. We conclude that LION also performs well even when training in the challenging low-data setting (details in Apps. F.5 and F.6).
+
+![](images/2ade93952b2c1687a4aed247344a7e707a70e0829b0b8547f8b519ee299a389b.jpg)  
+Figure 11: Voxel-guided synthesis. We show different methods with 0 and 50 steps of diffuse-denoise. Voxelizations of generated points are also shown: Yellow boxes indicate generated points correctly fill input voxels, green boxes indicate voxels should be filled but are left empty, red boxes indicate extra voxels.
+
+![](images/a5ec7783f89ce350c646589554680fe14dc18d0cd5cd5692ac372dc06bdb76d5.jpg)  
+Figure 13: Voxel-guided generation. Quality metrics for output points (lower is better) and voxel IOU with respect to input (higher is better). $x -$ - axes denote diffuse-denoise steps.
+
+![](images/78f765294bdff9804bb3ae06cde9832dbc910c39d07cd4c4c2e5c738c6154ef6.jpg)  
+Figure 12: Reconstruction metrics with respect to clean inputs for airplane category (lower is better) when guiding synthesis with voxelized or noisy inputs (using uniform, outlier, and normal noise, see App. F.7). $_ x$ -axes denote number of diffuse-denoise steps.
+
+# 5.4 Voxel-guided Shape Synthesis and Denoising with Fine-tuned Encoders
+
+Next, we test our strategy for multimodal voxel-guided shape synthesis (see Sec. 3.1) using the airplane class LION model (experiment details in App. E, more experiments in App. F.7). We first voxelize our training set and fine-tune our encoder networks to produce the correct encodings to decode back the original shapes. When processing voxelized shapes with our point-cloud networks, we sample points on the surface of the voxels. As discussed, we can use different numbers of diffusedenoise steps in latent space to generate various plausible shapes and correct for poor encodings. Instead of voxelizations, we can also consider different noisy inputs (we use normal, uniform, and outlier noise, see App. F.7) and achieve multimodal denoising with the same approach. The same tasks can be attempted with the important DDM-based baselines PVD and DPM, by directly—not in a latent space—diffusing and denoising voxelized (converted to point clouds) or noisy point clouds.
+
+Fig. 12 shows the reconstruction performance of LION, DPM and PVD for different numbers of diffuse-denoise steps (we voxelized or noised the validation set to measure this). We see that for almost all inputs—voxelized or different noises—LION performs best. PVD and DPM perform acceptably for normal and uniform noise, which is similar to the noise injected during training of their DDMs, but perform very poorly for outlier noise or voxel inputs, which is the most relevant case to us, because voxels can be easily placed by users. It is LION’s unique framework with additional fine-tuned encoders in its VAE and only latent DDMs that makes this possible. Performing more diffuse-denoise steps means that more independent, novel shapes are generated. These will be cleaner and of higher quality, but also correspond less to the noisy or voxel inputs used for guidance. In Fig. 13, we show this trade-off for the voxel-guidance experiment (other experiments in App. F.7), where (top) we measured the outputs’ synthesis quality by calculating 1-NNA with respect to the validation set, and (bottom) the average intersection over union (IOU) between the input voxels and the voxelized outputs. We generally see a trade-off: More diffuse-denoise steps result in lower 1-NNA (better quality), but also lower IOU. LION strikes the best balance by a large gap: Its additional encoder network directly generates plausible latent encodings from the perturbed inputs that are both high quality and also correspond well to the input. This trade-off is visualized in Fig. 11 for LION, DPM, and PVD, where we show generated point clouds and voxelizations (note that performing no diffuse-denoise at all for PVD and DPM corresponds to simply keeping the input, as these models’ DDMs operate directly on point clouds). We see that running 50 diffuse-denoise steps to generate diverse outputs for DPM and especially PVD results in a significant violation of the input voxelization. In contrast, LION generates realistic outputs that also obey the driving voxels. Overall, LION wins out both in this task and also in unconditional generation with large gaps over these previous DDM-based point cloud generative models. We conclude that LION does not only offer state-of-the-art 3D shape generation quality, but is also very versatile. Note that guided synthesis can also be combined with mesh reconstruction, as shown in Fig. 4.
+
+![](images/e439f9c4b4cd464eb59193841e41c04c3f2eabc99776306355fe2b44efd72e31.jpg)  
+Figure 14: We apply Text2Mesh [49] on meshes generated by LION. In Text2Mesh, textures are generated and meshes refined such that rendered images of the 3D objects are aligned with user-provided text prompts [105].
+
+![](images/81667f186ae68d9e39acfe123525b7508d90e9671430dcd91425046a6551bf91.jpg)  
+Figure 16: Text-driven shape generation of chairs and cars with LION. Bottom row is the text prompt used as input.
+
+# 5.5 Sampling Time
+
+While our main experiments use 1,000-step DDPM-based synthesis, which takes $\approx 2 7 . 1 2$ seconds, we can significantly accelerate generation without significant loss in quality. Using DDIM-based sampling [106], we can generate high quality shapes in under one second (Fig. 15), which would enable real-time interactive applications. More analyses in App. F.9.
+
+![](images/625795a9b501d90ded76d171008f1ad0b2a315ea890e5a177dc72ef8863de109.jpg)  
+Figure 15: 25-step DDIM [106] samples (0.89 seconds per shape).
+
+# 5.6 Overview of Additional Experiments in Appendix
+
+(i) In App. F.1, we perform various ablation studies. The experiments quantitatively validate LION’s architecture choices and the advantage of our hierarchical VAE setup with conditional latent DDMs. (ii) In App. F.8, we measure LION’s autoencoding performance. (iii) To demonstrate the value of directly outputting meshes, in App. F.10 we use Text2Mesh [49] to generate textures based on text prompts for synthesized LION samples (Fig. 14). This would not be possible, if we only generated point clouds. (iv) To qualitatively show that LION can be adapted easily to other relevant tasks, in App. F.11 we condition LION on CLIP embeddings of the shapes’ rendered images, following CLIP-Forge [34] (Fig. 16). This enables text-driven 3D shape generation and single view 3D reconstruction (Fig. 17). (v) We also show many more samples (Apps. F.2-F.6) and shape interpolations (App. F.12) from our models, more examples of voxel-guided and noise-guided synthesis (App. F.7), and we further analyze our 13-class LION model (App. F.3.2).
+
+# 6 Conclusions
+
+We introduced LION, a novel generative model of 3D shapes. LION uses a VAE framework with hierarchical DDMs in latent space and can be combined with SAP for mesh generation. LION achieves state-of-the-art shape generation performance and enables applications such as voxel-conditioned synthesis, multimodal shape denoising, and shape interpolation. LION is currently trained on 3D point clouds only and can not directly generate textured shapes. A promising extension would be to include image-based training by incorporating neural or differentiable rendering [17, 107–111] and to also synthesize textures [16, 112–114]. Furthermore, LION currently focuses on single object generation only. It would be interesting to extend it to full 3D scene synthesis. Moreover, synthesis could be further accelerated by building on works on accelerated sampling from DDMs [61, 62, 67, 106, 115–121].
+
+Broader Impact. We believe that LION can potentially improve 3D content creation and assist the workflow of digital artists. We designed LION with such applications in mind and hope that it can grow into a practical tool enhancing artists’ creativity. Although we do not see any immediate negative use-cases for LION, it is important that practitioners apply an abundance of caution to mitigate impacts given generative modeling more generally can also be used for malicious purposes, discussed for instance in Vaccari and Chadwick [122], Nguyen et al. [123], Mirsky and Lee [124].
+
+![](images/2a8dd3f6899aade67630f04ae5ec6f0108def53fe8be061056ed6a9d207f21b7.jpg)  
+Figure 17: Single view 3D reconstructions of a car from an RGB image. LION can generate multiple plausible outputs using our diffuse-denoise technique.
+
+References   
+[1] Jiajun Wu, Chengkai Zhang, Tianfan Xue, Bill Freeman, and Josh Tenenbaum. Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling. In Advances in Neural Information Processing Systems, 2016. [2] Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. Learning representations and generative models for 3D point clouds. In ICML, 2018.   
+[3] Chun-Liang Li, Manzil Zaheer, Yang Zhang, Barnabas Poczos, and Ruslan Salakhutdinov. Point cloud gan. arXiv preprint arXiv:1810.05795, 2018.   
+[4] Diego Valsesia, Giulia Fracastoro, and Enrico Magli. Learning localized generative models for 3d point clouds via graph convolution. In International Conference on Learning Representations (ICLR) 2019, 2019. [5] Wenlong Huang, Brian Lai, Weijian Xu, and Zhuowen Tu. 3d volumetric modeling with introspective neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33(01), pages 8481–8488, 2019. [6] Dong Wook Shu, Sung Woo Park, and Junseok Kwon. 3d point cloud generative adversarial network based on tree structured graph convolutions. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 3859–3868, 2019.   
+[7] Zhiqin Chen and Hao Zhang. Learning implicit fields for generative shape modeling. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019. [8] Michael Niemeyer and Andreas Geiger. Giraffe: Representing scenes as compositional generative neural feature fields. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), 2021. [9] Katja Schwarz, Yiyi Liao, Michael Niemeyer, and Andreas Geiger. Graf: Generative radiance fields for 3d-aware image synthesis. In Advances in Neural Information Processing Systems (NeurIPS), 2020.   
+[10] Yiyi Liao, Katja Schwarz, Lars Mescheder, and Andreas Geiger. Towards unsupervised learning of generative models for 3d controllable image synthesis. In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), 2020.   
+[11] Eric Chan, Marco Monteiro, Petr Kellnhofer, Jiajun Wu, and Gordon Wetzstein. pi-gan: Periodic implicit generative adversarial networks for 3d-aware image synthesis. In Proc. CVPR, 2021.   
+[12] Eric R. Chan, Connor Z. Lin, Matthew A. Chan, Koki Nagano, Boxiao Pan, Shalini De Mello, Orazio Gallo, Leonidas Guibas, Jonathan Tremblay, Sameh Khamis, Tero Karras, and Gordon Wetzstein. Efficient geometry-aware 3D generative adversarial networks. In arXiv, 2021.   
+[13] Roy Or-El, Xuan Luo, Mengyi Shan, Eli Shechtman, Jeong Joon Park, and Ira Kemelmacher-Shlizerman. Stylesdf: High-resolution 3d-consistent image and geometry generation. arXiv preprint arXiv:2112.11427, 2021.   
+[14] Jiatao Gu, Lingjie Liu, Peng Wang, and Christian Theobalt. Stylenerf: A style-based 3d aware generator for high-resolution image synthesis. In International Conference on Learning Representations, 2022.   
+[15] Peng Zhou, Lingxi Xie, Bingbing Ni, and Qi Tian. CIPS-3D: A 3D-Aware Generator of GANs Based on Conditionally-Independent Pixel Synthesis. arXiv preprint arXiv:2110.09788, 2021.   
+[16] Dario Pavllo, Jonas Kohler, Thomas Hofmann, and Aurelien Lucchi. Learning generative models of textured 3d meshes from real-world images. In IEEE/CVF International Conference on Computer Vision (ICCV), 2021.   
+[17] Yuxuan Zhang, Wenzheng Chen, Huan Ling, Jun Gao, Yinan Zhang, Antonio Torralba, and Sanja Fidler. Image {gan}s meet differentiable rendering for inverse graphics and interpretable 3d neural rendering. In International Conference on Learning Representations, 2021.   
+[18] Moritz Ibing, Isaak Lim, and Leif P. Kobbelt. 3d shape generation with grid-based implicit functions. In 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021.   
+[19] Ruihui Li, Xianzhi Li, Ke-Hei Hui, and Chi-Wing Fu. SP-GAN:sphere-guided 3d shape generation and manipulation. ACM Transactions on Graphics (Proc. SIGGRAPH), 40(4), 2021.   
+[20] A. Luo, T. Li, W. Zhang, and T. Lee. Surfgen: Adversarial 3d shape synthesis with explicit surface discriminators. In 2021 IEEE/CVF International Conference on Computer Vision (ICCV), 2021.   
+[21] Zhiqin Chen, Vladimir G. Kim, Matthew Fisher, Noam Aigerman, Hao Zhang, and Siddhartha Chaudhuri. Decor-gan: 3d shape detailization by conditional refinement. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2021.   
+[22] Cheng Wen, Baosheng Yu, and Dacheng Tao. Learning progressive point embeddings for 3d point cloud generation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10266–10275, 2021.   
+[23] Zekun Hao, Arun Mallya, Serge Belongie, and Ming-Yu Liu. GANcraft: Unsupervised 3D Neural Rendering of Minecraft Worlds. In ICCV, 2021.   
+[24] Or Litany, Alex Bronstein, Michael Bronstein, and Ameesh Makadia. Deformable shape completion with graph convolutional autoencoders. CVPR, 2018.   
+[25] Qingyang Tan, Lin Gao, Yu-Kun Lai, and Shihong Xia. Variational autoencoders for deforming 3d mesh models. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5841–5850, 2018.   
+[26] Kaichun Mo, Paul Guerrero, Li Yi, Hao Su, Peter Wonka, Niloy Mitra, and Leonidas J Guibas. Structurenet: Hierarchical graph networks for 3d shape generation. arXiv preprint arXiv:1908.00575, 2019.   
+[27] Lin Gao, Jie Yang, Tong Wu, Yu-Jie Yuan, Hongbo Fu, Yu-Kun Lai, and Hao(Richard) Zhang. SDM-NET: Deep generative network for structured deformable mesh. ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH Asia 2019), 38(6):243:1–243:15, 2019.   
+[28] Lin Gao, Tong Wu, Yu-Jie Yuan, Ming-Xian Lin, Yu-Kun Lai, and Hao Zhang. Tm-net: Deep generative networks for textured meshes. ACM Transactions on Graphics (TOG), 40(6):263:1–263:15, 2021.   
+[29] Jinwoo Kim, Jaehoon Yoo, Juho Lee, and Seunghoon Hong. Setvae: Learning hierarchical composition for generative modeling of set-structured data. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 15059–15068, June 2021.   
+[30] Paritosh Mittal, Yen-Chi Cheng, Maneesh Singh, and Shubham Tulsiani. AutoSDF: Shape priors for 3d completion, reconstruction and generation. In CVPR, 2022.   
+[31] Guandao Yang, Xun Huang, Zekun Hao, Ming-Yu Liu, Serge Belongie, and Bharath Hariharan. PointFlow: 3D point cloud generation with continuous normalizing flows. In ICCV, 2019.   
+[32] Hyeongju Kim, Hyeonseung Lee, Woo Hyun Kang, Joun Yeop Lee, and Nam Soo Kim. SoftFlow: Probabilistic framework for normalizing flow on manifolds. In NeurIPS, 2020.   
+[33] Roman Klokov, Edmond Boyer, and Jakob Verbeek. Discrete point flow networks for efficient point cloud generation. In ECCV, 2020.   
+[34] Aditya Sanghi, Hang Chu, Joseph G Lambourne, Ye Wang, Chin-Yi Cheng, and Marco Fumero. Clipforge: Towards zero-shot text-to-shape generation. arXiv preprint arXiv:2110.02624, 2021.   
+[35] Yongbin Sun, Yue Wang, Ziwei Liu, Joshua E Siegel, and Sanjay E Sarma. Pointgrow: Autoregressively learned point cloud generation with self-attention. In Winter Conference on Applications of Computer Vision, 2020.   
+[36] Charlie Nash, Yaroslav Ganin, S. M. Ali Eslami, and Peter W. Battaglia. Polygen: An autoregressive generative model of 3d meshes. ICML, 2020.   
+[37] Wei-Jan Ko, Hui-Yu Huang, Yu-Liang Kuo, Chen-Yi Chiu, Li-Heng Wang, and Wei-Chen Chiu. Rpg: Learning recursive point cloud generation. arXiv preprint arXiv:2105.14322, 2021.   
+[38] Moritz Ibing, Gregor Kobsik, and Leif Kobbelt. Octree transformer: Autoregressive 3d shape generation on hierarchically structured sequences. arXiv preprint arXiv:2111.12480, 2021.   
+[39] Jianwen Xie, Zilong Zheng, Ruiqi Gao, Wenguan Wang, Zhu Song-Chun, and Ying Nian Wu. Learning descriptor networks for 3d shape synthesis and analysis. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.   
+[40] Jianwen Xie, Yifei Xu, Zilong Zheng, Ruiqi Gao, Wenguan Wang, Zhu Song-Chun, and Ying Nian Wu. Generative pointnet: Deep energy-based learning on unordered point sets for 3d generation, reconstruction and classification. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2021.   
+[41] Tianchang Shen, Jun Gao, Kangxue Yin, Ming-Yu Liu, and Sanja Fidler. Deep marching tetrahedra: a hybrid representation for high-resolution 3d shape synthesis. In Advances in Neural Information Processing Systems (NeurIPS), 2021.   
+[42] Kangxue Yin, Jun Gao, Maria Shugrina, Sameh Khamis, and Sanja Fidler. 3dstylenet: Creating 3d shapes with geometric and texture style variations. In Proceedings of International Conference on Computer Vision (ICCV), 2021.   
+[43] Dongsu Zhang, Changwoon Choi, Jeonghwan Kim, and Young Min Kim. Learning to generate 3d shapes with generative cellular automata. In International Conference on Learning Representations, 2021.   
+[44] Chen Chao, Zhizhong Han, Yu-Shen Liu, and Matthias Zwicker. Unsupervised learning of fine structure generation for 3d point clouds by 2d projection matching. arXiv preprint arXiv:2108.03746, 2021.   
+[45] Ruojin Cai, Guandao Yang, Hadar Averbuch-Elor, Zekun Hao, Serge Belongie, Noah Snavely, and Bharath Hariharan. Learning gradient fields for shape generation. In Proceedings of the European Conference on Computer Vision (ECCV), 2020.   
+[46] Linqi Zhou, Yilun Du, and Jiajun Wu. 3d shape generation and completion through point-voxel diffusion. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2021.   
+[47] Shitong Luo and Wei Hu. Diffusion probabilistic models for 3d point cloud generation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021.   
+[48] Yu Deng, Jiaolong Yang, and Xin Tong. Deformed implicit field: Modeling 3d shapes with learned dense correspondence. In IEEE Computer Vision and Pattern Recognition, 2021.   
+[49] Oscar Michel, Roi Bar-On, Richard Liu, Sagie Benaim, and Rana Hanocka. Text2mesh: Text-driven neural stylization for meshes. arXiv preprint arXiv:2112.03221, 2021.   
+[50] Ajay Jain, Ben Mildenhall, Jonathan T. Barron, Pieter Abbeel, and Ben Poole. Zero-shot text-guided object generation with dream fields. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2022.   
+[51] Nasir Khalid, Tianhao Xie, Eugene Belilovsky, and Tiberiu Popa. Text to mesh without 3d supervision using limit subdivision. arXiv preprint arXiv:2203.13333, 2022.   
+[52] Le Hui, Rui Xu, Jin Xie, Jianjun Qian, and Jian Yang. Progressive point cloud deconvolution generation network. In ECCV, 2020.   
+[53] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In Advances in Neural Information Processing Systems, 2020.   
+[54] Jonathan Ho, Chitwan Saharia, William Chan, David J Fleet, Mohammad Norouzi, and Tim Salimans. Cascaded diffusion models for high fidelity image generation. arXiv preprint arXiv:2106.15282, 2021.   
+[55] Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In International Conference on Machine Learning, 2021.   
+[56] Prafulla Dhariwal and Alexander Quinn Nichol. Diffusion models beat GANs on image synthesis. In Advances in Neural Information Processing Systems, 2021.   
+[57] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High-resolution image synthesis with latent diffusion models. arXiv preprint arXiv:2112.10752, 2021.   
+[58] Arash Vahdat, Karsten Kreis, and Jan Kautz. Score-based generative modeling in latent space. In Advances in Neural Information Processing Systems, 2021.   
+[59] Alex Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob McGrew, Ilya Sutskever, and Mark Chen. Glide: Towards photorealistic image generation and editing with text-guided diffusion models. arXiv preprint arXiv:2112.10741, 2021.   
+[60] Konpat Preechakul, Nattanat Chatthee, Suttisak Wizadwongsa, and Supasorn Suwajanakorn. Diffusion autoencoders: Toward a meaningful and decodable representation. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2022.   
+[61] Tim Dockhorn, Arash Vahdat, and Karsten Kreis. Score-based generative modeling with critically-damped langevin diffusion. In International Conference on Learning Representations (ICLR), 2022.   
+[62] Zhisheng Xiao, Karsten Kreis, and Arash Vahdat. Tackling the generative learning trilemma with denoising diffusion GANs. In International Conference on Learning Representations (ICLR), 2022.   
+[63] Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen. Hierarchical text-conditional image generation with clip latents. arXiv preprint arXiv:2204.06125, 2022.   
+[64] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, S. Sara Mahdavi, Rapha Gontijo Lopes, Tim Salimans, Jonathan Ho, David J Fleet, and Mohammad Norouzi. Photorealistic text-to-image diffusion models with deep language understanding. arXiv preprint arXiv:2205.11487, 2022.   
+[65] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International Conference on Machine Learning, 2015.   
+[66] Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. In Proceedings of the 33rd Annual Conference on Neural Information Processing Systems, 2019.   
+[67] Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021.   
+[68] Songyou Peng, Chiyu "Max" Jiang, Yiyi Liao, Michael Niemeyer, Marc Pollefeys, and Andreas Geiger. Shape as points: A differentiable poisson solver. In Advances in Neural Information Processing Systems (NeurIPS), 2021.   
+[69] Diederik P Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models. In Advances in Neural Information Processing Systems, 2021.   
+[70] Diederik P Kingma and Max Welling. Auto-encoding variational bayes. In The International Conference on Learning Representations, 2014.   
+[71] Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In International Conference on Machine Learning, 2014.   
+[72] Jakub Tomczak and Max Welling. Vae with a vampprior. In International Conference on Artificial Intelligence and Statistics, pages 1214–1223, 2018.   
+[73] Hiroshi Takahashi, Tomoharu Iwata, Yuki Yamanaka, Masanori Yamada, and Satoshi Yagi. Variational autoencoder with implicit optimal priors. Proceedings of the AAAI Conference on Artificial Intelligence, 33(01):5066–5073, Jul. 2019.   
+[74] Matthias Bauer and Andriy Mnih. Resampled priors for variational autoencoders. In Kamalika Chaudhuri and Masashi Sugiyama, editors, Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, volume 89 of Proceedings of Machine Learning Research, pages 66–75. PMLR, 16–18 Apr 2019.   
+[75] Arash Vahdat and Jan Kautz. NVAE: A deep hierarchical variational autoencoder. In Advances in Neural Information Processing Systems, 2020.   
+[76] Jyoti Aneja, Alexander Schwing, Jan Kautz, and Arash Vahdat. NCP-VAE: Variational autoencoders with noise contrastive priors. In Advances in Neural Information Processing Systems, 2021.   
+[77] Abhishek Sinha, Jiaming Song, Chenlin Meng, and Stefano Ermon. D2c: Diffusion-denoising models for few-shot conditional generation. In Advances in Neural Information Processing Systems, 2021.   
+[78] Mihaela Rosca, Balaji Lakshminarayanan, and Shakir Mohamed. Distribution matching in variational inference. arXiv preprint arXiv:1802.06847, 2018.   
+[79] Matthew D Hoffman and Matthew J Johnson. Elbo surgery: yet another way to carve up the variational evidence lower bound. In Workshop in Advances in Approximate Bayesian Inference, NeurIPS, 2016.   
+[80] Zhijian Liu, Haotian Tang, Yujun Lin, and Song Han. Point-voxel cnn for efficient 3d deep learning. In Advances in Neural Information Processing Systems, 2019.   
+[81] Charles R Qi, Hao Su, Kaichun Mo, and Leonidas J Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017.   
+[82] Charles R Qi, Li Yi, Hao Su, and Leonidas J Guibas. Pointnet++: Deep hierarchical feature learning on point sets in a metric space. In Advances in Neural Information Processing Systems, 2017.   
+[83] Kaiming He, X. Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.   
+[84] Yuxin Wu and Kaiming He. Group normalization. In Proceedings of the European Conference on Computer Vision (ECCV), 2018.   
+[85] Chenlin Meng, Yutong He, Yang Song, Jiaming Song, Jiajun Wu, Jun-Yan Zhu, and Stefano Ermon. SDEdit: Guided image synthesis and editing with stochastic differential equations. In International Conference on Learning Representations, 2022.   
+[86] Nanxin Chen, Yu Zhang, Heiga Zen, Ron J Weiss, Mohammad Norouzi, and William Chan. Wavegrad: Estimating gradients for waveform generation. In International Conference on Learning Representations (ICLR), 2021.   
+[87] Zhifeng Kong, Wei Ping, Jiaji Huang, Kexin Zhao, and Bryan Catanzaro. DiffWave: A Versatile Diffusion Model for Audio Synthesis. In International Conference on Learning Representations, 2021.   
+[88] Myeonghun Jeong, Hyeongju Kim, Sung Jun Cheon, Byoung Jin Choi, and Nam Soo Kim. Diff-tts: A denoising diffusion model for text-to-speech. arXiv preprint arXiv:2104.01409, 2021.   
+[89] Nanxin Chen, Yu Zhang, Heiga Zen, Ron J. Weiss, Mohammad Norouzi, Najim Dehak, and William Chan. Wavegrad 2: Iterative refinement for text-to-speech synthesis. arXiv preprint arXiv:2106.09660, 2021.   
+[90] Vadim Popov, Ivan Vovk, Vladimir Gogoryan, Tasnima Sadekova, and Mikhail Kudinov. Grad-tts: A diffusion probabilistic model for text-to-speech. In International Conference on Machine Learning, 2021.   
+[91] Songxiang Liu, Dan Su, and Dong Yu. Diffgan-tts: High-fidelity and efficient text-to-speech with denoising diffusion gans. arXiv preprint arXiv:2201.11972, 2022.   
+[92] Gautam Mittal, Jesse Engel, Curtis Hawthorne, and Ian Simon. Symbolic music generation with diffusion models. In Proceedings of the 22nd International Society for Music Information Retrieval Conference, 2021.   
+[93] Kushagra Pandey, Avideep Mukherjee, Piyush Rai, and Abhishek Kumar. Diffusevae: Efficient, controllable and high-fidelity generation from low-dimensional latents. arXiv preprint arXiv:2201.00308, 2022.   
+[94] Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. In International Conference on Machine Learning, 2015.   
+[95] Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real NVP. In International Conference on Learning Representations ICLR, 2017.   
+[96] Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 2018.   
+[97] Will Grathwohl, Ricky T. Q. Chen, Jesse Bettencourt, Ilya Sutskever, and David Duvenaud. FFJORD: Free-form continuous dynamics for scalable reversible generative models. In International Conference on Learning Representations, 2019.   
+[98] Zhaoyang Lyu, Zhifeng Kong, Xudong XU, Liang Pan, and Dahua Lin. A conditional point diffusionrefinement paradigm for 3d point cloud completion. In International Conference on Learning Representations (ICLR), 2022.   
+[99] Thibault Groueix, Matthew Fisher, Vladimir G. Kim, Bryan Russell, and Mathieu Aubry. AtlasNet: A Papier-Mâché Approach to Learning 3D Surface Generation. In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), 2018.   
+[100] Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger. Occupancy networks: Learning 3d reconstruction in function space. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4460–4470, 2019.   
+[101] Songyou Peng, Michael Niemeyer, Lars Mescheder, and Andreas Geiger Marc Pollefeys. Convolutional occupancy networks. In European Conference on Computer Vision (ECCV), 2020.
+
+[102] Francis Williams, Matthew Trager, Joan Bruna, and Denis Zorin. Neural splines: Fitting 3d surfaces with infinitely-wide neural networks. In 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021.
+
+[103] Francis Williams, Zan Gojcic, Sameh Khamis, Denis Zorin, Joan Bruna, Sanja Fidler, and Or Litany. Neural fields as learnable kernels for 3d reconstruction. arXiv preprint arXiv:2111.13674, 2021.
+
+[104] Angel X. Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, Jianxiong Xiao, Li Yi, and Fisher Yu. ShapeNet: An Information-Rich 3D Model Repository. Technical Report arXiv:1512.03012 [cs.GR], Stanford University — Princeton University — Toyota Technological Institute at Chicago, 2015.
+
+[105] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. Learning transferable visual models from natural language supervision. In International Conference on Machine Learning, ICML, 2021.
+
+[106] Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In International Conference on Learning Representations, 2021.
+
+[107] Wenzheng Chen, Huan Ling, Jun Gao, Edward Smith, Jaakko Lehtinen, Alec Jacobson, and Sanja Fidler. Learning to predict 3d objects with an interpolation-based differentiable renderer. In Advances in Neural Information Processing Systems, 2019.
+
+[108] Samuli Laine, Janne Hellsten, Tero Karras, Yeongho Seol, Jaakko Lehtinen, and Timo Aila. Modular primitives for high-performance differentiable rendering. ACM Transactions on Graphics, 2020.
+
+[109] Ben Mildenhall, Pratul P. Srinivasan, Matthew Tancik, Jonathan T. Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. In ECCV, 2020.
+
+[110] Wenzheng Chen, Joey Litalien, Jun Gao, Zian Wang, Clement Fuji Tsang, Sameh Khamis, Or Litany, and Sanja Fidler. DIB- $\mathbf { \cdot R } { + + }$ : Learning to predict lighting and material with a hybrid differentiable renderer. In Advances in Neural Information Processing Systems (NeurIPS), 2021.
+
+[111] Ayush Tewari, Justus Thies, Ben Mildenhall, Pratul Srinivasan, Edgar Tretschk, Yifan Wang, Christoph Lassner, Vincent Sitzmann, Ricardo Martin-Brualla, Stephen Lombardi, Tomas Simon, Christian Theobalt, Matthias Niessner, Jonathan T. Barron, Gordon Wetzstein, Michael Zollhoefer, and Vladislav Golyanik. Advances in neural rendering. arXiv preprint arXiv:2111.05849, 2021.
+
+[112] Michael Oechsle, Lars Mescheder, Michael Niemeyer, Thilo Strauss, and Andreas Geiger. Texture fields: Learning texture representations in function space. In Proceedings IEEE International Conf. on Computer Vision (ICCV), 2019.
+
+[113] Zhiqin Chen, Kangxue Yin, and Sanja Fidler. Auv-net: Learning aligned uv maps for texture transfer and synthesis. In The Conference on Computer Vision and Pattern Recognition (CVPR), 2022.
+
+[114] Yawar Siddiqui, Justus Thies, Fangchang Ma, Qi Shan, Matthias Nießner, and Angela Dai. Texturify: Generating textures on 3d shape surfaces. arXiv preprint arXiv:2204.02411, 2022.
+
+[115] Daniel Watson, Jonathan Ho, Mohammad Norouzi, and William Chan. Learning to efficiently sample from diffusion probabilistic models. arXiv preprint arXiv:2106.03802, 2021.
+
+[116] Zhifeng Kong and Wei Ping. On fast sampling of diffusion probabilistic models. arXiv preprint arXiv:2106.00132, 2021.
+
+[117] Alexia Jolicoeur-Martineau, Ke Li, Rémi Piché-Taillefer, Tal Kachman, and Ioannis Mitliagkas. Gotta Go Fast When Generating Data with Score-Based Models. arXiv preprint arXiv:2105.14080, 2021.
+
+[118] Daniel Watson, William Chan, Jonathan Ho, and Mohammad Norouzi. Learning Fast Samplers for Diffusion Models by Differentiating Through Sample Quality. In International Conference on Learning Representations, 2022.
+
+[119] Luping Liu, Yi Ren, Zhijie Lin, and Zhou Zhao. Pseudo numerical methods for diffusion models on manifolds. In International Conference on Learning Representations (ICLR), 2022.
+
+[120] Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling of diffusion models. In International Conference on Learning Representations (ICLR), 2022.
+
+[121] Fan Bao, Chongxuan Li, Jun Zhu, and Bo Zhang. Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models. In International Conference on Learning Representations, 2022.   
+[122] Cristian Vaccari and Andrew Chadwick. Deepfakes and disinformation: Exploring the impact of synthetic political video on deception, uncertainty, and trust in news. Social Media $^ +$ Society, 6(1): 2056305120903408, 2020.   
+[123] Thanh Thi Nguyen, Quoc Viet Hung Nguyen, Cuong M. Nguyen, Dung Nguyen, Duc Thanh Nguyen, and Saeid Nahavandi. Deep learning for deepfakes creation and detection: A survey. arXiv preprint arXiv:1909.11573, 2021.   
+[124] Yisroel Mirsky and Wenke Lee. The creation and detection of deepfakes: A survey. ACM Comput. Surv., 54(1), 2021.   
+[125] Merlin Nimier-David, Delio Vicini, Tizian Zeltner, and Wenzel Jakob. Mitsuba 2: a retargetable forward and inverse renderer. ACM Transactions on Graphics, 38:1–17, 11 2019. doi: 10.1145/3355089.3356498.   
+[126] Jonathan Krause, Michael Stark, Jia Deng, and Li Fei-Fei. 3d object representations for fine-grained categorization. In 4th International IEEE Workshop on 3D Representation and Recognition (3dRR-13), Sydney, Australia, 2013.   
+[127] Sungjoon Choi, Qian-Yi Zhou, Stephen Miller, and Vladlen Koltun. A large dataset of object scans. arXiv:1602.02481, 2016.   
+[128] Xingyuan Sun, Jiajun Wu, Xiuming Zhang, Zhoutong Zhang, Chengkai Zhang, Tianfan Xue, Joshua B Tenenbaum, and William T Freeman. Pix3d: Dataset and methods for single-image 3d shape modeling. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.   
+[129] Brian DO Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12(3):313–326, 1982.   
+[130] Ulrich G Haussmann and Etienne Pardoux. Time reversal of diffusions. The Annals of Probability, pages 1188–1205, 1986.   
+[131] Pascal Vincent. A connection between score matching and denoising autoencoders. Neural Computation, 23(7):1661–1674, 2011.   
+[132] Radford M Neal and Geoffrey E Hinton. A view of the em algorithm that justifies incremental, sparse, and other variants. In Learning in graphical models, pages 355–368. Springer, 1998.   
+[133] Diederik P. Kingma and Max Welling. An introduction to variational autoencoders. Foundations and Trends in Machine Learning, 12(4):307–392, 2019.   
+[134] J. R. Dormand and P. J. Prince. A family of embedded runge–kutta formulae. Journal of Computational and Applied Mathematics, 6(1):19–26, 1980.   
+[135] Michael Kazhdan, Matthew Bolitho, and Hugues Hoppe. Poisson surface reconstruction. In Proceedings of the fourth Eurographics symposium on Geometry processing, volume 7, 2006.   
+[136] William E. Lorensen and Harvey E. Cline. Marching cubes: A high resolution 3d surface construction algorithm. In Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’87, page 163–169, New York, NY, USA, 1987. Association for Computing Machinery.   
+[137] Jie Hu, Li Shen, and Gang Sun. Squeeze-and-excitation networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7132–7141, 2018.   
+[138] Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral normalization for generative adversarial networks. In International Conference on Learning Representations, 2018.   
+[139] Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(11), 2008.   
+[140] Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E. Sarma, Michael M. Bronstein, and Justin M. Solomon. Dynamic graph cnn for learning on point clouds. ACM Transactions on Graphics (TOG), 2019.   
+[141] Hengshuang Zhao, Li Jiang, Jiaya Jia, Philip HS Torr, and Vladlen Koltun. Point transformer. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 16259–16268, 2021.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] Please see Sec. 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] Please see Sec. 6.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] We did not derive novel theoretical results. We rather propose a novel generative model of 3D shapes.   
+(b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] We will release code and instructions to reproduce all experiments upon acceptance of the manuscript. The internal guidelines of our institution prevent us from releasing code at this stage.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We provide all model training and evaluation details in the App. D, including all hyperparameters.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Following standard conventions in the related 3D generative modeling literature, we do not report error bars. Furthermore, we avoid running similar setups repeatedly to save computational resources (our main models are quite large and require substantial GPU resources for training, see App. E.9).   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Please see App. E.9.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] For baseline comparisons we run publicly available code from previous, publicly available papers, which we cite. We also use various datasets. Here, we present a summary. To compare to baselines, we use the following codes: • r-GAN, l-GAN [2]: https://github.com/optas/latent_3d_points (MIT License) • PointFlow [31]: https://github.com/stevenygd/PointFlow (MIT License) • SoftFlow [32]: https://github.com/ANLGBOY/SoftFlow • Set-VAE [29]: https://github.com/jw9730/setvae (MIT License) • DPF-NET [33]: https://github.com/Regenerator/dpf-nets • DPM [47]: https://github.com/luost26/diffusion-point-cloud (MIT License) • PVD [46]: https://github.com/alexzhou907/PVD (MIT License) • ShapeGF [45]: https://github.com/RuojinCai/ShapeGF (MIT License) • SP-GAN [19]: https://github.com/liruihui/sp-gan (MIT License) • PDGN [52]: https://github.com/fpthink/PDGN (MIT License) • IM-GAN [7]: https://github.com/czq142857/implicit-decoder (MIT license) and https://github.com/czq142857/IM-NET-pytorch (MIT license) • GCA [43]: https://github.com/96lives/gca (MIT license)
+
+We use further codebases in other places:
+
+• We use the MitSuba renderer for visualizations [125]: https: //github.com/mitsuba-renderer/mitsuba2 (License: https: //github.com/mitsuba-renderer/mitsuba2/blob/master/LICENSE),
+
+and the code to generate the scene discription files for MitSuba [31]: https://github.com/zekunhao1995/PointFlowRenderer.   
+• We rely on SAP [68] for mesh generation with the code at https://github.com/ autonomousvision/shape_as_points (MIT License).   
+• For calculating the evaluation metrics, we use the implementation for CD at https: //github.com/ThibaultGROUEIX/ChamferDistancePytorch (MIT License) and for EMD at https://github.com/daerduoCarey/PyTorchEMD.   
+• We use Text2Mesh [49] for per-sample text-driven texture synthesis: https:// github.com/threedle/text2mesh (MIT License)
+
+We also rely on the following datasets:
+
+• ShapeNet [104]. Its terms of use can be found at https://shapenet.org/ terms.   
+• The Cars dataset [126] from http://ai.stanford.edu/\~jkrause/cars/car_ dataset.html with ImageNet License: https://image-net.org/download. php.   
+• The TurboSquid data repository, https://www.turbosquid.com. We obtained a custom license from TurboSquid.   
+• Redwood 3DScan Dataset [127]: https://github.com/isl-org/ redwood-3dscan (Public Domain)   
+• Pix3D [128]: https://github.com/xingyuansun/pix3d. (Creative Commons Attribution 4.0 International License).   
+(b) Did you mention the license of the assets? [Yes] In App. E.8, we mention the licenses of the codes and other assets we are using.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] We are primarily using the publicly available ShapeNet [104] dataset, which has been widely used in the generative modeling literature as standard benchmark. It only consists of simple 3D models of shapes such as airplanes, chairs, cars, etc.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/xijYyYFlRIf/xijYyYFlRIf.md

@@ -0,0 +1,296 @@
+# GAUDI: A Neural Architect for Immersive 3D Scene Generation
+
+Miguel Angel Bautista⇤ Pengsheng Guo⇤ Samira Abnar Walter Talbott
+
+# Alexander Toshev Zhuoyuan Chen Laurent Dinh Shuangfei Zhai Hanlin Goh
+
+Daniel Ulbricht
+
+Afshin Dehghan
+
+Josh Susskind
+
+Apple https://github.com/apple/ml-gaudi
+
+# Abstract
+
+We introduce GAUDI, a generative model capable of capturing the distribution of complex and realistic 3D scenes that can be rendered immersively from a moving camera. We tackle this challenging problem with a scalable yet powerful approach, where we first optimize a latent representation that disentangles radiance fields and camera poses. This latent representation is then used to learn a generative model that enables both unconditional and conditional generation of 3D scenes. Our model generalizes previous works that focus on single objects by removing the assumption that the camera pose distribution can be shared across samples. We show that GAUDI obtains state-of-the-art performance in the unconditional generative setting across multiple datasets and allows for conditional generation of 3D scenes given conditioning variables like sparse image observations or text that describes the scene.
+
+# 1 Introduction
+
+In order for learning systems to be able to understand and create 3D spaces, progress in generative models for 3D is sorely needed. The quote "The creation continues incessantly through the media of man." is often attributed to Antoni Gaudí, who we pay homage to with our method’s name. In this paper we ask the question: can creation continue through the media of learning machines? We are interested in generative models that can capture the distribution of 3D scenes and then render views from scenes sampled from the learned distribution. Extensions of such generative models to conditional inference problems could have tremendous impact in a wide range of tasks in machine learning and computer vision. For example, one could sample plausible scene completions that are consistent with an image observation, or a text description (see Fig. 1 for 3D scenes sampled from GAUDI). In addition, such models would be of great practical use in model-based reinforcement learning and planning [14], SLAM [43], or 3D content creation.
+
+Recent works on generative modeling for 3D objects or scenes [60, 6, 8] employ a Generative Adversarial Network (GAN) where the generator explicitly encodes radiance fields — a parametric function that takes as input the coordinates of a point in 3D space and camera pose, and outputs a density scalar and RGB value for that 3D point. Images can be rendered from the radiance field generated by the model by passing the queried 3D points through the volume rendering equation to project onto any 2D camera view. While compelling on small or simple 3D datasets (e.g. single objects or a small number of indoor scenes), GANs suffer from training pathologies including mode collapse $\pm \pm \pm \pm \pm \pm \pm \pm \pm$ and are difficult to train on data for which a canonical coordinate system does not exist, as is the case for 3D scenes $\left[ \left[ 6 1 \right] \right]$ . In addition, one key difference between modeling distributions of 3D objects vs. scenes is that when modeling objects, the distribution of valid camera poses does not depend on each object and is defined per dataset (i.e. typically as $S O ( 3 )$ ), which is not true for scenes. See Fig. $\bar { \bigtriangledown ( \mathfrak { b } ) }$ and note how the two scene layouts define different areas of navigable space (different dark grey shaded areas) where valid camera poses can be placed, revealing a strong dependency between scenes and their camera pose distributions. GAUDI captures this dependency by modeling the joint distribution of scenes and camera poses.
+
+![](images/817e9e0b07bda48800f0017cb652036d1cffe99dbb4fab43512b87d33c1bfd6f.jpg)  
+Figure 1: GAUDI allows to model both conditional and unconditional distributions over complex 3D scenes. Sampled scenes and poses from (left) the unconditional distribution, and (right) a distribution conditioned on an image observation or a text prompt.
+
+In GAUDI, we map each trajectory (i.e. a sequence of posed images from a 3D scene) into a latent representation that encodes a radiance field (e.g. the 3D scene) and camera path in a completely disentangled way. We find these latent representations by interpreting them as free parameters and formulating an optimization problem where the latent representation for each trajectory is optimized via a reconstruction objective. This simple training process is scalable to thousands of trajectories. Interpreting the latent representation of each trajectory as a free parameter also makes it simple to handle a large and variable number of views for each trajectory rather than requiring a sophisticated encoder architecture to pool across a large number of views. After optimizing latent representations for an observed empirical distribution of trajectories, we learn a generative model over the set of latent representations. In the unconditional case, the model can sample radiance fields entirely from the prior distribution learned by the model, allowing it to synthesize scenes by interpolating within the latent space. In the conditional case, conditional variables available to the model at training time (e.g. images, text prompts, etc.) can be used to generate radiance fields consistent with those variables. Our contributions can be summarized as:
+
+We scale 3D scene generation to thousands of indoor scenes containing hundreds of thousands of images, without suffering from mode collapse or canonical orientation issues during training.
+
+• We introduce a novel denoising optimization objective to find latent representations that jointly model a radiance field and the camera poses in a disentangled manner.
+
+• Our approach obtains state-of-the-art generation performance across multiple datasets.
+
+• Our approach allows for various generative setups: unconditional generation as well as conditional on images or text.
+
+# 2 Related Work
+
+In recent years the field has witnessed outstanding progress in generative modeling for the 2D image domain, with most approaches focusing either on adversarial $\mathbb { \left[ \left[ 2 1 \right] , \left[ 2 2 \right] \right] }$ or auto-regressive models [69, 46, 11]. More recently, score matching based approaches [18, 62] have gained popularity. In particular, Denoising Diffusion Probabilistic Models (DDPMs) [ $\boxed { 1 7 } \boxed { 3 7 } \boxed { 5 2 } \boxed { 6 8 }$ have emerged as strong contenders to both adversarial and auto-regressive approaches. In DDPMs, the goal is to learn a step-by-step inversion of a fixed diffusion Markov Chain that gradually transforms an empirical data distribution to a fixed posterior, which typically takes the form of an isotropic Gaussian distribution. In parallel, the last couple of years have seen a revolution in how 3D data is represented within neural networks. By representing a 3D scene as a radiance field, NeRF $[ \left| 3 3 \right| ]$ introduces an approach to optimize the weights of a MLP to represent the radiance of 3D points that fall inside the field-of-view of a given set of posed RGB images. Given the radiance for a set of 3D points that lie on a ray shot from a given camera pose, NeRF $\mathbb { B } 3 \mathbb { I }$ uses volumetric rendering to compute the color for the corresponding pixel and optimizes the MLP weights via a reconstruction loss in image space.
+
+A few attempts have also been made at incorporating a radiance field representation within generative models. Most approaches have focused on the problem of single objects with known canonical orientations like faces or Shapenet objects with shared camera pose distributions across samples in a dataset [60, 6, 38, 25, 5, 12, 75, 47, 10]. Extending these approaches from single objects to completely unconstrained 3D scenes is an unsolved problem. One paper worth mentioning in this space is GSN $\textcircled { 8 }$ , which breaks the radiance field into a grid of local radiance fields that collectively represent a scene. While this decomposition of radiance fields endows the model with high representational capacity, GSN still suffers from the standard training pathologies of GANs, like mode collapse $\mathbb { \lVert \rVert }$ , which are exacerbated by the fact that unconstrained 3D scenes do not have a canonical orientation. As we show in our experiments (cf. Sect. 4), these issues become prominent as the training set size increases, impacting the capacity of the generative model to capture complex distributions. Separately, a line of recent approaches have also studied the problem of learning generative models of scenes without employing radiance fields [40, 70, 51]. These works assume that the model has access to room layouts and a database of object CAD models during training, simplifying the problem of scene generation to a selection of objects from the database and pose predictions for each object.
+
+Finally, approaches that learn to predict a target view given a single (or multiple) source view and relative pose transformation have been recently proposed [27, 74, 57, 9, 13]. The pure reconstruction objective employed by these approaches forces them to learn a deterministic conditional function that maps a source image and a relative camera transformation to a target image. The first is that this scene completion problem is ill-posed (e.g. given a single source view of a scene there are multiple target completions that are equally likely). Attempts at modeling the problem in a probabilistic manner have been proposed $\underline { { \lVert 5 3 \rVert } } , \boxed { 4 9 }$ . However, these approaches suffer from inconsistency in predicted scenes because they do not explicitly model a 3D consistent representation like a radiance field.
+
+# 3 GAUDI
+
+Our goal is to learn a generative model given an empirical distribution of trajectories over 3D scenes. Let $X = \{ x _ { i \in \{ 0 , . . . , n \} } \}$ denote a collection of examples defining an empirical distribution, where each example $x _ { i }$ is a trajectory. Every trajectory $x _ { i }$ is defined as a variable length sequence of corresponding RGB, depth images and 6DOF camera poses (see Fig. 3).
+
+We decompose the task of learning a generative model in two stages. First, we obtain a latent representation $\mathbf { z } = [ \mathbf { z } _ { \mathrm { s c e n e } } , \mathbf { z } _ { \mathrm { p o s e } } ]$ for each example $x \in X$ that represents the scene radiance field and pose in separate disentangled vectors. Second, given a set of latents $Z = \{ \mathbf { z } _ { i \in \{ 0 , \dots , n \} } \}$ we learn the distribution $p ( Z )$ .
+
+# 3.1 Optimizing latent representations for radiance fields and camera poses
+
+We now turn to the task of finding a latent representation $\mathbf { z } \in Z$ for each example $x \in X$ (i.e. for each trajectory in the empirical distribution). To obtain this latent representation we take an encoder-less view and interpret $\mathbf { z }$ ’s as free parameters to be found via an optimization problem $\mathbb { D } \mathbb { B }$ . To map latents $\mathbf { z }$ to trajectories $x$ , we design a network architecture (i.e. a decoder) that disentangles camera poses and radiance field parameterization. Our decoder architecture is composed of 3 networks (shown in Fig. 2):
+
+• The camera pose decoder network $c$ (parameterized by $\theta _ { c . }$ ), is responsible for predicting camera poses $\hat { \mathbf { T } } _ { s } \in S E ( 3 )$ at the normalized temporal position $s \in [ - 1 , 1 ]$ in the trajectory, conditioned on $\mathbf { z } _ { \mathrm { p o s e } }$ which represents the camera poses for the whole trajectory. To ensure that the output of $c$ is a valid camera pose (e.g. an element of $S E ( 3 )$ ), we output a 3D vector representing a normalized quaternion $\mathbf { q } _ { s }$ for the orientation and a 3D translation vector $\mathbf { t } _ { s }$ .
+
+![](images/e4bad21ddfcad7a7125072fe03e7aa0257fb5650fce1690227c0a92664eaf018.jpg)  
+Figure 2: Architecture of the decoder model that disentangles camera poses from 3D geometry and appearance of the scene. Our decoder is composed by 3 submodules. A decoder $d$ that takes as input a latent code representing the scene $\mathbf { z } _ { \mathrm { s c e n e } }$ and produces a factorized representation of 3D space via a tri-plane latent encoding $\mathbf { W }$ . A radiance field network $f$ that takes as input points $ { \mathbf { p } } \in  { \mathbf { R } } ^ { 3 }$ and is conditioned on W to predict a density $\sigma$ and a signal a to be rendered via volumetric rendering (Eq. 1). Finally, we decode the camera poses through a network $c$ that takes as input a normalized temporal position $s \in [ - 1 , 1 ]$ and is conditioned on $\mathbf { z } _ { \mathrm { p o s e } }$ which represents camera poses for the whole trajectory $x$ to predict the camera pose $\hat { \mathbf { T } } _ { s } \in S E ( 3 )$ .
+
+• The scene decoder network $d$ (parameterized by $\theta _ { d }$ ), is responsible for predicting a conditioning variable for the radiance field network $f$ . This network takes as input a latent code that represents the scene $\mathbf { z } _ { \mathrm { s c e n e } }$ and predicts an axis-aligned tri-plane representation $[ \mathbb { A } 1 ] , \bigstar ]$ $\mathbf { W } \in \mathbb { R } ^ { 3 \times S \times S ^ { \star } \times F }$ . Which correspond to 3 feature maps $[ \mathbf { W } _ { x y } , \mathbf { \bar { W } } _ { x z } , \mathbf { \bar { W } } _ { y z } ]$ of spatial dimension $S \times S$ and $F$ channels, one for each axis aligned plane: $x y$ , $x z$ and $y z$ .
+
+• The radiance field decoder network $f$ (parameterized by $\theta _ { f }$ ), is tasked with reconstructing image level targets using the volumetric rendering equation in Eq. $^ { 1 }$ . The input to $f$ is $ { \mathbf { p } } \in \mathbb { R } ^ { 3 }$ and the tri-plane representation $\mathbf { W } = [ \mathbf { W } _ { x y } , \mathbf { W } _ { x z } , \mathbf { \bar { W } } _ { y z } ]$ (we do not condition on the camera orientation to improve consistency $\pmb { \mathbb { I } } \mathbf { \varOmega }$ ). Given a 3D point $\overset { \circ } { \mathbf { p } } = [ i , j , k ]$ for which radiance is to be predicted, we orthogonally project $\mathbf { p }$ into each plane in W and perform bi-linear sampling. We concatenate the 3 bi-linearly sampled vectors into $\mathbf { \bar { w } } _ { x y z } = [ \mathbf { W } _ { x y } ( i , j ) , \mathbf { W } _ { x z } ( j , k ) , \mathbf { W } _ { y z } ( i , k ) ] \in \mathbb { R } ^ { 3 F }$ , which is used to condition the radiance field function $f$ . We implement $f$ as a MLP that outputs a density value $\sigma$ and a signal a. To predict the value $\mathbf { v }$ of a pixel, the volumetric rendering equation is used (cf. Eq. 1) where a 3D point is expressed as ray direction $\mathbf { r }$ (corresponding with the pixel location) at particular depth $u$ .
+
+$$
+\begin{array} { r l r } & { } & { { \bf v } ( { \bf r } , { \bf W } ) = \int _ { u _ { n } } ^ { u _ { f } } T r ( u ) \sigma \left( { \bf r } ( u ) , { \bf w } _ { x y z } \right) { \bf a } \left( { \bf r } ( u ) , { \bf w } _ { x y z } \right) d u } \\ & { } & { T r ( u ) = \exp \left( - \int _ { u _ { n } } ^ { u } \sigma ( { \bf r } ( u ) , { \bf w } _ { x y z } ) d u \right) . } \end{array}
+$$
+
+We formulate a reconstruction objective to jointly optimize for $\theta _ { d } , \theta _ { c } , \theta _ { f }$ and $\{ \mathbf { z } \} _ { i = \{ 0 , \ldots , n \} }$ , shown in Eq. 2. Note that while latents $\mathbf { z }$ are optimized for each example $x$ independently, the parameters of the networks $\theta _ { d } , \theta _ { c } , \theta _ { f }$ are amortized across all examples $x \in X$ . As opposed to previous auto-decoding approaches $\textcircled { 1 2 } , \textcircled { 3 9 } \textcircled { 1 }$ , each latent $\mathbf { z }$ is perturbed during training with additive noise that is proportional to the empirical standard deviation across all latents, $\mathbf { z } = \mathbf { z } + \beta \mathcal { N } ( 0 , \mathrm { s t d } ( Z ) )$ , inducing a contractive representation $\left[ \left[ 5 0 \right] \right]$ . In this setting, $\beta$ controls the trade-off between the entropy of the distribution of latents $\mathbf { z } \in Z$ and the reconstruction quality. With $\beta = 0$ the distribution of $\mathbf { z }$ ’s becomes a set of indicator functions (i.e. similar as one would get from a vanilla auto-encoder). For large $\beta > 0$ the structure in distribution of $\mathbf { z }$ ’s is destroyed, as latents are perturbed with large magnitudes of noise. We use a small $\beta > 0$ value to enforce a latent space in which interpolated samples (or samples that contain small deviations from the empirical distribution, as the ones that one might get from sampling a subsequent generative model) are included in the support of the decoder function, sacrificing a small cost in reconstruction fidelity.
+
+$$
+\operatorname* { m i n } _ { \theta _ { d } , \theta _ { f } , \theta _ { c } , Z } \mathbb { E } _ { x \sim X } \left[ \mathcal { L } _ { \mathrm { s c e n e } } ( \mathbf { x } _ { s } ^ { \mathrm { i m } } , \mathbf { z } _ { \mathrm { s c e n e } } , \mathbf { T } _ { s } ) + \lambda \mathcal { L } _ { \mathrm { p o s e } } ( \mathbf { T } _ { s } , \mathbf { z } _ { \mathrm { p o s e } } , s ) \right]
+$$
+
+We optimize parameters $\theta _ { d } , \theta _ { f } , \theta _ { c }$ and latents $\mathbf { z } \in Z$ with two different losses. The first loss function $\mathcal { L } _ { \mathrm { s c e n e } }$ measures the reconstruction between the radiance field encoded in $\mathbf { z } _ { \mathrm { s c e n e } }$ and the images in the trajectory $\mathbf { x } _ { s } ^ { \mathrm { { i m } } }$ (where $s$ denotes the normalized temporal position of the frame in the trajectory), given ground-truth camera poses $\mathbf { T } _ { s }$ required for rendering. We use an $l _ { 2 }$ loss for RGB and $l _ { 1 }$ for depth 1. The second loss function $\mathcal { L } _ { \mathrm { p o s e } }$ measures the camera pose reconstruction error between the poses $\hat { \mathbf { T } } _ { s }$ encoded in $\mathbf { z } _ { \mathrm { p o s e } }$ and the ground-truth poses. We employ an $l _ { 2 }$ loss on translation and $l _ { 1 }$ loss for the normalized quaternion part of the camera pose. Although theoretically normalized quaternions are not necessarily unique (e.g. q and $- \mathbf { q } )$ we do not observe any issues empirically during training.
+
+# 3.2 Prior Learning
+
+Given a set of latents $\mathbf { z } \in Z$ resulting from minimizing the objective in Eq. $2 ,$ our goal is to learn a generative model $p ( Z )$ that captures their distribution (i.e. after minimizing the objective in Eq. $2$ we interpret $\mathbf { z } \in Z$ as examples from an empirical distribution in latent space). In order to model $p ( Z )$ we employ a Denoising Diffusion Probabilistic Model (DDPM) [17], a recent score-matching $[ \overline { { 1 8 } } ]$ based model that learns to reverse a diffusion Markov Chain with a large but finite number of timesteps. In DDPMs $\mathbb { \lVert 1 7 \rVert }$ it is shown that this reverse process is equivalent to learning a sequence of denoising auto-encoders with tied weights. The supervised denoising objective in DDPMs makes learning $p ( Z )$ simple and scalable. This allows us to learn a powerful generative model that enables both unconditional and conditional generation of 3D scenes. For training our prior $p _ { \theta _ { p } } ( Z )$ we take the objective function in $ { \mathbb { I } } ^ { [ 1 2 ] }$ defined in Eq. $\textcircled { 3 }$ In Eq. $\sum t$ denotes the timestep, $\epsilon \sim \mathcal { N } ( 0 , \mathbf { I } )$ is the noise and $\bar { \alpha } _ { t }$ is a noise magnitude parameter with a fixed scheduling. Finally, $\epsilon _ { \theta _ { p } }$ denotes the denoising model.
+
+$$
+\operatorname* { m i n } _ { \theta _ { p } } \mathbb { E } _ { t , \mathbf { z } \sim Z , \epsilon \sim \mathcal { N } ( 0 , \mathbf { I } ) } \left[ \| \epsilon - \epsilon _ { \theta _ { p } } \big ( \sqrt { \bar { \alpha } _ { t } } \mathbf { z } + \sqrt { 1 - \bar { \alpha } _ { t } } \epsilon , t \big ) \| ^ { 2 } \right]
+$$
+
+At inference time, we sample $\mathbf { z } \sim p _ { \theta _ { p } } ( Z )$ by following the inference process in DDPMs. We start by sampling $\mathbf { z } _ { T } \sim \mathcal { N } ( 0 , \mathbf { I } )$ and iteratively apply $\epsilon _ { \theta _ { p } }$ to gradually denoise $\mathbf { z } _ { T }$ , thus reversing the diffusion Markov Chain to obtain $\mathbf { z } _ { 0 }$ . We then feed $\mathbf { z } _ { 0 }$ as input to the decoder architecture (cf. Fig. $2 )$ and reconstruct a radiance field and a camera path.
+
+If the goal is to learn a conditional distribution of the latents $p ( Z | Y )$ , given paired data $\{ \mathbf { z } \in Z , y \in$ $Y \}$ , the denoising model $\epsilon _ { \theta }$ is augmented with a conditioning variable $y$ , resulting in $\epsilon _ { \theta _ { p } } ( \mathbf { z } , t , y )$ , implementation details about how the conditioning variable is used in the denoising architecture can be found in the appendix C.
+
+# 4 Experiments
+
+In this section we show the applicability of GAUDI to multiple problems. First, we evaluate reconstruction quality and performance of the reconstruction stage. Then, we evaluate the performance of our model in generative tasks including unconditional and conditional inference, in which radiance fields are generated from conditioning variables corresponding to images or text prompts. Full experimental settings and details can be found in the appendix B
+
+# 4.1 Data
+
+We report results on 4 datasets: Vizdoom $\mathbb { \left| \left[ 2 . 3 \right] \right| }$ , Replica [64], VLN-CE [26] and ARKit Scenes [1], which vary in number of scenes and complexity (see Fig. 3 and Tab. 1).
+
+Vizdoom $\pmb { \mathbb { Z } 3 } \|$ : Vizdoom is a synthetic simulated environment with simple texture and geometry. We use the data provided by $\pmb { \mathbb { B } } ] \mathbf { l }$ to train our model. It is the simplest dataset in terms of number of scenes and trajectories, as well as texture, serving as a test bed to examine GAUDI in the simplest setting.
+
+Replica $\pmb { \mathbb { \lVert 6 4 \rVert } }$ : Replica is a dataset comprised of 18 realistic scenes from which trajectories are rendered via Habitat $\mathbb {  \boldsymbol {  9 }  }$ . We used the data provided by $\textcircled { 8 }$ to train our model.
+
+VLN-CE $\left[ \left[ 2 6 \right] \right]$ : VLN-CE is a dataset originally designed for vision and language navigation in continuous environments. This dataset is composed of 3.6K trajectories of an agent navigating between two points in a 3D scene from the Matterport 3D dataset $\mathbb { H }$ . We render observations via
+
+![](images/7097bcaa55df2e214b67e47ccadd3bb523e55c7c711e34fc104dfcb75226e874.jpg)  
+Figure 3: (a) Examples of the 4 datasets we use in this paper (from left to right): Vizdoom $[ [ 2 3 ] ]$ , Replica $\pmb { \| 6 4 \| }$ , VLN-CE $\left[ \left[ 2 6 \right] \right]$ , ARKitScenes [1]. (b) Layouts for two scenes in VLN-CE $\pmb { \Vert 2 6 \Vert }$ , where navigable areas are shaded in dark gray. Blue and red dots represent start-end positions and the camera path is highlighted in blue.
+
+Habitat $\mathbb { \left[ \left. 5 9 \right. \right. }$ . Notably, this dataset contains also textual descriptions of the trajectories taken by an agent. In Sect. 4.5 we train GAUDI in a conditional manner to generate 3D scenes given a description.
+
+ARKitScenes [1]: ARKitScenes is a dataset of scans of indoor spaces. This dataset contains more than 5K scans of about 1.6K different indoor spaces. As opposed to the previous datasets where RGB, depth and camera poses are obtained via rendering in a simulation (i.e. either Vizdoom $\mathbb { \left[ \left. 2 3 \right] \right. }$ or Habitat [59]), ARKitScenes provides raw RGB and depth of the scans and camera poses estimated using ARKit SLAM. In addition, whereas trajectories from the previous datasets are point-to-point, as typically done in navigation, the camera trajectories for ARKitScenes resembles a natural scan a of full indoor space. In our experiments we use a subset of 1K scans from ARKitScenes to train our models.
+
+# 4.2 Reconstruction
+
+We first validate the hypothesis that the optimization problem described in Eq. $2$ can find latent codes $\mathbf { z }$ that are able reconstruct the trajectories in the empirical distribution in a satisfactory way. In Tab. 1 we report reconstruction performance of our model across all datasets. Fig. $\sharp$ shows reconstructions of random trajectories for each dataset. For all our experiments we set the dimension of $\mathbf { z } _ { \mathrm { s c e n e } }$ and $\mathbf { z } _ { \mathrm { p o s e } }$ to 2048 and $\beta = 0 . 1$ unless otherwise stated. During training, we normalize camera poses for each trajectory so that the middle frame in a trajectory becomes the origin of the coordinate system. See appendix $\boxed { \ E }$ for ablation experiments.
+
+![](images/2a55d72050e7abf5dc63fe3d29bdd1e15b96487d65ea03ab1a1bcf5616c9bcc1.jpg)  
+Figure 4: Qualitative reconstruction results of random trajectories on different datasets (one for each column): Vizdoom $\pmb { \left. \pmb { \left. \bar { 2 3 } \right. } \right. }$ , Replica $\pmb { \mathbb { \lVert 6 4 \rVert } }$ , VLN-CE $\left[ \left[ 2 6 \right] \right]$ and ARKitScenes [1]. For each pair of images the left is ground-truth and right is reconstruction.
+
+<table><tr><td></td><td>#sc-#tr-#im</td><td>11↓</td><td>PSNR↑</td><td>SSIM个</td><td>Rot Err. ↓</td><td>Trans. Err ↓</td></tr><tr><td>Vizdoom 2</td><td>1-32-1k</td><td>0.004</td><td>44.42</td><td>0.98</td><td>0.01</td><td>1.26</td></tr><tr><td>Replica √</td><td>18-100-1k</td><td>0.006</td><td>38.86</td><td>0.99</td><td>0.03</td><td>0.01</td></tr><tr><td>VLN-CE [26</td><td>90-3.6k-600k</td><td>0.031</td><td>25.17</td><td>0.73</td><td>0.30</td><td>0.02</td></tr><tr><td>ARKitScenes</td><td>300-1k-600k</td><td>0.039</td><td>24.51</td><td>0.76</td><td>0.16</td><td>0.04</td></tr></table>
+
+Table 1: Reconstruction results of the optimization process described in Eq. 2. The first column shows the number of scenes (#sc), trajectories (#tr) and images (#im) per dataset. Due to the large number of images on VLN-CE $\mathbb { \left[ \left[ 2 6 \right] \right] }$ and ARKitScenes $\mathbb { I I }$ datasets we sample 10 random images per trajectory to compute the reconstruction metrics.
+
+# 4.3 Interpolation
+
+In addition, to evaluate the structure of the latent representation obtained from minimizing the optimization problem in Eq. $\bigstar ,$ we show interpolation results between pairs of latents $( \mathbf { z } _ { i } , \mathbf { z } _ { j } )$ in
+
+![](images/6cddcf51ce96aa1d9d58ca018cda77b4d7a30a8b1859865add9ed9c19a2902b3.jpg)  
+Fig. $\boxed { 5 }$ To render images while interpolating the scene we place a fixed camera at the origin of the coordinate system. We observe a smooth transition of scenes in both geometry (walls, ceilings) and texture (stairs, carpets). More visualizations are included in the appendix H.   
+Figure 5: Interpolation of 3D scenes in latent space (e.g. interpolating the encoded radiance field) for the VLN-CE dataset $\pmb { \mathbb { Z } } 6 \|$ . Each row corresponds to a different interpolation path.
+
+# 4.4 Unconditional generative modeling
+
+Given latent representations $\textbf { z } \in \ Z$ that can reconstruct samples $x \in X$ with high accuracy as shown in Sect. $\boxed { 4 . 2 }$ we now evaluate the capacity of the prior $p _ { \theta _ { p } } ( Z )$ to capture the empirical distribution $x \in \mathcal { X }$ by learning the distribution of latents $\mathbf { z } _ { i } \in Z$ . To do so we sample $\mathbf { z } \sim p _ { \theta _ { p } } ( Z )$ by following the inference process in DDPMs, and then feed $\mathbf { z }$ through the decoder network, which results in trajectories of RGB images that are then used for evaluation. We compare our approach with the following baselines: GRAF $\pmb { \| 6 0 \| }$ , $\pi$ -GAN $\pmb { \mathbb { H } }$ and GSN $\textcircled { 8 }$ , where all models have access to ground-truth depth information during training. We sample 5k images from predicted and target distributions for each model and dataset and report both FID $\mathbb { \left[ \left[ 1 6 \right] \right] }$ and SwAV-FID $\begin{array} { r l } {  { \bigl [ \bigl | 3 5 \bigr | \bigr ] } } \end{array}$ scores. We report quantitative results in Tab. $^ { 2 , }$ where we can see that GAUDI obtains state-of-the-art performance across all datasets and metrics. We attribute this performance improvement to the fact that GAUDI learns disentangled yet corresponding latents for radiance fields and camera poses, which is key when modeling scenes (see ablations in the appendix $\mathbf { E } )$ . We note that to obtain these great empirical results GAUDI needs to simultaneously find latents with high reconstruction fidelity while also efficiently learning their distribution.
+
+<table><tr><td rowspan="2"></td><td colspan="2">VizDoom 因</td><td colspan="2">Replica 国</td><td colspan="2">VLN-CE 回</td><td colspan="2">ARKitScenes</td></tr><tr><td>FID↓</td><td>SwAV-FID↓</td><td>FID↓</td><td>SwAV-FID↓</td><td>FID↓</td><td>SwAV-FID↓</td><td>FID↓</td><td>SwAV-FID↓</td></tr><tr><td>GRAF60</td><td>47.50 ± 2.13</td><td>5.44 ± 0.43</td><td>65.37 ± 1.64</td><td>5.76 ± 0.14</td><td>90.43±4.83</td><td>8.65±0.27</td><td>87.06±9.99</td><td>13.44±0.26</td></tr><tr><td>π-GAN</td><td>143.55 ± 4.81</td><td>15.26 ±0.15</td><td>166.55 ± 3.61</td><td>13.17 ±0.20</td><td>151.26 ± 4.19674 14.07 ±0.56</td><td></td><td>134.80±10.60</td><td>15.58 ±0.13</td></tr><tr><td>GSN</td><td>37.21 ± 1.17</td><td>4.56±0.19</td><td>41.75 ±1.33</td><td>4.14±0.02</td><td>43.32 ±8.86</td><td>6.19±0.49</td><td>79.54 ± 2.60</td><td>10.21 ± 0.15</td></tr><tr><td>GAUDI</td><td>33.70±1.27</td><td>3.24±0.12</td><td>18.75±0.63</td><td>1.76 ±0.05</td><td>18.52 ±0.11</td><td>3.63±0.65</td><td>37.35 ±0.38</td><td>4.14±0.03</td></tr></table>
+
+Table 2: Generative performance of state-of-the-art approaches for generative modelling of radiance fields on 4 scene datasets: Vizdoom $\mathbb { \left[ \left. 2 3 \right\| \right. }$ , Replica $\textcircled { 6 4 } \textcircled { 1 6 }$ , VLN-CE $\dot { \left\| 2 6 \right\| }$ and ARKitScenes [1], according to FID [16] and SwAV-FID $\pmb { \Vert 3 5 \Vert }$ metrics.
+
+In Fig. $\boxed { 6 }$ we show samples from the unconditional distribution learnt by GAUDI for different datasets. We observe that GAUDI is able to generate diverse and realistic 3D scenes from the empirical distribution which can be rendered from the sampled camera poses.
+
+# 4.5 Conditional Generative Modeling
+
+GAUDI can also tackle conditional generative problems of the form $p ( Z | Y )$ , where a conditioning variable $y \in Y$ is given to condition $p ( Z )$ . For a given conditional inference problem we assume the existence of paired data [46, 11, 45]. As an example, for training a text-conditional model we assume the existence of pairs $\left\{ \mathbf { z } _ { i } , y _ { i } \right\}$ , where $\mathbf { z } _ { i }$ is a latent scene representation and $y _ { i }$ is its corresponding text prompt. In this section we show both quantitative and qualitative results for conditional inference problems. The first conditioning variable we consider are textual descriptions of trajectories. Second, we consider a model where randomly sampled RGB images in a trajectory act as conditioning. Finally, we use a categorical variable that indicates the 3D environment (i.e. the particular indoor space) from which each trajectory was obtained. Tab. $3$ shows quantitative results for the different conditional inference problems. Details on the implementation of the conditional DDPM are given in Sect. D
+
+![](images/824f78214958dbf2b3c9b035017c70eb707ec7077e5001af194aaa5e4f2fdd8b.jpg)
+
+<table><tr><td>Avg.△Per-Environment</td><td></td></tr><tr><td>FID↓</td><td>SwAV-FID↓</td></tr><tr><td>-50.79</td><td>-4.10</td></tr></table>
+
+Table 3: Quantitative results of Conditional Generative Modeling on VLN-CE [26] dataset. GAUDI is able to produce high-quality scene renderings with low FID and SwAV-FID scores. In the right table we show the difference in average per-environment FID score between the conditional and unconditional models.   
+
+<table><tr><td colspan="2">Text Conditioning</td><td colspan="2">Image Conditioning</td><td colspan="2">Categorical Conditioning</td></tr><tr><td>FID↓</td><td>SwAV-FID↓</td><td>FID↓</td><td>SwAV-FID↓</td><td>FID↓</td><td>SwAV-FID↓</td></tr><tr><td>18.50</td><td>3.75</td><td>19.51</td><td>3.93</td><td>18.74</td><td>3.61</td></tr></table>
+
+# 4.5.1 Text Conditioning
+
+We tackle the challenging task of training a text conditional model for 3D scene generation. We use the navigation text descriptions provided in VLN-CE $\left[ \left[ 2 6 \right] \right]$ to condition our model. These text descriptions contain high level information about the scene as well as the navigation path (i.e. "Walk out of the bedroom and into the living room", "Exit the room through the swinging doors and then enter the bedroom"). We employ a pre-trained RoBERTa-base $\dot { \left\| 2 9 \right\| }$ text encoder and use its intermediate representation to condition the diffusion model. Fig. $^ { 7 }$ shows qualitative results of GAUDI for this task. To the best of our knowledge, this is the first model that allows for conditional 3D scene generation from text in an amortized manner (i.e. without distilling CLIP $\mathbb { \lVert H \ 4 \rVert }$ through a costly optimization problem [19, 32]).
+
+![](images/6a315b5a6d145501104248636bd4e02075a4f7ce8251740624ef11afacfcd0e1.jpg)  
+Figure 7: Text conditional 3D scene generation using GAUDI (one sample per row). Our model is able to capture the conditional distributions of scenes by generating multiple plausible scenes and camera paths that match the given text prompts.
+
+# 4.5.2 Image Conditioning
+
+We now analyze whether GAUDI is able to pick up information from the RGB images to predict a distribution over $Z$ . In this experiment we randomly pick images in a trajectory $x \in X$ and use it as a conditioning variable $y$ . For this experiment we use trajectories in the VLN-CE dataset $\pmb { \left. 2 6 \right. }$ . During each training iteration we sample a random image for each trajectory $x$ and use it as a conditioning variable. We employ a pre-trained ResNet-18 [15] as an image encoder. During inference, the resulting conditional GAUDI model is able to sample radiance fields where the given image is observed from a stochastic viewpoint. In Fig. $\textcircled { 8 }$ we show samples from the model conditioned on different RGB images.
+
+![](images/1bff990965fcbaf030c03ad42819efa0570a7bd8ebecadef306163172b4e5182.jpg)  
+Figure 8: Image conditional 3D scene generation using GAUDI (one sample per row). Given a conditioned image (top row), our model is able to sample scenes where the same or contextually similar view is observed from a stochastic viewpoint.
+
+![](images/c2a11d158e4378324735e7e1ea88f9415e6b1d9a49b5ed5bd6e2557b410a5e3c.jpg)  
+Figure 9: Samples from the GAUDI model conditioned on a categorical variable denoting the indoor scene (one sample per row).
+
+# 4.5.3 Categorical Conditioning
+
+Finally, we analyze how GAUDI performs when conditioned on a categorical variable that indicates the underlying 3D indoor environment in which each trajectory was recorded. We perform experiments in the VLN-CE $[ \overline { { 2 6 } } ]$ dataset, where we employ a trainable embedding layer to learn a representation for categorical variables indicating each environment. We compare the per-environment FID score of conditional model with its unconditional counterpart. This per-enviroment FID score is computed only on real images of the same indoor environment that the model is conditioned on. Our hypothesis is that if the model efficiently captures the information in the conditioning variable it should capture the environment specific distribution better than its unconditional counterpart trained on the same data. In Tab. $\textcircled { 3 }$ the last column shows difference (e.g. the $\Delta$ ) on the average per-environment FID score between the conditional and unconditional model on VLN-CE dataset. We observe that the conditional model consistently obtains a better FID score than the unconditional model across all indoor environments, resulting in a sharp reduction of average FID and SwAV-FID scores. In addition, in Fig. 9 we show samples from the model conditioned on a given categorical variable.
+
+# 5 Conclusion
+
+We have introduced GAUDI, a generative model that captures distributions of complex and realistic 3D scenes. GAUDI uses a scalable two-stage approach which first involves learning a latent representation that disentangles radiance fields and camera poses. The distribution of disentangled latent representations is then modeled with a powerful prior. Our model obtains state-of-the-art performance when compared with recent baselines across multiple 3D datasets and metrics. GAUDI can be used both for conditional and unconditional problems, and enabling new tasks like generating 3D scenes from text descriptions.
+
+# References
+
+[1] Gilad Baruch, Zhuoyuan Chen, Afshin Dehghan, Yuri Feigin, Peter Fu, Thomas Gebauer, Daniel Kurz, Tal Dimry, Brandon Joffe, Arik Schwartz, et al. Arkitscenes: A diverse real-world dataset for 3d indoor scene understanding using mobile rgb-d data. In Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 1), 2021.   
+[2] Piotr Bojanowski, Armand Joulin, David Lopez-Paz, and Arthur Szlam. Optimizing the latent space of generative networks. arXiv preprint arXiv:1707.05776, 2017.   
+[3] Marcus Carter and Ben Egliston. Ethical implications of emerging mixed reality technologies. 2020.   
+[4] Lluis Castrejon, Nicolas Ballas, and Aaron Courville. Improved conditional vrnns for video prediction. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 7608–7617, 2019.   
+[5] Eric R Chan, Connor Z Lin, Matthew A Chan, Koki Nagano, Boxiao Pan, Shalini De Mello, Orazio Gallo, Leonidas Guibas, Jonathan Tremblay, Sameh Khamis, et al. Efficient geometry-aware 3d generative adversarial networks. arXiv preprint arXiv:2112.07945, 2021.   
+[6] Eric R Chan, Marco Monteiro, Petr Kellnhofer, Jiajun Wu, and Gordon Wetzstein. pi-GAN: Periodic implicit generative adversarial networks for 3d-aware image synthesis. arXiv preprint arXiv:2012.00926, 2020.   
+[7] Angel Chang, Angela Dai, Thomas Funkhouser, Maciej Halber, Matthias Niessner, Manolis Savva, Shuran Song, Andy Zeng, and Yinda Zhang. Matterport3d: Learning from rgb-d data in indoor environments. arXiv preprint arXiv:1709.06158, 2017.   
+[8] Terrance DeVries, Miguel Angel Bautista, Nitish Srivastava, Graham W Taylor, and Joshua M Susskind. Unconstrained scene generation with locally conditioned radiance fields. ICCV, 2021.   
+[9] Emilien Dupont, Miguel Angel Bautista, Alex Colburn, Aditya Sankar, Carlos Guestrin, Josh Susskind, and Qi Shan. Equivariant neural rendering. In International Conference on Machine Learning, pages 2761–2770. PMLR, 2020.   
+[10] Emilien Dupont, Hyunjik Kim, SM Ali Eslami, Danilo Jimenez Rezende, and Dan Rosenbaum. From data to functa: Your data point is a function and you can treat it like one. In International Conference on Machine Learning, pages 5694–5725. PMLR, 2022.   
+[11] Patrick Esser, Robin Rombach, and Bjorn Ommer. Taming transformers for high-resolution image synthesis. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12873–12883, 2021.   
+[12] Jiatao Gu, Lingjie Liu, Peng Wang, and Christian Theobalt. Stylenerf: A style-based 3d-aware generator for high-resolution image synthesis. arXiv preprint arXiv:2110.08985, 2021.   
+[13] Pengsheng Guo, Miguel Angel Bautista, Alex Colburn, Liang Yang, Daniel Ulbricht, Joshua M Susskind, and Qi Shan. Fast and explicit neural view synthesis. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 3791–3800, 2022.   
+[14] David Ha and Jürgen Schmidhuber. World models. arXiv preprint arXiv:1803.10122, 2018.   
+[15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   
+[16] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. arXiv preprint arXiv:1706.08500, 2017.   
+[17] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in Neural Information Processing Systems, 33:6840–6851, 2020.   
+[18] Aapo Hyvärinen and Peter Dayan. Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6(4), 2005.   
+[19] Ajay Jain, Ben Mildenhall, Jonathan T Barron, Pieter Abbeel, and Ben Poole. Zero-shot text-guided object generation with dream fields. arXiv preprint arXiv:2112.01455, 2021.   
+[20] Niharika Jain, Alberto Olmo, Sailik Sengupta, Lydia Manikonda, and Subbarao Kambhampati. Imperfect imaganation: Implications of gans exacerbating biases on facial data augmentation and snapchat selfie lenses. arXiv preprint arXiv:2001.09528, 2020.   
+[21] Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4401–4410, 2019.   
+[22] Tero Karras, Samuli Laine, Miika Aittala, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. Analyzing and improving the image quality of stylegan. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8110–8119, 2020.   
+[23] Michał Kempka, Marek Wydmuch, Grzegorz Runc, Jakub Toczek, and Wojciech Jaskowski. Vizdoom: ´ A doom-based ai research platform for visual reinforcement learning. In 2016 IEEE Conference on Computational Intelligence and Games (CIG), pages 1–8. IEEE, 2016.   
+[24] Seung Wook Kim, Jonah Philion, Antonio Torralba, and Sanja Fidler. Drivegan: Towards a controllable high-quality neural simulation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5820–5829, 2021.   
+[25] Adam R Kosiorek, Heiko Strathmann, Daniel Zoran, Pol Moreno, Rosalia Schneider, Sona Mokrá, and Danilo Jimenez Rezende. Nerf-vae: A geometry aware 3d scene generative model. In International Conference on Machine Learning, pages 5742–5752. PMLR, 2021.   
+[26] Jacob Krantz, Erik Wijmans, Arjun Majundar, Dhruv Batra, and Stefan Lee. Beyond the nav-graph: Vision and language navigation in continuous environments. In European Conference on Computer Vision (ECCV), 2020.   
+[27] Zihang Lai, Sifei Liu, Alexei A Efros, and Xiaolong Wang. Video autoencoder: self-supervised disentanglement of static 3d structure and motion. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 9730–9740, 2021.   
+[28] Lingjie Liu, Jiatao Gu, Kyaw Zaw Lin, Tat-Seng Chua, and Christian Theobalt. Neural sparse voxel fields. arXiv preprint arXiv:2007.11571, 2020.   
+[29] Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692, 2019.   
+[30] Willi Menapace, Stéphane Lathuilière, Sergey Tulyakov, Aliaksandr Siarohin, and Elisa Ricci. Playable video generation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10061–10070, 2021.   
+[31] Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger. Occupancy networks: Learning 3d reconstruction in function space. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4460–4470, 2019.   
+[32] Oscar Michel, Roi Bar-On, Richard Liu, Sagie Benaim, and Rana Hanocka. Text2mesh: Text-driven neural stylization for meshes. arXiv preprint arXiv:2112.03221, 2021.   
+[33] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. arXiv preprint arXiv:2003.08934, 2020.   
+[34] Yisroel Mirsky and Wenke Lee. The creation and detection of deepfakes: A survey. ACM Computing Surveys (CSUR), 54(1):1–41, 2021.   
+[35] Stanislav Morozov, Andrey Voynov, and Artem Babenko. On self-supervised image representations for GAN evaluation. In International Conference on Learning Representations, 2021.   
+[36] Alex Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob McGrew, Ilya Sutskever, and Mark Chen. Glide: Towards photorealistic image generation and editing with text-guided diffusion models. arXiv preprint arXiv:2112.10741, 2021.   
+[37] Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In International Conference on Machine Learning, pages 8162–8171. PMLR, 2021.   
+[38] Michael Niemeyer and Andreas Geiger. Giraffe: Representing scenes as compositional generative neural feature fields. arXiv preprint arXiv:2011.12100, 2020.   
+[39] Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Lovegrove. Deepsdf: Learning continuous signed distance functions for shape representation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 165–174, 2019.   
+[40] Despoina Paschalidou, Amlan Kar, Maria Shugrina, Karsten Kreis, Andreas Geiger, and Sanja Fidler. Atiss: Autoregressive transformers for indoor scene synthesis. Advances in Neural Information Processing Systems, 34, 2021.   
+[41] Songyou Peng, Michael Niemeyer, Lars Mescheder, Marc Pollefeys, and Andreas Geiger. Convolutional occupancy networks. arXiv preprint arXiv:2003.04618, 2, 2020.   
+[42] Ethan Perez, Florian Strub, Harm De Vries, Vincent Dumoulin, and Aaron Courville. Film: Visual reasoning with a general conditioning layer. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.   
+[43] Benjamin Planche, Xuejian Rong, Ziyan Wu, Srikrishna Karanam, Harald Kosch, YingLi Tian, Jan Ernst, and Andreas Hutter. Incremental scene synthesis. In Advances in Neural Information Processing Systems, pages 1668–1678, 2019.   
+[44] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International Conference on Machine Learning, pages 8748–8763. PMLR, 2021.   
+[45] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation. In International Conference on Machine Learning, pages 8821–8831. PMLR, 2021.   
+[46] Ali Razavi, Aaron Van den Oord, and Oriol Vinyals. Generating diverse high-fidelity images with vq-vae-2. Advances in neural information processing systems, 32, 2019.   
+[47] Daniel Rebain, Mark Matthews, Kwang Moo Yi, Dmitry Lagun, and Andrea Tagliasacchi. Lolnerf: Learn from one look. arXiv preprint arXiv:2111.09996, 2021.   
+[48] Christian Reiser, Songyou Peng, Yiyi Liao, and Andreas Geiger. Kilonerf: Speeding up neural radiance fields with thousands of tiny mlps. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 14335–14345, 2021.   
+[49] Xuanchi Ren and Xiaolong Wang. Look outside the room: Synthesizing a consistent long-term 3d scene video from a single image. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2022.   
+[50] Salah Rifai, Grégoire Mesnil, Pascal Vincent, Xavier Muller, Yoshua Bengio, Yann Dauphin, and Xavier Glorot. Higher order contractive auto-encoder. In Joint European conference on machine learning and knowledge discovery in databases, pages 645–660. Springer, 2011.   
+[51] Daniel Ritchie, Kai Wang, and Yu-an Lin. Fast and flexible indoor scene synthesis via deep convolutional generative models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6182–6190, 2019.   
+[52] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High-resolution image synthesis with latent diffusion models. arXiv preprint arXiv:2112.10752, 2021.   
+[53] Robin Rombach, Patrick Esser, and Björn Ommer. Geometry-free view synthesis: Transformers and no 3d priors, 2021.   
+[54] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computer-assisted intervention, pages 234–241. Springer, 2015.   
+[55] Negar Rostamzadeh, Emily Denton, and Linda Petrini. Ethics and creativity in computer vision. arXiv preprint arXiv:2112.03111, 2021.   
+[56] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, S. Sara Mahdavi, Rapha Gontijo Lopes, Tim Salimans, Jonathan Ho, David J Fleet, and Mohammad Norouzi. Photorealistic text-to-image diffusion models with deep language understanding, 2022.   
+[57] Mehdi S. M. Sajjadi, Henning Meyer, Etienne Pot, Urs Bergmann, Klaus Greff, Noha Radwan, Suhani Vora, Mario Lucic, Daniel Duckworth, Alexey Dosovitskiy, Jakob Uszkoreit, Thomas Funkhouser, and Andrea Tagliasacchi. Scene Representation Transformer: Geometry-Free Novel View Synthesis Through Set-Latent Scene Representations. CVPR, 2022.   
+[58] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. NeurIPS’16, page 2234–2242, Red Hook, NY, USA, 2016. Curran Associates Inc.   
+[59] Manolis Savva, Abhishek Kadian, Oleksandr Maksymets, Yili Zhao, Erik Wijmans, Bhavana Jain, Julian Straub, Jia Liu, Vladlen Koltun, Jitendra Malik, et al. Habitat: A platform for embodied ai research. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 9339–9347, 2019.   
+[60] Katja Schwarz, Yiyi Liao, Michael Niemeyer, and Andreas Geiger. GRAF: Generative radiance fields for 3d-aware image synthesis. arXiv preprint arXiv:2007.02442, 2020.   
+[61] Edward J Smith and David Meger. Improved adversarial systems for 3d object generation and reconstruction. In Conference on Robot Learning, pages 87–96. PMLR, 2017.   
+[62] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International Conference on Machine Learning, pages 2256–2265. PMLR, 2015.   
+[63] Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502, 2020.   
+[64] Julian Straub, Thomas Whelan, Lingni Ma, Yufan Chen, Erik Wijmans, Simon Green, Jakob J Engel, Raul Mur-Artal, Carl Ren, Shobhit Verma, et al. The replica dataset: A digital replica of indoor spaces. arXiv preprint arXiv:1906.05797, 2019.   
+[65] Hoang Thanh-Tung and Truyen Tran. Catastrophic forgetting and mode collapse in gans. In 2020 International Joint Conference on Neural Networks (IJCNN), pages 1–10. IEEE, 2020.   
+[66] Patrick Tinsley, Adam Czajka, and Patrick Flynn. This face does not exist... but it might be yours! identity leakage in generative models. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 1320–1328, 2021.   
+[67] Thomas Unterthiner, Sjoerd van Steenkiste, Karol Kurach, Raphael Marinier, Marcin Michalski, and Sylvain Gelly. Towards accurate generative models of video: A new metric & challenges. arXiv preprint arXiv:1812.01717, 2018.   
+[68] Arash Vahdat, Karsten Kreis, and Jan Kautz. Score-based generative modeling in latent space. Advances in Neural Information Processing Systems, 34:11287–11302, 2021.   
+[69] Aaron Van Den Oord, Oriol Vinyals, et al. Neural discrete representation learning. Advances in neural information processing systems, 30, 2017.   
+[70] Xinpeng Wang, Chandan Yeshwanth, and Matthias Nießner. Sceneformer: Indoor scene generation with transformers. In 2021 International Conference on 3D Vision (3DV), pages 106–115. IEEE, 2021.   
+[71] Daniel Watson, William Chan, Jonathan Ho, and Mohammad Norouzi. Learning fast samplers for diffusion models by differentiating through sample quality. In International Conference on Learning Representations, 2021.   
+[72] Daniel Watson, Jonathan Ho, Mohammad Norouzi, and William Chan. Learning to efficiently sample from diffusion probabilistic models. arXiv preprint arXiv:2106.03802, 2021.   
+[73] Alex Yu, Sara Fridovich-Keil, Matthew Tancik, Qinhong Chen, Benjamin Recht, and Angjoo Kanazawa. Plenoxels: Radiance fields without neural networks. arXiv preprint arXiv:2112.05131, 2021.
+
+[74] Alex Yu, Vickie Ye, Matthew Tancik, and Angjoo Kanazawa. pixelnerf: Neural radiance fields from one or few images. arXiv preprint arXiv:2012.02190, 2020.
+
+[75] Peng Zhou, Lingxi Xie, Bingbing Ni, and Qi Tian. Cips-3d: A 3d-aware generator of gans based on conditionally-independent pixel synthesis. arXiv preprint arXiv:2110.09788, 2021.
+
+# Checklist
+
+The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes] See Section ??.   
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   
+• Did you include the license to the code and datasets? [N/A]
+
+Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] These will be discussed in appendix A   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] These will be discussed in appendix A   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We will release code publicly upon acceptance.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] These details will be clarified in B   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] These details will be clarified in B
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See Sect. 4   
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/xijYyYFlRIf/xijYyYFlRIf_content_list.json

@@ -0,0 +1,1212 @@
+[
+    {
+        "type": "text",
+        "text": "GAUDI: A Neural Architect for Immersive 3D Scene Generation ",
+        "text_level": 1,
+        "bbox": [
+            303,
+            122,
+            694,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Miguel Angel Bautista⇤ Pengsheng Guo⇤ Samira Abnar Walter Talbott ",
+        "bbox": [
+            192,
+            220,
+            797,
+            237
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Alexander Toshev Zhuoyuan Chen Laurent Dinh Shuangfei Zhai Hanlin Goh ",
+        "text_level": 1,
+        "bbox": [
+            196,
+            256,
+            808,
+            272
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Daniel Ulbricht ",
+        "bbox": [
+            251,
+            292,
+            362,
+            308
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Afshin Dehghan ",
+        "bbox": [
+            446,
+            292,
+            562,
+            308
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Josh Susskind ",
+        "bbox": [
+            645,
+            292,
+            746,
+            308
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Apple https://github.com/apple/ml-gaudi ",
+        "bbox": [
+            354,
+            329,
+            643,
+            358
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            392,
+            535,
+            409
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce GAUDI, a generative model capable of capturing the distribution of complex and realistic 3D scenes that can be rendered immersively from a moving camera. We tackle this challenging problem with a scalable yet powerful approach, where we first optimize a latent representation that disentangles radiance fields and camera poses. This latent representation is then used to learn a generative model that enables both unconditional and conditional generation of 3D scenes. Our model generalizes previous works that focus on single objects by removing the assumption that the camera pose distribution can be shared across samples. We show that GAUDI obtains state-of-the-art performance in the unconditional generative setting across multiple datasets and allows for conditional generation of 3D scenes given conditioning variables like sparse image observations or text that describes the scene. ",
+        "bbox": [
+            233,
+            424,
+            766,
+            589
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            616,
+            310,
+            633
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In order for learning systems to be able to understand and create 3D spaces, progress in generative models for 3D is sorely needed. The quote \"The creation continues incessantly through the media of man.\" is often attributed to Antoni Gaudí, who we pay homage to with our method’s name. In this paper we ask the question: can creation continue through the media of learning machines? We are interested in generative models that can capture the distribution of 3D scenes and then render views from scenes sampled from the learned distribution. Extensions of such generative models to conditional inference problems could have tremendous impact in a wide range of tasks in machine learning and computer vision. For example, one could sample plausible scene completions that are consistent with an image observation, or a text description (see Fig. 1 for 3D scenes sampled from GAUDI). In addition, such models would be of great practical use in model-based reinforcement learning and planning [14], SLAM [43], or 3D content creation. ",
+        "bbox": [
+            174,
+            647,
+            825,
+            800
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent works on generative modeling for 3D objects or scenes [60, 6, 8] employ a Generative Adversarial Network (GAN) where the generator explicitly encodes radiance fields — a parametric function that takes as input the coordinates of a point in 3D space and camera pose, and outputs a density scalar and RGB value for that 3D point. Images can be rendered from the radiance field generated by the model by passing the queried 3D points through the volume rendering equation to project onto any 2D camera view. While compelling on small or simple 3D datasets (e.g. single objects or a small number of indoor scenes), GANs suffer from training pathologies including mode collapse $\\pm \\pm \\pm \\pm \\pm \\pm \\pm \\pm \\pm$ and are difficult to train on data for which a canonical coordinate system does not exist, as is the case for 3D scenes $\\left[ \\left[ 6 1 \\right] \\right]$ . In addition, one key difference between modeling distributions of 3D objects vs. scenes is that when modeling objects, the distribution of valid camera poses does not depend on each object and is defined per dataset (i.e. typically as $S O ( 3 )$ ), which is not true for scenes. See Fig. $\\bar { \\bigtriangledown ( \\mathfrak { b } ) }$ and note how the two scene layouts define different areas of navigable space (different dark grey shaded areas) where valid camera poses can be placed, revealing a strong dependency between scenes and their camera pose distributions. GAUDI captures this dependency by modeling the joint distribution of scenes and camera poses. ",
+        "bbox": [
+            174,
+            806,
+            823,
+            877
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/817e9e0b07bda48800f0017cb652036d1cffe99dbb4fab43512b87d33c1bfd6f.jpg",
+        "image_caption": [
+            "Figure 1: GAUDI allows to model both conditional and unconditional distributions over complex 3D scenes. Sampled scenes and poses from (left) the unconditional distribution, and (right) a distribution conditioned on an image observation or a text prompt. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            89,
+            821,
+            247
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            328,
+            825,
+            467
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In GAUDI, we map each trajectory (i.e. a sequence of posed images from a 3D scene) into a latent representation that encodes a radiance field (e.g. the 3D scene) and camera path in a completely disentangled way. We find these latent representations by interpreting them as free parameters and formulating an optimization problem where the latent representation for each trajectory is optimized via a reconstruction objective. This simple training process is scalable to thousands of trajectories. Interpreting the latent representation of each trajectory as a free parameter also makes it simple to handle a large and variable number of views for each trajectory rather than requiring a sophisticated encoder architecture to pool across a large number of views. After optimizing latent representations for an observed empirical distribution of trajectories, we learn a generative model over the set of latent representations. In the unconditional case, the model can sample radiance fields entirely from the prior distribution learned by the model, allowing it to synthesize scenes by interpolating within the latent space. In the conditional case, conditional variables available to the model at training time (e.g. images, text prompts, etc.) can be used to generate radiance fields consistent with those variables. Our contributions can be summarized as: ",
+        "bbox": [
+            173,
+            473,
+            825,
+            666
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We scale 3D scene generation to thousands of indoor scenes containing hundreds of thousands of images, without suffering from mode collapse or canonical orientation issues during training. ",
+        "bbox": [
+            173,
+            672,
+            823,
+            700
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We introduce a novel denoising optimization objective to find latent representations that jointly model a radiance field and the camera poses in a disentangled manner. ",
+        "bbox": [
+            171,
+            707,
+            823,
+            736
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Our approach obtains state-of-the-art generation performance across multiple datasets. ",
+        "bbox": [
+            173,
+            741,
+            754,
+            756
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Our approach allows for various generative setups: unconditional generation as well as conditional on images or text. ",
+        "bbox": [
+            173,
+            762,
+            825,
+            790
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            810,
+            321,
+            827
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In recent years the field has witnessed outstanding progress in generative modeling for the 2D image domain, with most approaches focusing either on adversarial $\\mathbb { \\left[ \\left[ 2 1 \\right] , \\left[ 2 2 \\right] \\right] }$ or auto-regressive models [69, 46, 11]. More recently, score matching based approaches [18, 62] have gained popularity. In particular, Denoising Diffusion Probabilistic Models (DDPMs) [ $\\boxed { 1 7 } \\boxed { 3 7 } \\boxed { 5 2 } \\boxed { 6 8 }$ have emerged as strong contenders to both adversarial and auto-regressive approaches. In DDPMs, the goal is to learn a step-by-step inversion of a fixed diffusion Markov Chain that gradually transforms an empirical data distribution to a fixed posterior, which typically takes the form of an isotropic Gaussian distribution. In parallel, the last couple of years have seen a revolution in how 3D data is represented within neural networks. By representing a 3D scene as a radiance field, NeRF $[ \\left| 3 3 \\right| ]$ introduces an approach to optimize the weights of a MLP to represent the radiance of 3D points that fall inside the field-of-view of a given set of posed RGB images. Given the radiance for a set of 3D points that lie on a ray shot from a given camera pose, NeRF $\\mathbb { B } 3 \\mathbb { I }$ uses volumetric rendering to compute the color for the corresponding pixel and optimizes the MLP weights via a reconstruction loss in image space. ",
+        "bbox": [
+            174,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            825,
+            202
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A few attempts have also been made at incorporating a radiance field representation within generative models. Most approaches have focused on the problem of single objects with known canonical orientations like faces or Shapenet objects with shared camera pose distributions across samples in a dataset [60, 6, 38, 25, 5, 12, 75, 47, 10]. Extending these approaches from single objects to completely unconstrained 3D scenes is an unsolved problem. One paper worth mentioning in this space is GSN $\\textcircled { 8 }$ , which breaks the radiance field into a grid of local radiance fields that collectively represent a scene. While this decomposition of radiance fields endows the model with high representational capacity, GSN still suffers from the standard training pathologies of GANs, like mode collapse $\\mathbb { \\lVert \\rVert }$ , which are exacerbated by the fact that unconstrained 3D scenes do not have a canonical orientation. As we show in our experiments (cf. Sect. 4), these issues become prominent as the training set size increases, impacting the capacity of the generative model to capture complex distributions. Separately, a line of recent approaches have also studied the problem of learning generative models of scenes without employing radiance fields [40, 70, 51]. These works assume that the model has access to room layouts and a database of object CAD models during training, simplifying the problem of scene generation to a selection of objects from the database and pose predictions for each object. ",
+        "bbox": [
+            173,
+            208,
+            825,
+            416
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Finally, approaches that learn to predict a target view given a single (or multiple) source view and relative pose transformation have been recently proposed [27, 74, 57, 9, 13]. The pure reconstruction objective employed by these approaches forces them to learn a deterministic conditional function that maps a source image and a relative camera transformation to a target image. The first is that this scene completion problem is ill-posed (e.g. given a single source view of a scene there are multiple target completions that are equally likely). Attempts at modeling the problem in a probabilistic manner have been proposed $\\underline { { \\lVert 5 3 \\rVert } } , \\boxed { 4 9 }$ . However, these approaches suffer from inconsistency in predicted scenes because they do not explicitly model a 3D consistent representation like a radiance field. ",
+        "bbox": [
+            174,
+            421,
+            825,
+            534
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 GAUDI ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            554,
+            269,
+            570
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to learn a generative model given an empirical distribution of trajectories over 3D scenes. Let $X = \\{ x _ { i \\in \\{ 0 , . . . , n \\} } \\}$ denote a collection of examples defining an empirical distribution, where each example $x _ { i }$ is a trajectory. Every trajectory $x _ { i }$ is defined as a variable length sequence of corresponding RGB, depth images and 6DOF camera poses (see Fig. 3). ",
+        "bbox": [
+            174,
+            585,
+            825,
+            642
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We decompose the task of learning a generative model in two stages. First, we obtain a latent representation $\\mathbf { z } = [ \\mathbf { z } _ { \\mathrm { s c e n e } } , \\mathbf { z } _ { \\mathrm { p o s e } } ]$ for each example $x \\in X$ that represents the scene radiance field and pose in separate disentangled vectors. Second, given a set of latents $Z = \\{ \\mathbf { z } _ { i \\in \\{ 0 , \\dots , n \\} } \\}$ we learn the distribution $p ( Z )$ . ",
+        "bbox": [
+            174,
+            648,
+            825,
+            707
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Optimizing latent representations for radiance fields and camera poses ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            723,
+            702,
+            738
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We now turn to the task of finding a latent representation $\\mathbf { z } \\in Z$ for each example $x \\in X$ (i.e. for each trajectory in the empirical distribution). To obtain this latent representation we take an encoder-less view and interpret $\\mathbf { z }$ ’s as free parameters to be found via an optimization problem $\\mathbb { D } \\mathbb { B }$ . To map latents $\\mathbf { z }$ to trajectories $x$ , we design a network architecture (i.e. a decoder) that disentangles camera poses and radiance field parameterization. Our decoder architecture is composed of 3 networks (shown in Fig. 2): ",
+        "bbox": [
+            173,
+            750,
+            825,
+            834
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• The camera pose decoder network $c$ (parameterized by $\\theta _ { c . }$ ), is responsible for predicting camera poses $\\hat { \\mathbf { T } } _ { s } \\in S E ( 3 )$ at the normalized temporal position $s \\in [ - 1 , 1 ]$ in the trajectory, conditioned on $\\mathbf { z } _ { \\mathrm { p o s e } }$ which represents the camera poses for the whole trajectory. To ensure that the output of $c$ is a valid camera pose (e.g. an element of $S E ( 3 )$ ), we output a 3D vector representing a normalized quaternion $\\mathbf { q } _ { s }$ for the orientation and a 3D translation vector $\\mathbf { t } _ { s }$ . ",
+        "bbox": [
+            174,
+            839,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/e4bad21ddfcad7a7125072fe03e7aa0257fb5650fce1690227c0a92664eaf018.jpg",
+        "image_caption": [
+            "Figure 2: Architecture of the decoder model that disentangles camera poses from 3D geometry and appearance of the scene. Our decoder is composed by 3 submodules. A decoder $d$ that takes as input a latent code representing the scene $\\mathbf { z } _ { \\mathrm { s c e n e } }$ and produces a factorized representation of 3D space via a tri-plane latent encoding $\\mathbf { W }$ . A radiance field network $f$ that takes as input points $ { \\mathbf { p } } \\in  { \\mathbf { R } } ^ { 3 }$ and is conditioned on W to predict a density $\\sigma$ and a signal a to be rendered via volumetric rendering (Eq. 1). Finally, we decode the camera poses through a network $c$ that takes as input a normalized temporal position $s \\in [ - 1 , 1 ]$ and is conditioned on $\\mathbf { z } _ { \\mathrm { p o s e } }$ which represents camera poses for the whole trajectory $x$ to predict the camera pose $\\hat { \\mathbf { T } } _ { s } \\in S E ( 3 )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            87,
+            825,
+            219
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• The scene decoder network $d$ (parameterized by $\\theta _ { d }$ ), is responsible for predicting a conditioning variable for the radiance field network $f$ . This network takes as input a latent code that represents the scene $\\mathbf { z } _ { \\mathrm { s c e n e } }$ and predicts an axis-aligned tri-plane representation $[ \\mathbb { A } 1 ] , \\bigstar ]$ $\\mathbf { W } \\in \\mathbb { R } ^ { 3 \\times S \\times S ^ { \\star } \\times F }$ . Which correspond to 3 feature maps $[ \\mathbf { W } _ { x y } , \\mathbf { \\bar { W } } _ { x z } , \\mathbf { \\bar { W } } _ { y z } ]$ of spatial dimension $S \\times S$ and $F$ channels, one for each axis aligned plane: $x y$ , $x z$ and $y z$ . ",
+        "bbox": [
+            173,
+            380,
+            825,
+            450
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• The radiance field decoder network $f$ (parameterized by $\\theta _ { f }$ ), is tasked with reconstructing image level targets using the volumetric rendering equation in Eq. $^ { 1 }$ . The input to $f$ is $ { \\mathbf { p } } \\in \\mathbb { R } ^ { 3 }$ and the tri-plane representation $\\mathbf { W } = [ \\mathbf { W } _ { x y } , \\mathbf { W } _ { x z } , \\mathbf { \\bar { W } } _ { y z } ]$ (we do not condition on the camera orientation to improve consistency $\\pmb { \\mathbb { I } } \\mathbf { \\varOmega }$ ). Given a 3D point $\\overset { \\circ } { \\mathbf { p } } = [ i , j , k ]$ for which radiance is to be predicted, we orthogonally project $\\mathbf { p }$ into each plane in W and perform bi-linear sampling. We concatenate the 3 bi-linearly sampled vectors into $\\mathbf { \\bar { w } } _ { x y z } = [ \\mathbf { W } _ { x y } ( i , j ) , \\mathbf { W } _ { x z } ( j , k ) , \\mathbf { W } _ { y z } ( i , k ) ] \\in \\mathbb { R } ^ { 3 F }$ , which is used to condition the radiance field function $f$ . We implement $f$ as a MLP that outputs a density value $\\sigma$ and a signal a. To predict the value $\\mathbf { v }$ of a pixel, the volumetric rendering equation is used (cf. Eq. 1) where a 3D point is expressed as ray direction $\\mathbf { r }$ (corresponding with the pixel location) at particular depth $u$ . ",
+        "bbox": [
+            173,
+            455,
+            826,
+            595
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c5c3740e02abc025279027c84bb643a78d16a70a4c88a7d1faa95b3918e8125f.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { { \\bf v } ( { \\bf r } , { \\bf W } ) = \\int _ { u _ { n } } ^ { u _ { f } } T r ( u ) \\sigma \\left( { \\bf r } ( u ) , { \\bf w } _ { x y z } \\right) { \\bf a } \\left( { \\bf r } ( u ) , { \\bf w } _ { x y z } \\right) d u } \\\\ & { } & { T r ( u ) = \\exp \\left( - \\int _ { u _ { n } } ^ { u } \\sigma ( { \\bf r } ( u ) , { \\bf w } _ { x y z } ) d u \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            598,
+            671,
+            670
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We formulate a reconstruction objective to jointly optimize for $\\theta _ { d } , \\theta _ { c } , \\theta _ { f }$ and $\\{ \\mathbf { z } \\} _ { i = \\{ 0 , \\ldots , n \\} }$ , shown in Eq. 2. Note that while latents $\\mathbf { z }$ are optimized for each example $x$ independently, the parameters of the networks $\\theta _ { d } , \\theta _ { c } , \\theta _ { f }$ are amortized across all examples $x \\in X$ . As opposed to previous auto-decoding approaches $\\textcircled { 1 2 } , \\textcircled { 3 9 } \\textcircled { 1 }$ , each latent $\\mathbf { z }$ is perturbed during training with additive noise that is proportional to the empirical standard deviation across all latents, $\\mathbf { z } = \\mathbf { z } + \\beta \\mathcal { N } ( 0 , \\mathrm { s t d } ( Z ) )$ , inducing a contractive representation $\\left[ \\left[ 5 0 \\right] \\right]$ . In this setting, $\\beta$ controls the trade-off between the entropy of the distribution of latents $\\mathbf { z } \\in Z$ and the reconstruction quality. With $\\beta = 0$ the distribution of $\\mathbf { z }$ ’s becomes a set of indicator functions (i.e. similar as one would get from a vanilla auto-encoder). For large $\\beta > 0$ the structure in distribution of $\\mathbf { z }$ ’s is destroyed, as latents are perturbed with large magnitudes of noise. We use a small $\\beta > 0$ value to enforce a latent space in which interpolated samples (or samples that contain small deviations from the empirical distribution, as the ones that one might get from sampling a subsequent generative model) are included in the support of the decoder function, sacrificing a small cost in reconstruction fidelity. ",
+        "bbox": [
+            173,
+            671,
+            826,
+            851
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4818bbecc9e6b13ea0efd6ec48d45d28e4f76e9cb1bf63856fca7a720b6c4cb2.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta _ { d } , \\theta _ { f } , \\theta _ { c } , Z } \\mathbb { E } _ { x \\sim X } \\left[ \\mathcal { L } _ { \\mathrm { s c e n e } } ( \\mathbf { x } _ { s } ^ { \\mathrm { i m } } , \\mathbf { z } _ { \\mathrm { s c e n e } } , \\mathbf { T } _ { s } ) + \\lambda \\mathcal { L } _ { \\mathrm { p o s e } } ( \\mathbf { T } _ { s } , \\mathbf { z } _ { \\mathrm { p o s e } } , s ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            853,
+            714,
+            880
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We optimize parameters $\\theta _ { d } , \\theta _ { f } , \\theta _ { c }$ and latents $\\mathbf { z } \\in Z$ with two different losses. The first loss function $\\mathcal { L } _ { \\mathrm { s c e n e } }$ measures the reconstruction between the radiance field encoded in $\\mathbf { z } _ { \\mathrm { s c e n e } }$ and the images in the trajectory $\\mathbf { x } _ { s } ^ { \\mathrm { { i m } } }$ (where $s$ denotes the normalized temporal position of the frame in the trajectory), given ground-truth camera poses $\\mathbf { T } _ { s }$ required for rendering. We use an $l _ { 2 }$ loss for RGB and $l _ { 1 }$ for depth 1. The second loss function $\\mathcal { L } _ { \\mathrm { p o s e } }$ measures the camera pose reconstruction error between the poses $\\hat { \\mathbf { T } } _ { s }$ encoded in $\\mathbf { z } _ { \\mathrm { p o s e } }$ and the ground-truth poses. We employ an $l _ { 2 }$ loss on translation and $l _ { 1 }$ loss for the normalized quaternion part of the camera pose. Although theoretically normalized quaternions are not necessarily unique (e.g. q and $- \\mathbf { q } )$ we do not observe any issues empirically during training. ",
+        "bbox": [
+            171,
+            882,
+            825,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            825,
+            193
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Prior Learning ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            207,
+            318,
+            223
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given a set of latents $\\mathbf { z } \\in Z$ resulting from minimizing the objective in Eq. $2 ,$ our goal is to learn a generative model $p ( Z )$ that captures their distribution (i.e. after minimizing the objective in Eq. $2$ we interpret $\\mathbf { z } \\in Z$ as examples from an empirical distribution in latent space). In order to model $p ( Z )$ we employ a Denoising Diffusion Probabilistic Model (DDPM) [17], a recent score-matching $[ \\overline { { 1 8 } } ]$ based model that learns to reverse a diffusion Markov Chain with a large but finite number of timesteps. In DDPMs $\\mathbb { \\lVert 1 7 \\rVert }$ it is shown that this reverse process is equivalent to learning a sequence of denoising auto-encoders with tied weights. The supervised denoising objective in DDPMs makes learning $p ( Z )$ simple and scalable. This allows us to learn a powerful generative model that enables both unconditional and conditional generation of 3D scenes. For training our prior $p _ { \\theta _ { p } } ( Z )$ we take the objective function in $ { \\mathbb { I } } ^ { [ 1 2 ] }$ defined in Eq. $\\textcircled { 3 }$ In Eq. $\\sum t$ denotes the timestep, $\\epsilon \\sim \\mathcal { N } ( 0 , \\mathbf { I } )$ is the noise and $\\bar { \\alpha } _ { t }$ is a noise magnitude parameter with a fixed scheduling. Finally, $\\epsilon _ { \\theta _ { p } }$ denotes the denoising model. ",
+        "bbox": [
+            173,
+            232,
+            826,
+            400
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cad09aae2fc0f261fdd8eb7f9ba492672c3404732eec1c1102262061dc203f74.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta _ { p } } \\mathbb { E } _ { t , \\mathbf { z } \\sim Z , \\epsilon \\sim \\mathcal { N } ( 0 , \\mathbf { I } ) } \\left[ \\| \\epsilon - \\epsilon _ { \\theta _ { p } } \\big ( \\sqrt { \\bar { \\alpha } _ { t } } \\mathbf { z } + \\sqrt { 1 - \\bar { \\alpha } _ { t } } \\epsilon , t \\big ) \\| ^ { 2 } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            417,
+            684,
+            444
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "At inference time, we sample $\\mathbf { z } \\sim p _ { \\theta _ { p } } ( Z )$ by following the inference process in DDPMs. We start by sampling $\\mathbf { z } _ { T } \\sim \\mathcal { N } ( 0 , \\mathbf { I } )$ and iteratively apply $\\epsilon _ { \\theta _ { p } }$ to gradually denoise $\\mathbf { z } _ { T }$ , thus reversing the diffusion Markov Chain to obtain $\\mathbf { z } _ { 0 }$ . We then feed $\\mathbf { z } _ { 0 }$ as input to the decoder architecture (cf. Fig. $2 )$ and reconstruct a radiance field and a camera path. ",
+        "bbox": [
+            174,
+            446,
+            825,
+            505
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "If the goal is to learn a conditional distribution of the latents $p ( Z | Y )$ , given paired data $\\{ \\mathbf { z } \\in Z , y \\in$ $Y \\}$ , the denoising model $\\epsilon _ { \\theta }$ is augmented with a conditioning variable $y$ , resulting in $\\epsilon _ { \\theta _ { p } } ( \\mathbf { z } , t , y )$ , implementation details about how the conditioning variable is used in the denoising architecture can be found in the appendix C. ",
+        "bbox": [
+            174,
+            511,
+            825,
+            569
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            585,
+            312,
+            603
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section we show the applicability of GAUDI to multiple problems. First, we evaluate reconstruction quality and performance of the reconstruction stage. Then, we evaluate the performance of our model in generative tasks including unconditional and conditional inference, in which radiance fields are generated from conditioning variables corresponding to images or text prompts. Full experimental settings and details can be found in the appendix B ",
+        "bbox": [
+            174,
+            617,
+            825,
+            688
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 Data ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            703,
+            246,
+            718
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We report results on 4 datasets: Vizdoom $\\mathbb { \\left| \\left[ 2 . 3 \\right] \\right| }$ , Replica [64], VLN-CE [26] and ARKit Scenes [1], which vary in number of scenes and complexity (see Fig. 3 and Tab. 1). ",
+        "bbox": [
+            171,
+            727,
+            825,
+            757
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Vizdoom $\\pmb { \\mathbb { Z } 3 } \\|$ : Vizdoom is a synthetic simulated environment with simple texture and geometry. We use the data provided by $\\pmb { \\mathbb { B } } ] \\mathbf { l }$ to train our model. It is the simplest dataset in terms of number of scenes and trajectories, as well as texture, serving as a test bed to examine GAUDI in the simplest setting. ",
+        "bbox": [
+            174,
+            762,
+            825,
+            805
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Replica $\\pmb { \\mathbb { \\lVert 6 4 \\rVert } }$ : Replica is a dataset comprised of 18 realistic scenes from which trajectories are rendered via Habitat $\\mathbb {  \\boldsymbol {  9 }  }$ . We used the data provided by $\\textcircled { 8 }$ to train our model. ",
+        "bbox": [
+            174,
+            810,
+            823,
+            839
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "VLN-CE $\\left[ \\left[ 2 6 \\right] \\right]$ : VLN-CE is a dataset originally designed for vision and language navigation in continuous environments. This dataset is composed of 3.6K trajectories of an agent navigating between two points in a 3D scene from the Matterport 3D dataset $\\mathbb { H }$ . We render observations via ",
+        "bbox": [
+            174,
+            844,
+            825,
+            887
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/7097bcaa55df2e214b67e47ccadd3bb523e55c7c711e34fc104dfcb75226e874.jpg",
+        "image_caption": [
+            "Figure 3: (a) Examples of the 4 datasets we use in this paper (from left to right): Vizdoom $[ [ 2 3 ] ]$ , Replica $\\pmb { \\| 6 4 \\| }$ , VLN-CE $\\left[ \\left[ 2 6 \\right] \\right]$ , ARKitScenes [1]. (b) Layouts for two scenes in VLN-CE $\\pmb { \\Vert 2 6 \\Vert }$ , where navigable areas are shaded in dark gray. Blue and red dots represent start-end positions and the camera path is highlighted in blue. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            88,
+            820,
+            190
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Habitat $\\mathbb { \\left[ \\left. 5 9 \\right. \\right. }$ . Notably, this dataset contains also textual descriptions of the trajectories taken by an agent. In Sect. 4.5 we train GAUDI in a conditional manner to generate 3D scenes given a description. ",
+        "bbox": [
+            173,
+            279,
+            823,
+            308
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "ARKitScenes [1]: ARKitScenes is a dataset of scans of indoor spaces. This dataset contains more than 5K scans of about 1.6K different indoor spaces. As opposed to the previous datasets where RGB, depth and camera poses are obtained via rendering in a simulation (i.e. either Vizdoom $\\mathbb { \\left[ \\left. 2 3 \\right] \\right. }$ or Habitat [59]), ARKitScenes provides raw RGB and depth of the scans and camera poses estimated using ARKit SLAM. In addition, whereas trajectories from the previous datasets are point-to-point, as typically done in navigation, the camera trajectories for ARKitScenes resembles a natural scan a of full indoor space. In our experiments we use a subset of 1K scans from ARKitScenes to train our models. ",
+        "bbox": [
+            173,
+            313,
+            826,
+            424
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Reconstruction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            439,
+            318,
+            454
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We first validate the hypothesis that the optimization problem described in Eq. $2$ can find latent codes $\\mathbf { z }$ that are able reconstruct the trajectories in the empirical distribution in a satisfactory way. In Tab. 1 we report reconstruction performance of our model across all datasets. Fig. $\\sharp$ shows reconstructions of random trajectories for each dataset. For all our experiments we set the dimension of $\\mathbf { z } _ { \\mathrm { s c e n e } }$ and $\\mathbf { z } _ { \\mathrm { p o s e } }$ to 2048 and $\\beta = 0 . 1$ unless otherwise stated. During training, we normalize camera poses for each trajectory so that the middle frame in a trajectory becomes the origin of the coordinate system. See appendix $\\boxed { \\ E }$ for ablation experiments. ",
+        "bbox": [
+            173,
+            464,
+            826,
+            561
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/2a55d72050e7abf5dc63fe3d29bdd1e15b96487d65ea03ab1a1bcf5616c9bcc1.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative reconstruction results of random trajectories on different datasets (one for each column): Vizdoom $\\pmb { \\left. \\pmb { \\left. \\bar { 2 3 } \\right. } \\right. }$ , Replica $\\pmb { \\mathbb { \\lVert 6 4 \\rVert } }$ , VLN-CE $\\left[ \\left[ 2 6 \\right] \\right]$ and ARKitScenes [1]. For each pair of images the left is ground-truth and right is reconstruction. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            573,
+            823,
+            636
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/a09dcc79a31770431fc08d0afb30c075a93746572b63c805c0e6cc65b2bfe061.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>#sc-#tr-#im</td><td>11↓</td><td>PSNR↑</td><td>SSIM个</td><td>Rot Err. ↓</td><td>Trans. Err ↓</td></tr><tr><td>Vizdoom 2</td><td>1-32-1k</td><td>0.004</td><td>44.42</td><td>0.98</td><td>0.01</td><td>1.26</td></tr><tr><td>Replica √</td><td>18-100-1k</td><td>0.006</td><td>38.86</td><td>0.99</td><td>0.03</td><td>0.01</td></tr><tr><td>VLN-CE [26</td><td>90-3.6k-600k</td><td>0.031</td><td>25.17</td><td>0.73</td><td>0.30</td><td>0.02</td></tr><tr><td>ARKitScenes</td><td>300-1k-600k</td><td>0.039</td><td>24.51</td><td>0.76</td><td>0.16</td><td>0.04</td></tr></table>",
+        "bbox": [
+            279,
+            707,
+            715,
+            767
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Reconstruction results of the optimization process described in Eq. 2. The first column shows the number of scenes (#sc), trajectories (#tr) and images (#im) per dataset. Due to the large number of images on VLN-CE $\\mathbb { \\left[ \\left[ 2 6 \\right] \\right] }$ and ARKitScenes $\\mathbb { I I }$ datasets we sample 10 random images per trajectory to compute the reconstruction metrics. ",
+        "bbox": [
+            173,
+            771,
+            826,
+            827
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 Interpolation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            857,
+            305,
+            872
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In addition, to evaluate the structure of the latent representation obtained from minimizing the optimization problem in Eq. $\\bigstar ,$ we show interpolation results between pairs of latents $( \\mathbf { z } _ { i } , \\mathbf { z } _ { j } )$ in ",
+        "bbox": [
+            174,
+            883,
+            823,
+            912
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/6cddcf51ce96aa1d9d58ca018cda77b4d7a30a8b1859865add9ed9c19a2902b3.jpg",
+        "image_caption": [
+            "Fig. $\\boxed { 5 }$ To render images while interpolating the scene we place a fixed camera at the origin of the coordinate system. We observe a smooth transition of scenes in both geometry (walls, ceilings) and texture (stairs, carpets). More visualizations are included in the appendix H. ",
+            "Figure 5: Interpolation of 3D scenes in latent space (e.g. interpolating the encoded radiance field) for the VLN-CE dataset $\\pmb { \\mathbb { Z } } 6 \\|$ . Each row corresponds to a different interpolation path. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            143,
+            823,
+            315
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 Unconditional generative modeling ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            380,
+            457,
+            395
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Given latent representations $\\textbf { z } \\in \\ Z$ that can reconstruct samples $x \\in X$ with high accuracy as shown in Sect. $\\boxed { 4 . 2 }$ we now evaluate the capacity of the prior $p _ { \\theta _ { p } } ( Z )$ to capture the empirical distribution $x \\in \\mathcal { X }$ by learning the distribution of latents $\\mathbf { z } _ { i } \\in Z$ . To do so we sample $\\mathbf { z } \\sim p _ { \\theta _ { p } } ( Z )$ by following the inference process in DDPMs, and then feed $\\mathbf { z }$ through the decoder network, which results in trajectories of RGB images that are then used for evaluation. We compare our approach with the following baselines: GRAF $\\pmb { \\| 6 0 \\| }$ , $\\pi$ -GAN $\\pmb { \\mathbb { H } }$ and GSN $\\textcircled { 8 }$ , where all models have access to ground-truth depth information during training. We sample 5k images from predicted and target distributions for each model and dataset and report both FID $\\mathbb { \\left[ \\left[ 1 6 \\right] \\right] }$ and SwAV-FID $\\begin{array} { r l } {  { \\bigl [ \\bigl | 3 5 \\bigr | \\bigr ] } } \\end{array}$ scores. We report quantitative results in Tab. $^ { 2 , }$ where we can see that GAUDI obtains state-of-the-art performance across all datasets and metrics. We attribute this performance improvement to the fact that GAUDI learns disentangled yet corresponding latents for radiance fields and camera poses, which is key when modeling scenes (see ablations in the appendix $\\mathbf { E } )$ . We note that to obtain these great empirical results GAUDI needs to simultaneously find latents with high reconstruction fidelity while also efficiently learning their distribution. ",
+        "bbox": [
+            173,
+            405,
+            825,
+            599
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/eaee20d37ec6a637e2d74ff12b2900b2de01d59c027ed32ba75d6bb6b934a52d.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Generative performance of state-of-the-art approaches for generative modelling of radiance fields on 4 scene datasets: Vizdoom $\\mathbb { \\left[ \\left. 2 3 \\right\\| \\right. }$ , Replica $\\textcircled { 6 4 } \\textcircled { 1 6 }$ , VLN-CE $\\dot { \\left\\| 2 6 \\right\\| }$ and ARKitScenes [1], according to FID [16] and SwAV-FID $\\pmb { \\Vert 3 5 \\Vert }$ metrics. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">VizDoom 因</td><td colspan=\"2\">Replica 国</td><td colspan=\"2\">VLN-CE 回</td><td colspan=\"2\">ARKitScenes</td></tr><tr><td>FID↓</td><td>SwAV-FID↓</td><td>FID↓</td><td>SwAV-FID↓</td><td>FID↓</td><td>SwAV-FID↓</td><td>FID↓</td><td>SwAV-FID↓</td></tr><tr><td>GRAF60</td><td>47.50 ± 2.13</td><td>5.44 ± 0.43</td><td>65.37 ± 1.64</td><td>5.76 ± 0.14</td><td>90.43±4.83</td><td>8.65±0.27</td><td>87.06±9.99</td><td>13.44±0.26</td></tr><tr><td>π-GAN</td><td>143.55 ± 4.81</td><td>15.26 ±0.15</td><td>166.55 ± 3.61</td><td>13.17 ±0.20</td><td>151.26 ± 4.19674 14.07 ±0.56</td><td></td><td>134.80±10.60</td><td>15.58 ±0.13</td></tr><tr><td>GSN</td><td>37.21 ± 1.17</td><td>4.56±0.19</td><td>41.75 ±1.33</td><td>4.14±0.02</td><td>43.32 ±8.86</td><td>6.19±0.49</td><td>79.54 ± 2.60</td><td>10.21 ± 0.15</td></tr><tr><td>GAUDI</td><td>33.70±1.27</td><td>3.24±0.12</td><td>18.75±0.63</td><td>1.76 ±0.05</td><td>18.52 ±0.11</td><td>3.63±0.65</td><td>37.35 ±0.38</td><td>4.14±0.03</td></tr></table>",
+        "bbox": [
+            176,
+            614,
+            821,
+            689
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Fig. $\\boxed { 6 }$ we show samples from the unconditional distribution learnt by GAUDI for different datasets. We observe that GAUDI is able to generate diverse and realistic 3D scenes from the empirical distribution which can be rendered from the sampled camera poses. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            799
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.5 Conditional Generative Modeling ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            815,
+            446,
+            832
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "GAUDI can also tackle conditional generative problems of the form $p ( Z | Y )$ , where a conditioning variable $y \\in Y$ is given to condition $p ( Z )$ . For a given conditional inference problem we assume the existence of paired data [46, 11, 45]. As an example, for training a text-conditional model we assume the existence of pairs $\\left\\{ \\mathbf { z } _ { i } , y _ { i } \\right\\}$ , where $\\mathbf { z } _ { i }$ is a latent scene representation and $y _ { i }$ is its corresponding text prompt. In this section we show both quantitative and qualitative results for conditional inference problems. The first conditioning variable we consider are textual descriptions of trajectories. Second, we consider a model where randomly sampled RGB images in a trajectory act as conditioning. Finally, we use a categorical variable that indicates the 3D environment (i.e. the particular indoor space) from which each trajectory was obtained. Tab. $3$ shows quantitative results for the different conditional inference problems. Details on the implementation of the conditional DDPM are given in Sect. D ",
+        "bbox": [
+            174,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/824f78214958dbf2b3c9b035017c70eb707ec7077e5001af194aaa5e4f2fdd8b.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            88,
+            820,
+            299
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            311,
+            826,
+            382
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/80d267a4bf2bfe73ea720c9664d903b480c62c5f6bd55112247243fda5b8cd91.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Avg.△Per-Environment</td><td></td></tr><tr><td>FID↓</td><td>SwAV-FID↓</td></tr><tr><td>-50.79</td><td>-4.10</td></tr></table>",
+        "bbox": [
+            630,
+            395,
+            763,
+            445
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/bb02b0e496c30ab10c3f3442aad13fe2259888df2155a1b387e0e8fb41c46a14.jpg",
+        "table_caption": [
+            "Table 3: Quantitative results of Conditional Generative Modeling on VLN-CE [26] dataset. GAUDI is able to produce high-quality scene renderings with low FID and SwAV-FID scores. In the right table we show the difference in average per-environment FID score between the conditional and unconditional models. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Text Conditioning</td><td colspan=\"2\">Image Conditioning</td><td colspan=\"2\">Categorical Conditioning</td></tr><tr><td>FID↓</td><td>SwAV-FID↓</td><td>FID↓</td><td>SwAV-FID↓</td><td>FID↓</td><td>SwAV-FID↓</td></tr><tr><td>18.50</td><td>3.75</td><td>19.51</td><td>3.93</td><td>18.74</td><td>3.61</td></tr></table>",
+        "bbox": [
+            235,
+            396,
+            601,
+            445
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.5.1 Text Conditioning ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            536,
+            351,
+            553
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We tackle the challenging task of training a text conditional model for 3D scene generation. We use the navigation text descriptions provided in VLN-CE $\\left[ \\left[ 2 6 \\right] \\right]$ to condition our model. These text descriptions contain high level information about the scene as well as the navigation path (i.e. \"Walk out of the bedroom and into the living room\", \"Exit the room through the swinging doors and then enter the bedroom\"). We employ a pre-trained RoBERTa-base $\\dot { \\left\\| 2 9 \\right\\| }$ text encoder and use its intermediate representation to condition the diffusion model. Fig. $^ { 7 }$ shows qualitative results of GAUDI for this task. To the best of our knowledge, this is the first model that allows for conditional 3D scene generation from text in an amortized manner (i.e. without distilling CLIP $\\mathbb { \\lVert H \\ 4 \\rVert }$ through a costly optimization problem [19, 32]). ",
+        "bbox": [
+            173,
+            560,
+            825,
+            686
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/6a315b5a6d145501104248636bd4e02075a4f7ce8251740624ef11afacfcd0e1.jpg",
+        "image_caption": [
+            "Figure 7: Text conditional 3D scene generation using GAUDI (one sample per row). Our model is able to capture the conditional distributions of scenes by generating multiple plausible scenes and camera paths that match the given text prompts. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            702,
+            820,
+            843
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.5.2 Image Conditioning ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            92,
+            364,
+            106
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We now analyze whether GAUDI is able to pick up information from the RGB images to predict a distribution over $Z$ . In this experiment we randomly pick images in a trajectory $x \\in X$ and use it as a conditioning variable $y$ . For this experiment we use trajectories in the VLN-CE dataset $\\pmb { \\left. 2 6 \\right. }$ . During each training iteration we sample a random image for each trajectory $x$ and use it as a conditioning variable. We employ a pre-trained ResNet-18 [15] as an image encoder. During inference, the resulting conditional GAUDI model is able to sample radiance fields where the given image is observed from a stochastic viewpoint. In Fig. $\\textcircled { 8 }$ we show samples from the model conditioned on different RGB images. ",
+        "bbox": [
+            173,
+            114,
+            825,
+            227
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/1bff990965fcbaf030c03ad42819efa0570a7bd8ebecadef306163172b4e5182.jpg",
+        "image_caption": [
+            "Figure 8: Image conditional 3D scene generation using GAUDI (one sample per row). Given a conditioned image (top row), our model is able to sample scenes where the same or contextually similar view is observed from a stochastic viewpoint. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            236,
+            823,
+            416
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/c2a11d158e4378324735e7e1ea88f9415e6b1d9a49b5ed5bd6e2557b410a5e3c.jpg",
+        "image_caption": [
+            "Figure 9: Samples from the GAUDI model conditioned on a categorical variable denoting the indoor scene (one sample per row). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            488,
+            821,
+            633
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.5.3 Categorical Conditioning ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            694,
+            401,
+            708
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finally, we analyze how GAUDI performs when conditioned on a categorical variable that indicates the underlying 3D indoor environment in which each trajectory was recorded. We perform experiments in the VLN-CE $[ \\overline { { 2 6 } } ]$ dataset, where we employ a trainable embedding layer to learn a representation for categorical variables indicating each environment. We compare the per-environment FID score of conditional model with its unconditional counterpart. This per-enviroment FID score is computed only on real images of the same indoor environment that the model is conditioned on. Our hypothesis is that if the model efficiently captures the information in the conditioning variable it should capture the environment specific distribution better than its unconditional counterpart trained on the same data. In Tab. $\\textcircled { 3 }$ the last column shows difference (e.g. the $\\Delta$ ) on the average per-environment FID score between the conditional and unconditional model on VLN-CE dataset. We observe that the conditional model consistently obtains a better FID score than the unconditional model across all indoor environments, resulting in a sharp reduction of average FID and SwAV-FID scores. In addition, in Fig. 9 we show samples from the model conditioned on a given categorical variable. ",
+        "bbox": [
+            173,
+            718,
+            826,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            299,
+            106
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We have introduced GAUDI, a generative model that captures distributions of complex and realistic 3D scenes. GAUDI uses a scalable two-stage approach which first involves learning a latent representation that disentangles radiance fields and camera poses. The distribution of disentangled latent representations is then modeled with a powerful prior. Our model obtains state-of-the-art performance when compared with recent baselines across multiple 3D datasets and metrics. GAUDI can be used both for conditional and unconditional problems, and enabling new tasks like generating 3D scenes from text descriptions. ",
+        "bbox": [
+            174,
+            121,
+            826,
+            218
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            236,
+            266,
+            252
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "[1] Gilad Baruch, Zhuoyuan Chen, Afshin Dehghan, Yuri Feigin, Peter Fu, Thomas Gebauer, Daniel Kurz, Tal Dimry, Brandon Joffe, Arik Schwartz, et al. Arkitscenes: A diverse real-world dataset for 3d indoor scene understanding using mobile rgb-d data. In Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 1), 2021.   \n[2] Piotr Bojanowski, Armand Joulin, David Lopez-Paz, and Arthur Szlam. Optimizing the latent space of generative networks. arXiv preprint arXiv:1707.05776, 2017.   \n[3] Marcus Carter and Ben Egliston. Ethical implications of emerging mixed reality technologies. 2020.   \n[4] Lluis Castrejon, Nicolas Ballas, and Aaron Courville. Improved conditional vrnns for video prediction. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 7608–7617, 2019.   \n[5] Eric R Chan, Connor Z Lin, Matthew A Chan, Koki Nagano, Boxiao Pan, Shalini De Mello, Orazio Gallo, Leonidas Guibas, Jonathan Tremblay, Sameh Khamis, et al. Efficient geometry-aware 3d generative adversarial networks. arXiv preprint arXiv:2112.07945, 2021.   \n[6] Eric R Chan, Marco Monteiro, Petr Kellnhofer, Jiajun Wu, and Gordon Wetzstein. pi-GAN: Periodic implicit generative adversarial networks for 3d-aware image synthesis. arXiv preprint arXiv:2012.00926, 2020.   \n[7] Angel Chang, Angela Dai, Thomas Funkhouser, Maciej Halber, Matthias Niessner, Manolis Savva, Shuran Song, Andy Zeng, and Yinda Zhang. Matterport3d: Learning from rgb-d data in indoor environments. arXiv preprint arXiv:1709.06158, 2017.   \n[8] Terrance DeVries, Miguel Angel Bautista, Nitish Srivastava, Graham W Taylor, and Joshua M Susskind. Unconstrained scene generation with locally conditioned radiance fields. ICCV, 2021.   \n[9] Emilien Dupont, Miguel Angel Bautista, Alex Colburn, Aditya Sankar, Carlos Guestrin, Josh Susskind, and Qi Shan. Equivariant neural rendering. In International Conference on Machine Learning, pages 2761–2770. PMLR, 2020.   \n[10] Emilien Dupont, Hyunjik Kim, SM Ali Eslami, Danilo Jimenez Rezende, and Dan Rosenbaum. From data to functa: Your data point is a function and you can treat it like one. In International Conference on Machine Learning, pages 5694–5725. PMLR, 2022.   \n[11] Patrick Esser, Robin Rombach, and Bjorn Ommer. Taming transformers for high-resolution image synthesis. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12873–12883, 2021.   \n[12] Jiatao Gu, Lingjie Liu, Peng Wang, and Christian Theobalt. Stylenerf: A style-based 3d-aware generator for high-resolution image synthesis. arXiv preprint arXiv:2110.08985, 2021.   \n[13] Pengsheng Guo, Miguel Angel Bautista, Alex Colburn, Liang Yang, Daniel Ulbricht, Joshua M Susskind, and Qi Shan. Fast and explicit neural view synthesis. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 3791–3800, 2022.   \n[14] David Ha and Jürgen Schmidhuber. World models. arXiv preprint arXiv:1803.10122, 2018.   \n[15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   \n[16] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. arXiv preprint arXiv:1706.08500, 2017.   \n[17] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in Neural Information Processing Systems, 33:6840–6851, 2020.   \n[18] Aapo Hyvärinen and Peter Dayan. Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6(4), 2005.   \n[19] Ajay Jain, Ben Mildenhall, Jonathan T Barron, Pieter Abbeel, and Ben Poole. Zero-shot text-guided object generation with dream fields. arXiv preprint arXiv:2112.01455, 2021.   \n[20] Niharika Jain, Alberto Olmo, Sailik Sengupta, Lydia Manikonda, and Subbarao Kambhampati. Imperfect imaganation: Implications of gans exacerbating biases on facial data augmentation and snapchat selfie lenses. arXiv preprint arXiv:2001.09528, 2020.   \n[21] Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4401–4410, 2019.   \n[22] Tero Karras, Samuli Laine, Miika Aittala, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. Analyzing and improving the image quality of stylegan. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8110–8119, 2020.   \n[23] Michał Kempka, Marek Wydmuch, Grzegorz Runc, Jakub Toczek, and Wojciech Jaskowski. Vizdoom: ´ A doom-based ai research platform for visual reinforcement learning. In 2016 IEEE Conference on Computational Intelligence and Games (CIG), pages 1–8. IEEE, 2016.   \n[24] Seung Wook Kim, Jonah Philion, Antonio Torralba, and Sanja Fidler. Drivegan: Towards a controllable high-quality neural simulation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5820–5829, 2021.   \n[25] Adam R Kosiorek, Heiko Strathmann, Daniel Zoran, Pol Moreno, Rosalia Schneider, Sona Mokrá, and Danilo Jimenez Rezende. Nerf-vae: A geometry aware 3d scene generative model. In International Conference on Machine Learning, pages 5742–5752. PMLR, 2021.   \n[26] Jacob Krantz, Erik Wijmans, Arjun Majundar, Dhruv Batra, and Stefan Lee. Beyond the nav-graph: Vision and language navigation in continuous environments. In European Conference on Computer Vision (ECCV), 2020.   \n[27] Zihang Lai, Sifei Liu, Alexei A Efros, and Xiaolong Wang. Video autoencoder: self-supervised disentanglement of static 3d structure and motion. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 9730–9740, 2021.   \n[28] Lingjie Liu, Jiatao Gu, Kyaw Zaw Lin, Tat-Seng Chua, and Christian Theobalt. Neural sparse voxel fields. arXiv preprint arXiv:2007.11571, 2020.   \n[29] Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692, 2019.   \n[30] Willi Menapace, Stéphane Lathuilière, Sergey Tulyakov, Aliaksandr Siarohin, and Elisa Ricci. Playable video generation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10061–10070, 2021.   \n[31] Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger. Occupancy networks: Learning 3d reconstruction in function space. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4460–4470, 2019.   \n[32] Oscar Michel, Roi Bar-On, Richard Liu, Sagie Benaim, and Rana Hanocka. Text2mesh: Text-driven neural stylization for meshes. arXiv preprint arXiv:2112.03221, 2021.   \n[33] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. arXiv preprint arXiv:2003.08934, 2020.   \n[34] Yisroel Mirsky and Wenke Lee. The creation and detection of deepfakes: A survey. ACM Computing Surveys (CSUR), 54(1):1–41, 2021.   \n[35] Stanislav Morozov, Andrey Voynov, and Artem Babenko. On self-supervised image representations for GAN evaluation. In International Conference on Learning Representations, 2021.   \n[36] Alex Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob McGrew, Ilya Sutskever, and Mark Chen. Glide: Towards photorealistic image generation and editing with text-guided diffusion models. arXiv preprint arXiv:2112.10741, 2021.   \n[37] Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In International Conference on Machine Learning, pages 8162–8171. PMLR, 2021.   \n[38] Michael Niemeyer and Andreas Geiger. Giraffe: Representing scenes as compositional generative neural feature fields. arXiv preprint arXiv:2011.12100, 2020.   \n[39] Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Lovegrove. Deepsdf: Learning continuous signed distance functions for shape representation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 165–174, 2019.   \n[40] Despoina Paschalidou, Amlan Kar, Maria Shugrina, Karsten Kreis, Andreas Geiger, and Sanja Fidler. Atiss: Autoregressive transformers for indoor scene synthesis. Advances in Neural Information Processing Systems, 34, 2021.   \n[41] Songyou Peng, Michael Niemeyer, Lars Mescheder, Marc Pollefeys, and Andreas Geiger. Convolutional occupancy networks. arXiv preprint arXiv:2003.04618, 2, 2020.   \n[42] Ethan Perez, Florian Strub, Harm De Vries, Vincent Dumoulin, and Aaron Courville. Film: Visual reasoning with a general conditioning layer. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.   \n[43] Benjamin Planche, Xuejian Rong, Ziyan Wu, Srikrishna Karanam, Harald Kosch, YingLi Tian, Jan Ernst, and Andreas Hutter. Incremental scene synthesis. In Advances in Neural Information Processing Systems, pages 1668–1678, 2019.   \n[44] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International Conference on Machine Learning, pages 8748–8763. PMLR, 2021.   \n[45] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation. In International Conference on Machine Learning, pages 8821–8831. PMLR, 2021.   \n[46] Ali Razavi, Aaron Van den Oord, and Oriol Vinyals. Generating diverse high-fidelity images with vq-vae-2. Advances in neural information processing systems, 32, 2019.   \n[47] Daniel Rebain, Mark Matthews, Kwang Moo Yi, Dmitry Lagun, and Andrea Tagliasacchi. Lolnerf: Learn from one look. arXiv preprint arXiv:2111.09996, 2021.   \n[48] Christian Reiser, Songyou Peng, Yiyi Liao, and Andreas Geiger. Kilonerf: Speeding up neural radiance fields with thousands of tiny mlps. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 14335–14345, 2021.   \n[49] Xuanchi Ren and Xiaolong Wang. Look outside the room: Synthesizing a consistent long-term 3d scene video from a single image. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2022.   \n[50] Salah Rifai, Grégoire Mesnil, Pascal Vincent, Xavier Muller, Yoshua Bengio, Yann Dauphin, and Xavier Glorot. Higher order contractive auto-encoder. In Joint European conference on machine learning and knowledge discovery in databases, pages 645–660. Springer, 2011.   \n[51] Daniel Ritchie, Kai Wang, and Yu-an Lin. Fast and flexible indoor scene synthesis via deep convolutional generative models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6182–6190, 2019.   \n[52] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High-resolution image synthesis with latent diffusion models. arXiv preprint arXiv:2112.10752, 2021.   \n[53] Robin Rombach, Patrick Esser, and Björn Ommer. Geometry-free view synthesis: Transformers and no 3d priors, 2021.   \n[54] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computer-assisted intervention, pages 234–241. Springer, 2015.   \n[55] Negar Rostamzadeh, Emily Denton, and Linda Petrini. Ethics and creativity in computer vision. arXiv preprint arXiv:2112.03111, 2021.   \n[56] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, S. Sara Mahdavi, Rapha Gontijo Lopes, Tim Salimans, Jonathan Ho, David J Fleet, and Mohammad Norouzi. Photorealistic text-to-image diffusion models with deep language understanding, 2022.   \n[57] Mehdi S. M. Sajjadi, Henning Meyer, Etienne Pot, Urs Bergmann, Klaus Greff, Noha Radwan, Suhani Vora, Mario Lucic, Daniel Duckworth, Alexey Dosovitskiy, Jakob Uszkoreit, Thomas Funkhouser, and Andrea Tagliasacchi. Scene Representation Transformer: Geometry-Free Novel View Synthesis Through Set-Latent Scene Representations. CVPR, 2022.   \n[58] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. NeurIPS’16, page 2234–2242, Red Hook, NY, USA, 2016. Curran Associates Inc.   \n[59] Manolis Savva, Abhishek Kadian, Oleksandr Maksymets, Yili Zhao, Erik Wijmans, Bhavana Jain, Julian Straub, Jia Liu, Vladlen Koltun, Jitendra Malik, et al. Habitat: A platform for embodied ai research. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 9339–9347, 2019.   \n[60] Katja Schwarz, Yiyi Liao, Michael Niemeyer, and Andreas Geiger. GRAF: Generative radiance fields for 3d-aware image synthesis. arXiv preprint arXiv:2007.02442, 2020.   \n[61] Edward J Smith and David Meger. Improved adversarial systems for 3d object generation and reconstruction. In Conference on Robot Learning, pages 87–96. PMLR, 2017.   \n[62] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International Conference on Machine Learning, pages 2256–2265. PMLR, 2015.   \n[63] Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502, 2020.   \n[64] Julian Straub, Thomas Whelan, Lingni Ma, Yufan Chen, Erik Wijmans, Simon Green, Jakob J Engel, Raul Mur-Artal, Carl Ren, Shobhit Verma, et al. The replica dataset: A digital replica of indoor spaces. arXiv preprint arXiv:1906.05797, 2019.   \n[65] Hoang Thanh-Tung and Truyen Tran. Catastrophic forgetting and mode collapse in gans. In 2020 International Joint Conference on Neural Networks (IJCNN), pages 1–10. IEEE, 2020.   \n[66] Patrick Tinsley, Adam Czajka, and Patrick Flynn. This face does not exist... but it might be yours! identity leakage in generative models. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 1320–1328, 2021.   \n[67] Thomas Unterthiner, Sjoerd van Steenkiste, Karol Kurach, Raphael Marinier, Marcin Michalski, and Sylvain Gelly. Towards accurate generative models of video: A new metric & challenges. arXiv preprint arXiv:1812.01717, 2018.   \n[68] Arash Vahdat, Karsten Kreis, and Jan Kautz. Score-based generative modeling in latent space. Advances in Neural Information Processing Systems, 34:11287–11302, 2021.   \n[69] Aaron Van Den Oord, Oriol Vinyals, et al. Neural discrete representation learning. Advances in neural information processing systems, 30, 2017.   \n[70] Xinpeng Wang, Chandan Yeshwanth, and Matthias Nießner. Sceneformer: Indoor scene generation with transformers. In 2021 International Conference on 3D Vision (3DV), pages 106–115. IEEE, 2021.   \n[71] Daniel Watson, William Chan, Jonathan Ho, and Mohammad Norouzi. Learning fast samplers for diffusion models by differentiating through sample quality. In International Conference on Learning Representations, 2021.   \n[72] Daniel Watson, Jonathan Ho, Mohammad Norouzi, and William Chan. Learning to efficiently sample from diffusion probabilistic models. arXiv preprint arXiv:2106.03802, 2021.   \n[73] Alex Yu, Sara Fridovich-Keil, Matthew Tancik, Qinhong Chen, Benjamin Recht, and Angjoo Kanazawa. Plenoxels: Radiance fields without neural networks. arXiv preprint arXiv:2112.05131, 2021. ",
+        "bbox": [
+            178,
+            253,
+            826,
+            910
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            71,
+            828,
+            917
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            53,
+            828,
+            915
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            60,
+            828,
+            921
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[74] Alex Yu, Vickie Ye, Matthew Tancik, and Angjoo Kanazawa. pixelnerf: Neural radiance fields from one or few images. arXiv preprint arXiv:2012.02190, 2020. ",
+        "bbox": [
+            171,
+            92,
+            825,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[75] Peng Zhou, Lingxi Xie, Bingbing Ni, and Qi Tian. Cips-3d: A 3d-aware generator of gans based on conditionally-independent pixel synthesis. arXiv preprint arXiv:2110.09788, 2021. ",
+        "bbox": [
+            173,
+            126,
+            823,
+            154
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Checklist ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            178,
+            254,
+            194
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example: ",
+        "bbox": [
+            174,
+            209,
+            825,
+            265
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "• Did you include the license to the code and datasets? [Yes] See Section ??.   \n• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   \n• Did you include the license to the code and datasets? [N/A] ",
+        "bbox": [
+            217,
+            275,
+            825,
+            342
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below. ",
+        "bbox": [
+            173,
+            352,
+            823,
+            395
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            406,
+            339,
+            420
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] These will be discussed in appendix A   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] These will be discussed in appendix A   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            424,
+            825,
+            542
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            215,
+            546,
+            493,
+            561
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ",
+        "bbox": [
+            238,
+            565,
+            736,
+            595
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            212,
+            599,
+            393,
+            614
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We will release code publicly upon acceptance.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] These details will be clarified in B   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] These details will be clarified in B ",
+        "bbox": [
+            238,
+            618,
+            825,
+            750
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            215,
+            755,
+            825,
+            768
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] See Sect. 4   \n(b) Did you mention the license of the assets? [Yes]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            238,
+            772,
+            825,
+            893
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            212,
+            897,
+            705,
+            911
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            90,
+            826,
+            180
+        ],
+        "page_idx": 14
+    }
+]