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2024/Diffusion Generative Flow Samplers_ Improving learning signals through partial trajectory optimization/full.md

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+# DIFFUSION GENERATIVE FLOW SAMPLERS: IMPROVING LEARNING SIGNALS THROUGH PARTIAL TRAJECTORY OPTIMIZATION
+
+Dinghuai Zhang $^{1,2}$ ; Ricky T. Q. Chen $^{3}$ , Cheng-Hao Liu $^{1,4}$ , Aaron Courville $^{1,2}$  & Yoshua Bengio $^{1,2}$
+
+<sup>1</sup>Mila - Quebec AI Institute, <sup>2</sup>Université de Montréal, <sup>3</sup>FAIR, Meta AI, <sup>4</sup>McGill University
+
+# ABSTRACT
+
+We tackle the problem of sampling from intractable high-dimensional density functions, a fundamental task that often appears in machine learning and statistics. We extend recent sampling-based approaches that leverage controlled stochastic processes to model approximate samples from these target densities. The main drawback of these approaches is that the training objective requires full trajectories to compute, resulting in sluggish credit assignment issues due to use of entire trajectories and a learning signal present only at the terminal time. In this work, we present Diffusion Generative Flow Samplers (DGFS), a sampling-based framework where the learning process can be tractably broken down into short partial trajectory segments, via parameterizing an additional "flow function". Our method takes inspiration from the theory developed for generative flow networks (GFlowNets), allowing us to make use of intermediate learning signals. Through various challenging experiments, we demonstrate that DGFS achieves more accurate estimates of the normalization constant than closely-related prior methods.
+
+# 1 INTRODUCTION
+
+While diffusion models for generative modeling can be trained with efficient simulation-free training algorithms, their application in solving sampling problems—where no data is given but only an unnormalized density function—remains challenging. This is a very general problem formulation that appears across a diverse range of research fields including machine learning (ML) (Duane et al., 1987; Neal, 1995; Hernández-Lobato & Adams, 2015) and statistics (Neal, 1998; Andrieu et al., 2004; Zhang et al., 2021), and has wide application in scientific fields such as physics (Wu et al., 2018; Albergo et al., 2019; Noé et al., 2019) and chemistry (Frenkel & Smit, 1996; Hollingsworth & Dror, 2018; Holdijk et al., 2022).
+
+Two main lines of work to tackle this sampling problems are Monte Carlo (MC) methods and variational inference (VI). MC methods achieve sampling typically via a carefully designed Markov chain (MCMC) or ancestrally sample from sequential targets such as a series of annealed distributions (Moral & Doucet, 2002). Despite their wide application, MC methods can easily get stuck in local modes or suffer from long mixing time (Tieleman & Hinton, 2009; Desjardins et al., 2010). On the other hand, VI methods (Blei et al., 2016) use a parametric model to approximate the target distribution by minimizing some divergence. However, these models often possess limited expressiveness, and optimizing the divergence can lead to bad learning dynamics (Minka, 2001).
+
+Recent works have formulated sampling from unnormalized density functions as stochastic optimal control problems, where a diffusion model is trained as a stochastic process to generate the target distribution. At present, the difficulty in using these algorithms is in part due to the need to sequentially sample long trajectories during training, while only the final step is actually treated as a sample from the model. The popular Kullback-Leibler (KL) problem formulation only provides a learning signal for the terminal state, which can cause credit assignment problems in learning, especially for diffusion models which require fairly long sequential chains for sampling.
+
+In this work, we build on top of Lahlou et al. (2023), who noted that recent methods that train diffusion models given an unnormalized density function (Zhang & Chen, 2022; Vargas et al., 2023a) can be interpreted from a GFlowNet perspective (Bengio et al., 2023). Although most GFlowNet works perform amortized inference on combinatorial discrete domains, previous work (Lahlou et al., 2023) establishes a theoretical foundation for continuous space GFlowNets and conducts numerical analysis to show their feasibility. Based on this framework, we propose the Diffusion Generative Flow Sampler (DGFS), an enhanced training method of diffusion models for sampling when given an unnormalized density function. In particular, we formulate a training approach that learns intermediate "flow<sup>1</sup> functions" that aggregate information for training intermediate states. This allows us to further define partial trajectory-based training objectives, where we have access to only partial specification of the sampling path. We find that this approach significantly reduces gradient noise and helps stabilize convergence. Furthermore, our training framework enables us to receive learning signals at intermediate steps, even before the sequential sampling is completed, in contrast to previous methods that can only receive signals at the terminal steps.
+
+In summary, our contributions are as follows,
+
+- We propose DGFS, an effective algorithm that trains stochastic processes to sample from given unnormalized target densities.  
+- We show that, contrary to other diffusion-based sampling algorithms, DGFS can (1) update its parameters without having full trajectory specification; (2) receive intermediate signals even before a sampling path is finished.  
+- We show through extensive empirical study that these special advantages enable DGFS to benefit from more stable and informative training signals, with DGFS generating samples accurately from the target distribution.
+
+# 2 PRELIMINARIES
+
+# 2.1 SAMPLING AS STOCHASTIC OPTIMAL CONTROL
+
+We aim to sample from a  $D$ -dimensional target distribution  $\pi(\cdot)$  in  $\mathbb{R}^D$  defined by a given unnormalized target density  $\mu(\cdot)$  where  $\pi(\cdot) = \frac{\mu(\cdot)}{Z}$  and  $Z = \int \mu(\mathbf{x}) \, \mathrm{d}\mathbf{x}$ . Following a similar setup to prior works such as Tzen & Raginsky (2019); Zhang & Chen (2022); Vargas et al. (2023a), we consider a sequential latent variable model with a joint distribution denoted  $\mathcal{Q}$  of which samples follow a Markov process,
+
+$$
+\mathcal {Q} \left(\mathbf {x} _ {0}, \dots , \mathbf {x} _ {N}\right): \quad \mathbf {x} _ {n + 1} \sim P _ {F} (\cdot | \mathbf {x} _ {n}), \quad \mathbf {x} _ {0} \sim p _ {0} ^ {\text {r e f}} \left(\mathbf {x} _ {0}\right). \tag {1}
+$$
+
+Through this sequence model, we use the marginal distribution  $\mathcal{Q}(\mathbf{x}_N)$  at the terminal step  $N$  to approximate  $\pi (\mathbf{x}_N)$ . Here  $p_0^{\mathrm{ref}}(\cdot)$  is a reference distribution at time  $n = 0$ , and  $P_F(\cdot |\cdot)$  denotes a forward transition probability (or policy) distribution. This forward policy depends on  $n$ , i.e. it is  $P_F^{n + 1|n}(\cdot |\mathbf{x}_n)$ , but we omit the step index  $n$  out of simplicity or assume  $n$  is encoded in  $\mathbf{x}_n$ . We thus refer to  $\mathcal{Q}$  as the forward process in the following text<sup>2</sup>. In this work, we constrain the transition to be a unimodal Gaussian distribution:
+
+$$
+P _ {F} (\cdot | \mathbf {x} _ {n}) = \mathcal {N} (\cdot ; \mathbf {x} _ {n} + h \mathbf {f} (\mathbf {x} _ {n}, n), h \sigma^ {2} \mathbf {I}), \tag {2}
+$$
+
+where  $\mathbf{f}$  is a control drift parameterized by a neural network and  $h$  is a constant step size. We also define a reference process with the forward policy  $P_F^{\mathrm{ref}}(\cdot |\mathbf{x}_n) = \mathcal{N}(\cdot ;\mathbf{x}_n,h\sigma^2\mathbf{I})$
+
+$$
+\mathcal {Q} ^ {\text {r e f}} \left(\mathbf {x} _ {0}, \dots , \mathbf {x} _ {N}\right): \quad \mathbf {x} _ {n + 1} \sim P _ {F} ^ {\text {r e f}} (\cdot | \mathbf {x} _ {n}), \quad \mathbf {x} _ {0} \sim p _ {0} ^ {\text {r e f}} \left(\mathbf {x} _ {0}\right). \tag {3}
+$$
+
+Since  $P_F^{\mathrm{ref}}$  has no nonlinear  $\mathbf{f}$ , the marginal distributions of  $\mathcal{Q}^{\mathrm{ref}}$  at step  $n$  are known in closed form, denoted  $p_n^{\mathrm{ref}}(\cdot)$ . We can then construct a target process that corresponds to this reference process,
+
+$$
+\mathcal {P} \left(\mathbf {x} _ {0}, \dots , \mathbf {x} _ {N}\right) := \mathcal {Q} ^ {\text {r e f}} \left(\mathbf {x} _ {0}, \dots , \mathbf {x} _ {N}\right) \frac {\pi \left(\mathbf {x} _ {N}\right)}{p _ {N} ^ {\text {r e f}} \left(\mathbf {x} _ {N}\right)}. \tag {4}
+$$
+
+It can be shown that the marginal distribution of  $\mathcal{P}$  at the terminal time  $N$  is exactly proportional to the target  $\mu (\cdot)$  by marginalizing out the variables  $\mathbf{x}_0,\dots ,\mathbf{x}_{N - 1}$  in  $\mathcal{P}$ ; see Theorem 1 of Zhang & Chen (2022). Hence it is natural to consider learning the drift  $\mathbf{f}(\cdot ,\cdot)$  in Equation 2 by minimizing the KL divergence  $\mathcal{D}_{\mathrm{KL}}(\mathcal{Q}||\mathcal{P})$  between the forward process and this target process, which is equivalent to the following optimal control problem:
+
+$$
+\mathcal {Q} \left(\mathbf {x} _ {0: N}\right): \quad \mathbf {x} _ {n + 1} = \mathbf {x} _ {n} + h \mathbf {f} \left(\mathbf {x} _ {n}, n\right) + \sqrt {h} \sigma \boldsymbol {\epsilon} _ {n}, \quad \mathbf {x} _ {0} \sim p _ {0} ^ {\text {r e f}} (\cdot), \tag {5}
+$$
+
+$$
+\min  _ {\mathbf {f}} \mathbb {E} _ {\mathcal {Q}} \left[ \sum_ {n = 0} ^ {N - 1} \frac {h}{2 \sigma^ {2}} \| \mathbf {f} \left(\mathbf {x} _ {n}, n\right) \| ^ {2} + \log \Psi \left(\mathbf {x} _ {N}\right) \right], \Psi (\cdot) := \frac {p _ {N} ^ {\text {r e f}} (\cdot)}{\mu (\cdot)}, \tag {6}
+$$
+
+where  $\epsilon_{n}\sim \mathcal{N}(\mathbf{0},\mathbf{I}),\forall n$  . See Appendix B.1 for derivation.
+
+If we keep the total time  $T = hN$  fixed and push the number of steps  $N \to \infty$ , this formulation recovers the continuous-time stochastic optimal control formulation seen in prior works (Zhang & Chen, 2022):
+
+$$
+\mathcal {Q} \left(\mathbf {x} _ {[ 0, T ]}\right): \mathrm {d} \mathbf {x} _ {t} = \mathbf {f} \left(\mathbf {x} _ {t}, t\right) \mathrm {d} t + \sigma \mathrm {d} \mathbf {W} _ {t}, \quad \mathbf {x} _ {0} \sim p _ {0} ^ {\text {r e f}} \left(\mathbf {x} _ {0}\right), \tag {7}
+$$
+
+$$
+\min  _ {\mathbf {f}} \mathbb {E} _ {\mathcal {Q}} \left[ \int_ {0} ^ {T} \frac {1}{2 \sigma^ {2}} \| \mathbf {f} \left(\mathbf {x} _ {t}, t\right) \| ^ {2} \mathrm {d} t + \log \Psi \left(\mathbf {x} _ {T}\right) \right], \tag {8}
+$$
+
+where the forward process is defined by a variance exploding stochastic differential equation (VESDE) $^3$ , and  $\mathrm{dW}_t$  is a standard Wiener process. This formulation appears in standard stochastic optimal control theory (Kappen, 2005). The seminal path integral sampler (PIS) (Zhang & Chen, 2022) proposes to solve such optimal control problems with deep learning to tackle the sampling problem. Its follow-up, the denoising diffusion sampler (DDS) (Vargas et al., 2023a), takes a similar approach but adopts a variance preserving (VP) SDE rather than the variance exploding ones in Equation 5 and 7. We defer the derivation of VP modeling to Appendix B.2. The practical implementations of these prior works are performed in discrete time with constant step sizes, equivalent to the discrete formulation outlined above. Hence we also adopt the discrete-time formulation in the following sections for simplicity and better amenability to our proposed algorithm.
+
+# 2.2 GFLOWNETS
+
+Generative flow networks (Bengio et al., 2023, GFlowNets) are a family of amortized inference algorithms that treat the problem of generation with probability proportional to given rewards in an Markov decision process (MDP) as a sequential decision-making process.
+
+Let  $\mathcal{G} = (\mathcal{S},\mathcal{A})$  be a directed acyclic graph (DAG), where the set of vertices  $\mathbf{s}\in S$  are referred to as states, and the set of directed edges  $(\mathbf{s}\rightarrow \mathbf{s}^{\prime})\in \mathcal{A}\subseteq \mathcal{S}\times \mathcal{S}$  represent the transitions between the states and are called actions. Given a designated initial state  $\mathbf{s}_0$  with no incoming edges and a set of terminal states  $s_N$  without outgoing edges, a complete trajectory can be defined as a sequence of states  $\tau = (\mathbf{s}_0\to \mathbf{s}_1\to \dots \to \mathbf{s}_N)\in \mathcal{T}$ . We define the forward policy  $P_F(\mathbf{s}'|\mathbf{s})$  as a distribution over the children of every non-terminal state  $\mathbf{s}$ , which induces a distribution over complete trajectories,  $P_F(\tau) = \prod_{n = 0}^{N - 1}P_F(\mathbf{s}_{n + 1}|\mathbf{s}_n)$ . This in turn defines a marginal distribution over terminating states  $\mathbf{x}$  in the sense that  $P_T(\mathbf{x}) = \sum_{\tau \to \mathbf{x}}P_F(\tau)$ , where we call  $P_T(\cdot)$  the GFlowNet's terminating distribution. The aim of GFlowNet learning is to obtain a policy such that  $P_T(\cdot)\propto R(\cdot)$ , where  $R(\cdot):\mathcal{X}\to \mathbb{R}_{+}$  is a given reward function or unnormalized density, where we do not know the normalizing factor  $Z = \sum_{\mathbf{x}}R(\mathbf{x})$ . We also use the trajectory flow function  $F(\tau) = ZP_F(\tau)$  to incorporate the effect of the partition function. Accordingly, the state flow function is the marginalization of the trajectory flow function  $F(\mathbf{s}) = \sum_{\tau \ni \mathbf{s}}F(\tau):S\to \mathbb{R}_{+}$ . Most previous GFlowNet works are about composite discrete objects such as molecules; on the other hand, Lahlou et al. (2023) provide theoretical support and initial numerical demonstration for GFlowNets on continuous and hybrid (continuous and discrete) spaces. Based on this, our work aims to further tackle more complex sampling problems in continuous domains, i.e.  $S\subseteq \mathbb{R}^{D}$ .
+
+Detailed balance (DB). DB is one of the most important GFlowNet training losses. Apart from the forward policy and the state flow function, DB also requires the practitioners to parameterize an additional backward policy  $P_B(\mathbf{s}|\mathbf{s}';\pmb {\theta})$ , which is a distribution over the parents of any noninitial state  $\mathbf{s}^{\prime}\in S$ . The DB objective is defined for a single transition  $(\mathbf{s}\rightarrow \mathbf{s}^{\prime})$  as follows,
+
+$$
+\ell_ {\mathrm {D B}} (\mathbf {s}, \mathbf {s} ^ {\prime}; \boldsymbol {\theta}) = \left(\log \frac {F (\mathbf {s} ; \boldsymbol {\theta}) P _ {F} \left(\mathbf {s} ^ {\prime} \mid \mathbf {s} ; \boldsymbol {\theta}\right)}{F \left(\mathbf {s} ^ {\prime} ; \boldsymbol {\theta}\right) P _ {B} \left(\mathbf {s} \mid \mathbf {s} ^ {\prime} ; \boldsymbol {\theta}\right)}\right) ^ {2}. \tag {9}
+$$
+
+According to the theory of GFlowNets (Bengio et al., 2023), if the DB loss reaches 0 for any transition pair, then the forward policy samples correctly from the target distribution defined by  $R(\cdot)$ .
+
+# 3 DIFFUSION GENERATIVE FLOW SAMPLERS
+
+# 3.1 AMORTIZING TARGET INFORMATION INTO INTERMEDIATE STEPS
+
+Like many other probabilistic methods, the training objective of PIS is a KL divergence between the model and the target distribution (Equation 5). In particular, PIS's KL is calculated at the trajectory level, which means each computation relies on a complete trajectory  $\tau = (\mathbf{x}_0,\dots ,\mathbf{x}_N)$  sampled from the current model's forward process. Since the training signal  $\mu (\mathbf{x}_N)$  is only affected by the terminal state  $\mathbf{x}_N$ , long trajectories can potentially yield less informative credit assignment for the intermediate steps, which manifests as high gradient variance and can sometimes lead to more difficult optimization. Therefore, we propose to exploit intermediate learning signals that do not necessarily rely on the complete trajectory information. What if we could know or approximate the target distribution at any step  $n$ ? Note that the target process (Equation 4) can be decomposed into a product of conditional distributions
+
+$$
+\mathcal {P} \left(\mathbf {x} _ {0: N}\right) = \pi \left(\mathbf {x} _ {N}\right) \prod_ {n = 0} ^ {N - 1} P _ {B} \left(\mathbf {x} _ {n} \mid \mathbf {x} _ {n + 1}\right), \tag {10}
+$$
+
+where  $P_B(\cdot|\cdot)$  denotes the backward transition probability (or backward policy) derived from the joint of the target process. Then we have the expression for the marginal distribution at step  $n$ :
+
+$$
+p _ {n} \left(\mathbf {x} _ {n}\right) = \int \pi \left(\mathbf {x} _ {N}\right) \prod_ {l = n} ^ {N - 1} P _ {B} \left(\mathbf {x} _ {l} \mid \mathbf {x} _ {l + 1}\right) \mathrm {d} \mathbf {x} _ {n + 1: N}, \forall n \in \{0, \dots , N \}, \tag {11}
+$$
+
+where we set  $p_N(\cdot) = \pi (\cdot)$  for consistency. If we know the form of  $p_n(\cdot)$  then we could directly learn from it with shorter trajectories and thus achieve more efficient training. Unfortunately, although we know the form of  $P_B(\cdot|\cdot)$ , for general target distribution there is generally no known analytical expression for it. As a result, we propose to use a deep neural network  $F_{n}(\cdot;\pmb{\theta})$  with parameter  $\pmb{\theta}$  as a "helper" to approximate the unnormalized density of the  $n$ -th step target  $p_n(\cdot)$ .
+
+One naive way to train the  $F_{n}(\cdot ;\pmb{\theta})$  network is to fit its value to the MC quadrature estimate of the integral in Equation 11. However, every training step would then have to include such computationally expensive quadrature calculation, making the resulting algorithm extremely inefficient. Instead, we propose turning this quadrature calculation into an optimization problem in an amortized way, and instead train  $F_{n}(\cdot ;\pmb{\theta})$  to achieve the following constraint:
+
+$$
+F _ {n} \left(\mathbf {x} _ {n}; \boldsymbol {\theta}\right) \prod_ {l = n} ^ {N - 1} P _ {F} \left(\mathbf {x} _ {l + 1} \mid \mathbf {x} _ {l}; \boldsymbol {\theta}\right) = \mu \left(\mathbf {x} _ {N}\right) \prod_ {l = n} ^ {N - 1} P _ {B} \left(\mathbf {x} _ {l} \mid \mathbf {x} _ {l + 1}\right), \tag {12}
+$$
+
+for all partial trajectories  $\mathbf{x}_{n:N}$ . Here, we would parameterize both the function  $F_{n}$  and the distribution  $P_F$  from Equation 2 by deep neural networks. Once Equation 12 is achieved, integrating  $\mathbf{x}_{n+1:N}$  on both sides would lead to the fact that  $F_{n}(\mathbf{x}_{n};\boldsymbol{\theta})$  equals the integral in Equation 11. To put it another way,  $F_{n}(\cdot;\boldsymbol{\theta})$  amortizes the integration computation into the learning of  $\boldsymbol{\theta}$ . Notice that we only use  $\mu(\cdot)$  in Equation 12 because our problem setting provides only  $\mu$ . The unknown normalization constant is thus also absorbed into  $F_{n}$ . We construct a training objective by simply regressing the left hand side of Equation 12 to the right hand side. Furthermore, to ensure training stability, this regression is performed in the log space. In practice, the flow function at different steps share the same set of parameters; this is achieved by introducing an additional step embedding input for the  $F(\cdot,n;\boldsymbol{\theta})$  neural network. As a result, our method learns two neural networks (the forward policy, which is the same as in PIS, and the flow function) in the meantime.
+
+# 3.2 UPDATING PARAMETERS WITH INCOMPLETE TRAJECTORIES
+
+If we compare Equation 12 with  $n$  and  $n + 1$ , we obtain a formula with no dependence on  $\mu$ :
+
+$$
+F _ {n} \left(\mathbf {x} _ {n}; \boldsymbol {\theta}\right) P _ {F} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n}; \boldsymbol {\theta}\right) = F _ {n + 1} \left(\mathbf {x} _ {n + 1}; \boldsymbol {\theta}\right) P _ {B} \left(\mathbf {x} _ {n} \mid \mathbf {x} _ {n + 1}\right), \tag {13}
+$$
+
+where the details are deferred to Appendix C.1.
+
+GFlowNet perspective. This formulation is similar to the GFlowNet DB loss in Equation 9. Actually, it is not hard to see that the above modeling is a variant of GFlowNets. As Zhang et al. (2022a) has pointed out and as later adopted by Lahlou et al. (2023), there is a connection between diffusion modeling and GFlowNets. In the GFlowNet language, the target density  $\mu(\cdot)$  is the terminal reward function, the temporal-spatial specification space  $\{(n,\mathbf{x}_n):n\in \mathbb{N},\mathbf{x}_n\in \mathbb{R}^D\}$  is the GFlowNet state space  $S$ , the forward transition probability  $P_F(\cdot|\cdot)$  of the diffusion process  $\mathcal{Q}$  is the GFlowNet forward policy, the backward transition probability  $P_B(\cdot|\cdot)$  of the reference process  $\mathcal{Q}^{\mathrm{ref}}$  is the GFlowNet backward policy, and the helper network  $F_n(\cdot)$  is the state flow  $F(\mathbf{s})$ . Notice here that the backward policy is fixed and does not contain learnable parameters. In our setting, we use the step index  $n$  to distinguish between intermediate states  $\{(n,\mathbf{x}_n):n < N,\mathbf{x}_n\in \mathbb{R}^D\}$  and terminating states  $\{(N,\mathbf{x}_N),\mathbf{x}_N\in \mathbb{R}^D\}$ . What's more, the goal of GFlowNets, which is to generate terminal states with probability proportional to the reward value, matches the goal of sampling in our setting.
+
+It is thus worthwhile to investigate how GFlowNets could be trained in this context. Apart from the DB constraint, Madan et al. (2022) propose a more general subtrajectory balance constraint:  $F_{m}(\mathbf{x}_{m};\boldsymbol {\theta})\prod_{l = m}^{n - 1}P_{F}(\mathbf{x}_{l + 1}|\mathbf{x}_{l};\boldsymbol {\theta}) = F_{n}(\mathbf{x}_{n};\boldsymbol {\theta})\prod_{l = m}^{n - 1}P_{B}(\mathbf{x}_{l}|\mathbf{x}_{l + 1})$ , where  $\mathbf{x}_{m:n}$  is a subtrajectory. Notice that Equation 12 is actually a special case of this constraint if we specify  $n = N$ . The resulting objective for our method is
+
+$$
+\ell_ {\text {S u b T B}} \left(\mathbf {x} _ {m: n}; \boldsymbol {\theta}\right) = \left(\log \frac {F _ {m} \left(\mathbf {x} _ {m} ; \boldsymbol {\theta}\right) \prod_ {l = m} ^ {n - 1} P _ {F} \left(\mathbf {x} _ {l + 1} \mid \mathbf {x} _ {l} ; \boldsymbol {\theta}\right)}{F _ {n} \left(\mathbf {x} _ {n} ; \boldsymbol {\theta}\right) \prod_ {l = m} ^ {n - 1} P _ {B} \left(\mathbf {x} _ {l} \mid \mathbf {x} _ {l + 1}\right)}\right) ^ {2}. \tag {14}
+$$
+
+A training loss based on transitions and more generally partial trajectories makes it possible to update parameters without entire trajectory specification. This can significantly reduce the variance in the optimization, as shown by Madan et al. (2022) in a similar context. To better combine the training effect from subtrajectories with different lengths, the adopted objective is
+
+$$
+\mathcal {L} (\tau ; \boldsymbol {\theta}) = \frac {\sum_ {0 \leq m <   n \leq N} \lambda^ {n - m} \ell_ {\mathrm {S u b T B}} \left(\mathbf {x} _ {m : n}\right)}{\sum_ {0 \leq m <   n \leq N} \lambda^ {n - m}}, \tau = \left(\mathbf {x} _ {0}, \dots , \mathbf {x} _ {N}\right), \tag {15}
+$$
+
+where  $\lambda \in \mathbb{R}_+$  is a scalar for controlling assigned weights to different partial trajectories. For the modeling of the drift of the forward stochastic process, we use  $\mathbf{f}(\cdot ,n) / \sigma = \mathrm{NN}_1(\cdot ,n), + \mathrm{NN}_2(n)$ $\nabla \log \mu (\cdot)$ , which includes two different neural networks and gradient information from unnormalized target densities in a similar form to Langevin dynamics (Zhang & Chen, 2022). We have conduct extensive ablation study about the parameterization, training objectives, exploration techniques, and other considerations, and display them in Appendix C.
+
+Improved credit assignment with local signals. In both PIS and DDS algorithms, the only learning signal  $\mu (\mathbf{x}_N)$  comes at the end of complete trajectories  $\mathbf{x}_{0:N}$  when the density function is queried with  $\mathbf{x}_N$ . This is also the case for general GFlowNet methods, where we set  $F_{N}(\mathbf{x})\gets \mu (\mathbf{x}),\forall \mathbf{x}\in \mathbb{R}^{D}$ . In this way, the propagation of density information from terminal states to early states would be slow, thus affecting the training efficiency. We thus adopt the forward-looking trick proposed by Pan et al. (2023a) to incorporate intermediate learning signals for GFlowNets. The forward-looking technique essentially parameterizes the logarithm of the state flow function as  $\log F_n(\mathbf{x}_n;\pmb {\theta}) = \log \tilde{R}_n(\mathbf{x}_n) + \mathrm{NN}(\mathbf{x}_n;\pmb {\theta})$ , where  $\tilde{R}_n(\mathbf{x}_n)$  could be an arbitrary "forward-looking" estimate of how much energy  $\mathbf{x}_n$  could contribute to the final reward  $\mu (\mathbf{x}_N)$ . The details of the "NN" neural network are deferred to Appendix C.4. In the proposed DGFS algorithm, we take the form of the (log) partial reward to be
+
+$$
+\log \tilde {R} _ {n} (\cdot) = \left(1 - \frac {n}{N}\right) \log p _ {n} ^ {\text {r e f}} (\cdot) + \frac {n}{N} \log \mu (\cdot), \tag {16}
+$$
+
+which combines information from the target density and the reference marginal density.
+
+![](images/fefd87a29ba6e66fcb840d08c305b9c4322a87e2dde809919336de19323a8cb0.jpg)  
+Figure 1: Illustration of the DGFS algorithm. The proposed method can update from partial trajectories (colored segments) as well as intermediate signals (gray arrows).
+
+![](images/9f6396bc9e96c3fd180266e11b6172ff3cae4c2bb3b83eec93b8194d200dccc7.jpg)  
+Figure 2: Gradient variance of DGFS and PIS, explaining the better training effects of DGFS.
+
+We illustrate the resulting DGFS algorithm in Figure 1. Different from prior works, DGFS can update parameters with incomplete subtrajectories (colored curve segments in the figure) with the help of amortized flow functions, as well as intermediate signals at non-terminal step (grey arrows in the figure), which fundamentally improves the credit assignment process.
+
+Estimation of the partition function. For an arbitrary trajectory  $\tau = \{\mathbf{x}_n\}_{n=0}^N$ , one could define its (log) importance weight to be  $S(\tau; \boldsymbol{\theta}) = \log \mathcal{P}(\mathbf{x}_{0:N}) - \log \mathcal{Q}(\mathbf{x}_{0:N}; \boldsymbol{\theta})$ , where  $\mathcal{Q}(\mathbf{x}_{0:N}; \boldsymbol{\theta}) = p_0^{\mathrm{ref}}(\mathbf{x}_0) \prod_{n=0}^{N-1} P_F(\mathbf{x}_{n+1} | \mathbf{x}_n; \boldsymbol{\theta})$ . We obtain the following log-partition function ( $\log Z$ ) estimator:
+
+$$
+\log \sum_ {b = 1} ^ {B} \exp (S (\tau^ {(b)}; \boldsymbol {\theta})) - \log B \leq \log Z, \quad \tau^ {(b)} \sim \mathcal {Q} (\cdot ; \boldsymbol {\theta}), \tag {17}
+$$
+
+where  $B \in \mathbb{N}_{+}$  is the number of particles (i.e., trajectories) used for the estimation. Similar to the importance weighted auto-encoders (Burda et al., 2016), our estimator provides a valid lower bound of the logarithm of the normalizing factor with any forward process  $\mathcal{Q}$ . In this work, we use the difference between estimated log partition function and the ground truth as an evaluation metric. See the Appendix C.2 for additional analysis. We also remark that this approach is similar to the neural importance sampling in Nicoli et al. (2020); Wirnsberger et al. (2020) in the sense that we use the forward process  $\mathcal{Q}$  as the proposal distribution.
+
+# 3.3 DISCUSSION
+
+Variance of gradient updates. Figure 2 shows that the variance of stochastic gradients with DGFS is much smaller than that with PIS; notice that the comparison is valid as we take the same neural network architecture for both algorithms. We hypothesize that DGFS benefits from the low variance behavior due to the following two reasons: (1) From the perspective of temporal difference learning (Sutton & Barto, 2018; Kearns & Singh, 2000), it has long been discussed that policy learning from shorter trajectory data would lower the variance at the cost of potentially higher bias. Before this work, it was impossible for sampling with stochastic process algorithms to update with partial trajectories, while in the proposed DGFS method, we use an additional flow network to make this feasible. In Figure 3 we also show that the flow network can be properly learned and thus does not introduce much bias, indicating that DGFS achieves a better bias-variance trade-off. (2) Probabilistic ML researchers have shown that for KL divergence based distribution learning, the gradient will not vanish even if the optimal distribution has been reached; see Roeder et al. (2017) and Section 4 of Xu et al. (2022). Later works including Vaitl et al. (2022a,b) propose a different path-gradient estimator to tackle this issue with a similar insight with Roeder et al. (2017); see the related work section. On the other hand, the DGFS objective does achieve zero gradient with the optimal solution (c.f. Appendix C.3), making it a more stable algorithm.
+
+Modeling considerations. Notice that the DGFS derivation in Section 3 does not rely on the specific form of the stochastic process. Therefore, DGFS could take both the variance exploding (VE) (Zhang & Chen, 2022) or variance preserving (VP) (Vargas et al., 2023a) formulation. In our numerical study, we find the former to be more stable, and thus we default to it unless otherwise specified. We defer related ablations to the Appendix C.5.
+
+Table 1: Experimental results on benchmarking target densities. For each algorithm, we report its absolute estimation bias for the log normalizing factor and the corresponding standard deviation. Our method achieves the best performance among the diffusion modeling-based samplers.  
+
+<table><tr><td></td><td>MOG</td><td>FUNNEL</td><td>MANYWELL</td><td>VAE</td><td>Cox</td></tr><tr><td>SMC</td><td>0.289±0.112</td><td>0.307±0.076</td><td>22.36±7.536</td><td>14.34±2.604</td><td>99.61±8.382</td></tr><tr><td>VI-NF</td><td>1.354±0.473</td><td>0.272±0.048</td><td>2.677±0.016</td><td>6.961±2.501</td><td>83.49±2.434</td></tr><tr><td>CRAFT</td><td>0.348±0.210</td><td>0.454±0.485</td><td>0.987±0.599</td><td>0.521±0.239</td><td>13.79±2.351</td></tr><tr><td>FAB w/ BUFFER5</td><td>0.003±0.0005</td><td>0.0022±0.0005</td><td>0.032±0.004</td><td>N/A</td><td>0.19±0.04</td></tr><tr><td>PIS</td><td>0.036±0.007</td><td>0.305±0.013</td><td>1.391±1.014</td><td>2.049±2.826</td><td>11.28±1.365</td></tr><tr><td>DDS</td><td>0.028±0.013</td><td>0.416±0.094</td><td>1.154±0.626</td><td>1.740±1.158</td><td>N/A6</td></tr><tr><td>DGFS</td><td>0.019±0.008</td><td>0.274±0.014</td><td>0.904±0.067</td><td>0.180±0.083</td><td>8.974±1.169</td></tr></table>
+
+Convergence guarantees. Previous theoretical studies (Bortoli, 2022; Chen et al., 2022; Lee et al., 2022) have demonstrated that the distribution of terminal samples generated by a diffusion model can converge to the target distribution under mild assumptions if the control term is well learned. These findings are also applicable to DGFS since these proofs are independent of how the networks are trained. Additionally, Zhang et al. (2022a) have proven that a perfectly learned score term corresponds to a zero GFlowNet training loss under mild conditions, following theory of GFlowNets (Bengio et al., 2021; 2023). This provides assurance that a sufficiently well-trained DGFS can accurately sample from the target distribution.
+
+# 4 RELATED WORKS
+
+Sampling methods. MCMC methods (Brooks et al., 2011) are the most classical sampling algorithms, including Langevin dynamics (Welling & Teh, 2011), Hamiltonian Monte Carlo (Duane et al., 1987; Hoffman & Gelman, 2011; Chen et al., 2014), and others (Huang et al., 2023). With MCMC operators as building blocks, sequential MC methods (Liu, 2001; Doucet et al., 2001) generate samples ancestrally through a series of annealed targets. On the other hand, people turn to parametric models such as deep networks to learn samplers via variational inference (Rezende & Mohamed, 2015). One the most commonly adopted models is the normalizing flow (Dinh et al., 2014) due to its tractability and expressiveness. Various work have been conducted to improve normalizing flow based sampling (Müller et al., 2018; Gao et al., 2020; Wu et al., 2020; Chen et al., 2020; Gabri'e et al., 2021; Arbel et al., 2021; Midgley et al., 2022; de G. Matthews et al., 2022; Caselle et al., 2022; Gao et al., 2023). These methods support asymptotically unbiased estimation as shown in Nicoli et al. (2020). What's more, Vaitl et al. (2022a) propose to improve the gradient estimation of normalizing flow parameters with the path-gradient technique; Vaitl et al. (2022b) improve continuous normalizing flows based methods (Chen et al., 2018) with a similar technique. These methods have wide applications in natural sciences, for example in lattice field theories (Nicoli et al., 2021; 2023) and targeted free energy estimation (Wirnsberger et al., 2020).
+
+Recently, many researchers have developed powerful diffusion process to model distributions in the generative modeling community (Vincent, 2011; Sohl-Dickstein et al., 2015; Ho et al., 2020a; Song et al., 2020; Du et al., 2022). This inspires people to study how to utilize diffusion process as samplers to learn distribution from given unnormalized target densities (Tzen & Raginsky, 2019; Zhang & Chen, 2022; Vargas et al., 2021; Geffner & Domke, 2022; Doucet et al., 2022; Vargas et al., 2023a; Richter et al., 2023; Vargas & Nusken, 2023; Vargas et al., 2023c;b). Our work also falls into this category. Specifically, Vargas et al. (2023a) can be seen as a special case of the method proposed in Berner et al. (2022).
+
+GFlowNets. GFlowNet (Bengio et al., 2023) is a family of generalized variational inference algorithms with a focus on treating the sampling process as a sequential decision-making process.
+
+It is original proposed to sample diversity-seeking objects in structured scientific domains such as biological seque and molecule design (Bengio et al., 2021; Jain et al., 2022; 2023b; a; Shen et al., 2023; Zhang et al., 2023c; Hernandez-Garcia et al., 2023; Kim et al., 2023). However, people have developed GFlowNet theories and methodologies to be suitable for generating general objects with composite structures. GFlowNet has rich connection with existing probabilistic ML methods (Zhang et al., 2022b; Zimmermann et al., 2022; Malkin et al., 2023; Zhang et al., 2022a; Hu et al., 2023; Ma et al.). Malkin et al. (2023) also show that GFlowNet has the capability of training with off-policy data compared to previous amortized inference methods. As a policy learning framework under Markov decision processes (MDPs), researchers have also drawn fruitful connections between GFlowNet and reinforcement learning (Pan et al., 2023b;c;a). GFlowNet has also been demonstrated to have wide application in causal discovery (Deleu et al., 2022; 2023; Atanackovic et al., 2023), feature attribution (Li et al., 2023a), network structure learning (Liu et al., 2022), job scheduling (Zhang et al., 2023a), and graph combinatorial optimization (Zhang et al., 2023b). Most prior works focus on structured discrete data space. Lahlou et al. (2023) first provides a sound theoretical framework and initial numerical demonstration for using GFlowNet on continuous space. Our work follows this framework and scales up the performance. A prior study (Li et al., 2023b) initially explored GFlowNets in a continuous space, whereas their theory is demonstrated to have non-trivial flaws when considering integration under continuous measure as indicated by Lahlou et al. (2023).
+
+# 5 EXPERIMENTS
+
+# 5.1 BENCHMARKING TARGET DISTRIBUTIONS
+
+In this section, we conduct extensive experiments on a diverse set of challenging continuous sampling benchmarks to corroborate the effectiveness of DGFS against previous methods.
+
+Mixture of Gaussians (MoG) is a 2-dimensional Gaussian mixture where there are 9 modes designed to be well-separated from each other. The modes share the same variance of 0.3 and the means are located in the grid of  $\{-5,0,5\} \times \{-5,0,5\}$ .
+
+Funnel is a classical sampling benchmark problem from Neal (2003); Hoffman & Gelman (2011). This 10-dimensional density is defined by  $\mu (\mathbf{x}) = \mathcal{N}(x^{(0)};0,9)\mathcal{N}(\mathbf{x}^{(1:9)};\mathbf{0},\exp (x^{(0)})\mathbf{I})$
+
+Manywell (Midgley et al., 2022; Noé et al., 2019; Wu et al., 2020) is a 32-dimensional distribution defined by the product of 16 same 2-dimensional "double well" distributions, which share the unnormalized density of  $\mu(x,y) = \exp(-x^4 + 6x^2 + 0.5x - 0.5y^2)$ .
+
+VAE is a 30-dimensional sampling task where we use the latent posterior of a pretrained variational autoencoder (Kingma & Welling, 2014; Rezende et al., 2014, VAE) to be the target distribution.
+
+Cox denotes the 1600-dimensional target distribution of the log Gaussian cox process introduced in Møller et al. (1998) which models the positions of pine saplings. Its unnormalized density is  $\mu (\mathbf{x}) = \mathcal{N}(\mathbf{x};\mu ,\mathbf{K}^{-1})\prod_{d = 1}^{D}\exp (x^{(d)}y^{(d)} - \alpha x^{(d)})$ , where  $\mu, \mathbf{K}, \mathbf{y}, \alpha$  are constants.
+
+Baselines. We benchmark DGFS performance against a wide range of strong baseline methods. As an MCMC methods we include the sequential Monte Carlo sampler (Moral & Doucet, 2002, SMC), which is often considered the state-of-the-art sampling method. As for normalizing flow methods we include the variational inference with normalizing flows (Rezende & Mohamed, 2015, VI-NF) approach as well as more recent de G. Matthews et al. (2022, CRAFT) which is combined with MCMC. We also provide results of Midgley et al. (2022, FAB) with a replay buffer that utilizes the annealed importance sampling technique to augment the training. Last but not least, we compare with PIS (Zhang & Chen, 2022) and DDS (Vargas et al., 2023a) algorithms which are closest to our framework but trained with the KL divergence formulation (Equation 5) without the improved training techniques described in Section 3.
+
+Evaluation protocol. We measure the performance of different algorithms by their estimation bias of the log normalizing factor of unnormalized target distributions. Contrary to Zhang & Chen (2022)
+
+![](images/53672457945425069250796bff1170c1311e2803a5d5f0e720922c76845a4523.jpg)  
+Figure 3: The learned DGFS flow function and the ground truth samples from target process at different diffusion steps. This shows DGFS is able to learn flow functions correctly.
+
+![](images/d4c6ed2ab38ccef9cb2389fcf179b8e0c6354e2648b1433bb029255023a75db5.jpg)  
+Figure 4: Manywell plots. DGFS and DDS but not PIS recover all modes.
+
+![](images/0a0f651908c092f243dd73fdb66b347008f06365dba196f6c4f7c9856a27f7d7.jpg)
+
+which take the best checkpoint across the run, in this work we compute the average of the last ten checkpoints for a more robust comparison. All results are conducted with 5 random seeds.
+
+We display the benchmarking results in Table 1. We fail to achieve reasonable performance with DDS on Cox target where we mark it as "N/A". The advantage of the proposed DGFS method can be observed across different tasks, indicating it could indeed benefit from its design providing more informative training signals to achieve better convergence and capturing diverse modes even in high dimensional spaces. FAB with buffer produces highly stable results, which can be attributed to both annealed importance sampling, the use of buffers, larger networks, and other tricks. It is also remarkable that DGFS obtains very stable results compared to other baselines with diffusion modeling (PIS, DDS). We defer related experimental details and extra experiments to Appendix C.
+
+# 5.2 ANALYSIS
+
+First, we investigate the learning of the flow function introduced in Equation 12. In the first row of Figure 3, we visualize the learned flow function  $F_{n}(\mathbf{x}_{n})$  with  $n = 20,40,\dots ,100$  in the 2D MoG task. In the second row, we simulate samples from  $p_n(\mathbf{x}_n)$  by sampling according to Equation 11, i.e., moving ground truth  $\pi (\cdot)$  samples in the backward direction with policy  $P_B$ . The consistency between these two rows indicate that DGFS flow function can successfully approximate the intermediate marginal  $p_n(\cdot)$ , which makes it possible to provide an effective training signal for learning from partial trajectory pieces.
+
+For the MoG task, we demonstrate the samples together with ground truth contours in Figure 5. This visualization result matches the log partition function estimation bias in Table 1, where DGFS generates the nine modes more uniformly than other baselines. This corroborates the advantage of DGFS to capture all different modes in a complex landscape. We then show visualization results on the Manywell task in Figure 4. The Manywell distribution is 32-dimensional, so we display the joint of the first and third dimension. We plot the samples generated from the true distribution and different algorithms, together with the true density contours. This figure implies that PIS can model half of the modes in these two dimensions well, but miss the other two modes, whereas both DGFS and DDS could cover all four modes. Further, DGFS achieves slightly better mode capturing than DDS in the sense that its modes are more separate and thus closer to the true distribution, while for DDS there are considerable samples appear between the upper right mode and the bottom right mode, which is not ideal. We defer other visualizations to the Appendix C.5.
+
+To look into why DGFS outperforms PIS with similar architecture, we plot the learned drift network magnitude in Figure 6 in the Appendix. The result shows that the solution that DGFS achieved has more homogeneous magnitude outputs across different diffusion steps. This may explain one reason
+
+![](images/1b06aac924f7adc2895f80e52f23f1f2f41c068a7d2d6cf93159a9e66f344071.jpg)  
+Figure 5: MoG visualization of DGFS and other diffusion-based samplers shows that DGFS could capture the diverse modes well. The contours display the landscape of the target density.
+
+![](images/7d66d16976488592e6446def74178cf931b9a112d6614919c88b77e4c6ef2645.jpg)
+
+![](images/1dbb98f06fabd4be1af40a275c9cfea1c6b13c560d13bf262e19a0d296058774.jpg)
+
+why DGFS is easier to train than
+
+PIS, since the policy probabili-
+
+ties at different steps  $\{P_F(\cdot |\mathbf{x}_n)\}_{n}$  are actually parameterized by a single neural network with temporal embedding input. This makes the learning task more stationary with DGFS. We also conduct in-depth ablation studies to evaluate the design choices of DGFS in Appendix C.5 and C.7.
+
+# 6 CONCLUSION
+
+We propose the Diffusion Generative Flow Sampler (DGFS), a novel algorithm that trains diffusion models to sample from given unnormalized target densities. Different from prior works that could only learn from complete diffusion chains, DGFS could update its parameter with only partial specification of the stochastic process trajectory; what's more, DGFS can receive intermediate signals before completing the entire path. These features help DGFS benefit from efficient credit assignment and thus achieve better partition function estimation bias in the sampling benchmarks.
+
+The DGFS framework presents various opportunities for further research, fueled by its inherent limitations. To name a few, how can we better design the intermediate local signals in ways more sophisticated than the current straightforward approach? Additionally, can we harness DGFS's exploration capabilities to succeed in high dimensional tasks, pushing the boundaries of its performance? Can we combine DGFS with a prioritized replay buffer? This would largely reduce the number of querying  $\mu(\cdot)$ . From the perspective of application, there are also many interesting research questions. Can we deploy DGFS on scientific biological or chemical tasks such as protein conformation modeling, with equivariant modeling? We expect that exploring these future directions will lead to valuable insights and bring fruit follow-ups of DGFS.
+
+# ACKNOWLEDGEMENT
+
+The authors are grateful to the discussions with Francisco Vargas, Julius Berner, Lorenz Richter, Laurence Midgley, Minsu Kim, Yihang Chen, and Qinsheng Zhang.
+
+# ETHICS STATEMENT
+
+We commit to the ICLR Code of Ethics and affirm that our work only utilizes public datasets for experimentation. While our empirical results are largely based on synthetic data, we acknowledge the potential for misuse and urge the responsible application of the proposed methods with real-world data. We welcome any related discussions and feedback.
+
+# REPRODUCIBILITY STATEMENT
+
+We provide detailed algorithmic and experimental description in Section 3 and Appendix C, and we have open sourced the code accompanying this research in this GitHub link.
+
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+
+# A NOTATIONS
+
+<table><tr><td>Symbol</td><td>Description</td></tr><tr><td>RD</td><td>D-dimensional real space</td></tr><tr><td>N∈N+</td><td>number of steps in the discrete-time stochastic process</td></tr><tr><td>S</td><td>GFlowNet state space, equals {(n, xn): n ∈{0,1,...,N}, xn∈RD}; (n, xn) is often written as xn for simplicity</td></tr><tr><td>A</td><td>action / transition space (edges s→s&#x27;)</td></tr><tr><td>s</td><td>state in S</td></tr><tr><td>s0</td><td>initial state, equals (0, 0) in VE modeling</td></tr><tr><td>τ</td><td>complete trajectory (x0→...→xN), or x0:N</td></tr><tr><td>θ</td><td>learnable parameter of DGFS</td></tr><tr><td>Fn: RD→R+</td><td>state flow function at step n</td></tr><tr><td>PF</td><td>forward policy (distribution over children / next step)</td></tr><tr><td>PB</td><td>backward policy (distribution over parents / previous step)</td></tr><tr><td>Q</td><td>forward process defined in Equation 1</td></tr><tr><td>Qref</td><td>reference process defined in Equation 3</td></tr><tr><td>P</td><td>target process defined in Equation 4</td></tr><tr><td>pn ref: RD→R+</td><td>marginal distribution of reference process at step n</td></tr><tr><td>h∈R+</td><td>discretized step size for the stochastic process</td></tr><tr><td>f</td><td>drift term in the VE stochastic process (Equation 5)</td></tr><tr><td>σ</td><td>variance coefficient in the stochastic process</td></tr><tr><td>dWt</td><td>standard Wiener process in continuous-time stochastic process</td></tr><tr><td>R: RD→R+</td><td>reward function (unnormized target density)</td></tr><tr><td>Rn: RD→R+</td><td>intermediate forward-looking signal defined in Equation 16</td></tr><tr><td>Z∈R+</td><td>scalar, equal to ∑x∈RD R(x)</td></tr><tr><td>μ: RD→R+</td><td>same thing as R(·)</td></tr><tr><td>π: RD→R+</td><td>normalized density, equal to μ(·)/Z</td></tr></table>
+
+# B DETAILS ABOUT STOCHASTIC PROCESSES
+
+# B.1 VARIANCE EXPLODING (VE) PROCESS
+
+Song et al. (2020) and Zhang & Chen (2022) take a variance exploding stochastic process modeling. In Zhang & Chen (2022), the authors use Gisarnov theorem to derive an optimal control objective for KL divergence minimization. In this work, we present a discrete-time derivation for completeness.
+
+The reference process starts at  $p_0^{\mathrm{ref}}(\cdot)$ , which is a Dirac distribution at 0, and writes
+
+$$
+\begin{array}{l} \mathcal {Q} ^ {\text {r e f}} \left(\mathbf {x} _ {0: N}\right): \mathbf {x} _ {n + 1} = \mathbf {x} _ {n} + \sqrt {h} \sigma \boldsymbol {\epsilon} _ {n}, \boldsymbol {\epsilon} _ {n} \sim \mathcal {N} (\mathbf {0}, \mathbf {I}), \mathbf {x} _ {0} = \mathbf {0}, (18) \\ \Rightarrow Q ^ {\text {r e f}} \left(\mathbf {x} _ {0: N}\right) = p _ {0} ^ {\text {r e f}} \left(\mathbf {x} _ {0}\right) \prod_ {n = 0} ^ {N - 1} P _ {F} ^ {\text {r e f}} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n}\right) = p _ {N} ^ {\text {r e f}} \left(\mathbf {x} _ {N}\right) \prod_ {n = 0} ^ {N - 1} P _ {B} \left(\mathbf {x} _ {n} \mid \mathbf {x} _ {n + 1}\right). (19) \\ \end{array}
+$$
+
+Therefore,  $p_n^{\mathrm{ref}}(\cdot) = \mathcal{N}(\cdot ;\mathbf{0},hn\sigma^2\mathbf{I})$  . According to Bayesian formula,
+
+$$
+\begin{array}{l} P _ {B} \left(\mathbf {x} _ {n} \mid \mathbf {x} _ {n + 1}\right) = \frac {p _ {n} ^ {\text {r e f}} \left(\mathbf {x} _ {n}\right) P _ {F} ^ {\text {r e f}} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n}\right)}{p _ {n + 1} ^ {\text {r e f}} \left(\mathbf {x} _ {n + 1}\right)} = \frac {\mathcal {N} \left(\mathbf {x} _ {n} ; \mathbf {0} , h n \sigma^ {2} \mathbf {I}\right) \mathcal {N} \left(\mathbf {x} _ {n + 1} ; \mathbf {x} _ {n} , h \sigma^ {2} \mathbf {I}\right)}{\mathcal {N} \left(\mathbf {x} _ {n + 1} ; \mathbf {0} , h (n + 1) \sigma^ {2} \mathbf {I}\right)} (20) \\ = \mathcal {N} \left(\mathbf {x} _ {n}; \frac {n}{n + 1} \mathbf {x} _ {n + 1}, \frac {n}{n + 1} h \sigma^ {2} \mathbf {I}\right). (21) \\ \end{array}
+$$
+
+The target process is then
+
+$$
+\mathcal {P} \left(\mathbf {x} _ {0}, \dots , \mathbf {x} _ {N}\right) = Q ^ {\text {r e f}} \left(\mathbf {x} _ {0}, \dots , \mathbf {x} _ {N}\right) \frac {\pi \left(\mathbf {x} _ {N}\right)}{p _ {N} ^ {\text {r e f}} \left(\mathbf {x} _ {N}\right)} = \pi \left(\mathbf {x} _ {N}\right) \prod_ {n = 0} ^ {N - 1} P _ {B} \left(\mathbf {x} _ {n} \mid \mathbf {x} _ {n + 1}\right). \tag {22}
+$$
+
+The trajectory level log probability difference between model and target is
+
+$$
+\log \frac {Q}{P} = \log \frac {Q}{Q ^ {\text {r e f}}} + \log \frac {Q ^ {\text {r e f}}}{P}. \tag {23}
+$$
+
+It is obvious that the second term is  $\frac{\mathcal{Q}^{\mathrm{ref}}}{\mathcal{P}} = \frac{p_N^{\mathrm{ref}}(\mathbf{x}_N)}{\pi(\mathbf{x}_N)}$ , thus we only need to look at the first term
+
+$$
+\begin{array}{l} \log \frac {Q}{Q _ {\text {r e f}}} = \sum_ {n = 0} ^ {N - 1} \log \frac {P _ {F} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n} ; \boldsymbol {\theta}\right)}{P _ {F} ^ {\text {r e f}} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n}\right)} = \sum_ {n = 0} ^ {N - 1} \log \frac {\mathcal {N} (\cdot ; \mathbf {x} _ {n} + h \mathbf {f} \left(\mathbf {x} _ {n} , n\right) , h \sigma^ {2} \mathbf {I})}{\mathcal {N} (\cdot ; \mathbf {x} _ {n} , h \sigma^ {2} \mathbf {I})} (24) \\ = \sum_ {n = 0} ^ {N - 1} \left(\frac {h}{2 \sigma^ {2}} \| \mathbf {f} \left(\mathbf {x} _ {n}, n; \boldsymbol {\theta}\right) \| ^ {2} + \frac {\sqrt {h}}{\sigma^ {2}} \mathbf {f} \left(\mathbf {x} _ {n}, n; \boldsymbol {\theta}\right) ^ {T} \boldsymbol {\epsilon} _ {n}\right). (25) \\ \end{array}
+$$
+
+Notice that under the expectation of  $\mathcal{Q}$
+
+$$
+\mathbb {E} _ {\mathcal {Q}} \left[ \mathbf {f} \left(\mathbf {x} _ {n}, n; \boldsymbol {\theta}\right) ^ {T} \boldsymbol {\epsilon} _ {n} \right] = \mathbf {f} \left(\mathbf {x} _ {n}, n; \boldsymbol {\theta}\right) ^ {T} \mathbb {E} _ {\mathcal {Q}} \left[ \boldsymbol {\epsilon} _ {n} \right] = 0. \tag {26}
+$$
+
+Accordingly,
+
+$$
+\mathcal {D} _ {\mathrm {K L}} \left(\mathcal {Q} \| \mathcal {P}\right) = E _ {\mathcal {Q}} \left[ \log \frac {\mathcal {Q}}{\mathcal {P}} \right] = E _ {\mathcal {Q}} \left[ \sum_ {n = 0} ^ {N - 1} \frac {h}{2 \sigma^ {2}} \| \mathbf {f} \left(\mathbf {x} _ {n}, n; \boldsymbol {\theta}\right) \| ^ {2} + \log \frac {p _ {N} ^ {\operatorname {r e f}} \left(\mathbf {x} _ {N}\right)}{\pi \left(\mathbf {x} _ {N}\right)} \right], \tag {27}
+$$
+
+which leads us to Equation 6.
+
+# B.2 VARIANCE PRESERVING (VP) PROCESS
+
+Ho et al. (2020b) and Vargas et al. (2023a) take this VP formulation of diffusion process. The derivation is similar to the last section, just with a different variance schedule.
+
+We first define the backward policy
+
+$$
+P _ {B} \left(\mathbf {x} _ {n} \mid \mathbf {x} _ {n + 1}\right) = \mathcal {N} \left(\mathbf {x} _ {n}; \sqrt {1 - \beta_ {n}} \mathbf {x} _ {n + 1}; \beta_ {n} \sigma^ {2} \mathbf {I}\right). \tag {28}
+$$
+
+Then The target and reference process are
+
+$$
+\mathcal {P} \left(\mathbf {x} _ {0: N}\right) = \pi \left(\mathbf {x} _ {N}\right) \prod_ {n = 0} ^ {N - 1} P _ {B} \left(\mathbf {x} _ {n} \mid \mathbf {x} _ {n + 1}\right), \tag {29}
+$$
+
+$$
+\mathcal {Q} ^ {\text {r e f}} \left(\mathbf {x} _ {0: N}\right) = p _ {N} ^ {\text {r e f}} \left(\mathbf {x} _ {N}\right) \prod_ {n = 0} ^ {N - 1} P _ {B} \left(\mathbf {x} _ {n} \mid \mathbf {x} _ {n + 1}\right), \tag {30}
+$$
+
+where  $p_N^{\mathrm{ref}}(\cdot) = \mathcal{N}(\cdot ;\mathbf{0},\sigma^2\mathbf{I})$  . This guarantees that  $p_n^{\mathrm{ref}}(\cdot) = \mathcal{N}(\cdot ;\mathbf{0},\sigma^2\mathbf{I}),\forall n.$
+
+From using Bayesian formula on  $\mathcal{Q}^{\mathrm{ref}}$
+
+$$
+P _ {F} ^ {\operatorname {r e f}} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n}\right) = \frac {p _ {n + 1} ^ {\operatorname {r e f}} \left(\mathbf {x} _ {n + 1}\right) P _ {B} \left(\mathbf {x} _ {n} \mid \mathbf {x} _ {n + 1}\right)}{p _ {n} ^ {\operatorname {r e f}} \left(\mathbf {x} _ {n}\right)} = \mathcal {N} \left(\mathbf {x} _ {n + 1}; \sqrt {1 - \beta_ {n}} \mathbf {x} _ {n}, \beta_ {n} \sigma^ {2} \mathbf {I}\right). \tag {31}
+$$
+
+Inspired by this, let us define the (learnable) forward process to be
+
+$$
+P _ {F} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n}; \boldsymbol {\theta}\right) = \mathcal {N} \left(\mathbf {x} _ {n + 1}; \sqrt {1 - \beta_ {n}} \mathbf {x} _ {n} + \mathbf {f} \left(\mathbf {x} _ {n}, n; \boldsymbol {\theta}\right), \beta_ {n} \sigma^ {2} \mathbf {I}\right), \tag {32}
+$$
+
+$$
+\mathcal {Q} \left(\mathbf {x} _ {0: N}\right) = p _ {0} ^ {\text {r e f}} \left(\mathbf {x} _ {0}\right) \prod_ {n = 0} ^ {N - 1} P _ {F} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n}; \boldsymbol {\theta}\right). \tag {33}
+$$
+
+Similar to the derivation in VE,
+
+$$
+\mathbb {E} _ {\mathcal {Q}} \left[ \log \frac {\mathcal {Q}}{\mathcal {Q} ^ {\text {r e f}}} \right] = \mathbb {E} _ {\mathcal {Q}} \left[ \sum_ {n = 0} ^ {N - 1} \log \frac {P _ {F} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n} ; \boldsymbol {\theta}\right)}{P _ {F} ^ {\text {r e f}} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n}\right)} \right] = \sum_ {n = 0} ^ {N - 1} E _ {\mathcal {Q}} \left[ \frac {1}{2 \beta_ {n} \sigma^ {2}} \| \mathbf {f} \left(\mathbf {x} _ {n}, n; \boldsymbol {\theta}\right) \| ^ {2} \right]. \tag {34}
+$$
+
+Therefore,
+
+$$
+\mathcal {D} _ {\mathrm {K L}} (\mathcal {Q} \| \mathcal {P}) = E _ {\mathcal {Q}} \left[ \log \frac {\mathcal {Q}}{\mathcal {P}} \right] = E _ {\mathcal {Q}} \left[ \sum_ {n = 0} ^ {N - 1} \frac {1}{2 \beta_ {n} \sigma^ {2}} \| \mathbf {f} \left(\mathbf {x} _ {n}, n; \boldsymbol {\theta}\right) \| ^ {2} + \log \frac {p _ {N} ^ {\text {r e f}} \left(\mathbf {x} _ {N}\right)}{\pi \left(\mathbf {x} _ {N}\right)} \right], \tag {35}
+$$
+
+which is the learning objective of DDS algorithm.
+
+# C ALGORITHMIC DETAILS
+
+# C.1 RE-DERIVING DETAILED BALANCE CONSTRAINT
+
+Writing down Equation 12 with  $n$  and  $n + 1$ , we have
+
+$$
+F _ {n} \left(\mathbf {x} _ {n}; \boldsymbol {\theta}\right) \prod_ {l = n} ^ {N - 1} P _ {F} \left(\mathbf {x} _ {l + 1} \mid \mathbf {x} _ {l}; \boldsymbol {\theta}\right) = \mu \left(\mathbf {x} _ {N}\right) \prod_ {l = n} ^ {N - 1} P _ {B} \left(\mathbf {x} _ {l} \mid \mathbf {x} _ {l + 1}\right), \tag {36}
+$$
+
+$$
+\begin{array}{l} F _ {n + 1} \left(\mathbf {x} _ {n + 1}; \boldsymbol {\theta}\right) \prod_ {l = n + 1} ^ {N - 1} P _ {F} \left(\mathbf {x} _ {l + 1} \mid \mathbf {x} _ {l}; \boldsymbol {\theta}\right) = \mu \left(\mathbf {x} _ {N}\right) \prod_ {l = n + 1} ^ {N - 1} P _ {B} \left(\mathbf {x} _ {l} \mid \mathbf {x} _ {l + 1}\right), (37) \\ \Rightarrow \frac {F _ {n} \left(\mathbf {x} _ {n} ; \boldsymbol {\theta}\right) P _ {F} \left(\mathbf {x} _ {n + 1} \mid \mathbf {x} _ {n} ; \boldsymbol {\theta}\right)}{F _ {n + 1} \left(\mathbf {x} _ {n + 1} ; \boldsymbol {\theta}\right)} = P _ {B} \left(\mathbf {x} _ {n} \mid \mathbf {x} _ {n + 1}\right) (38) \\ \end{array}
+$$
+
+where the last equation is obtained by comparing the first two equations on both sides, which resonates the GFlowNet DB criterion.
+
+# C.2 log Z ESTIMATOR ANALYSIS
+
+The bound in Equation 17 could be derived in the following way:
+
+$$
+\begin{array}{l} \log Z = \log \int \mu (\mathbf {x} _ {N}) \mathrm {d} \mathbf {x} _ {N} = \log \int \mathcal {P} (\mathbf {x} _ {0: N}) \mathrm {d} \mathbf {x} _ {0: N} = \log \mathbb {E} _ {\mathcal {Q}} \left[ \frac {\mathcal {P} (\mathbf {x} _ {0 : N})}{\mathcal {Q} (\mathbf {x} _ {0 : N})} \right] (39) \\ = \log \mathbb {E} _ {\mathcal {Q}} \left[ \frac {1}{B} \sum_ {b = 1} ^ {B} \frac {\mathcal {P} \left(\mathbf {x} _ {0 : N} ^ {(b)}\right)}{\mathcal {Q} \left(\mathbf {x} _ {0 : N} ^ {(b)}\right)} \right] \leq \mathbb {E} _ {\mathcal {Q}} \left[ \log \frac {1}{B} \sum_ {b = 1} ^ {B} \frac {\mathcal {P} \left(\mathbf {x} _ {0 : N} ^ {(b)}\right)}{\mathcal {Q} \left(\mathbf {x} _ {0 : N} ^ {(b)}\right)} \right] (40) \\ \approx \log \frac {1}{B} \sum_ {b = 1} ^ {B} \exp (S (\tau^ {(b)})), \quad \tau^ {(b)} \sim \mathcal {Q}. (41) \\ \end{array}
+$$
+
+Notice that this bound is valid without any assumption on  $\mathcal{Q}$ . According to analysis in Burda et al. (2016), this lower bound estimator is monotonically increase as  $B$  goes larger; what's more, it is a tight bound which means that it would converge to  $\log Z$  when  $B \to \infty$ . We remark that this is actually also the estimator used in Zhang & Chen (2022).
+
+# C.3 GRADIENT VARIANCE ANALYSIS
+
+In Figure 2, we conduct experiment on Funnel and report the gradient variance of DGFS and PIS. The variance of DDS is in similar scale to PIS thus we ignore it. According to the analysis in Section 3.3, there are two reasons of why DGFS is a more robust algorithm: (1) DGFS utilize short trajectory which is beneficial for efficient credit assignment (2) DGFS does not take the KL divergence minimizing formulation which has non-zero gradient variance. To disentangle these two factors, we run a variant of DGFS which only utilize full trajectories to update the parameters in Figure 7. This variant is shown to have gradient variance higher than the original DGFS but still much lower than PIS. This confirms that both factors are valid reasons for why DGFS has a lower gradient variance.
+
+For general probabilistic methods with KL between model  $q_{\theta}(\cdot)$  and target distribution  $p(\cdot)$ ,
+
+$$
+\nabla_ {\boldsymbol {\theta}} \mathcal {D} _ {\mathrm {K L}} \left(q _ {\boldsymbol {\theta}} (\mathbf {x}) \| p (\mathbf {x})\right) = \mathbb {E} _ {q} \left[ \nabla_ {\boldsymbol {\theta}} \log q _ {\boldsymbol {\theta}} (\mathbf {x}) \log \frac {q _ {\boldsymbol {\theta}} (\mathbf {x})}{p (\mathbf {x})} \right] + \mathbb {E} _ {q} \left[ \nabla_ {\boldsymbol {\theta}} \log q _ {\boldsymbol {\theta}} (\mathbf {x}) \right]. \tag {42}
+$$
+
+When the model reaches its optimal solution, i.e.,  $q_{\theta} = p$ , the first term is zero as it contains  $\log \frac{q_{\theta}(\mathbf{x})}{p(\mathbf{x})} = 0$ . However, the second term  $\nabla_{\theta}\log q_{\theta}(\mathbf{x})$  is only zero under expectation of  $q_{\theta}$ . That is to say, there will be a stochastic non-zero gradient term with KL formulation. This observation is the core motivation of Roeder et al. (2017).
+
+![](images/552098914d984ccab1f8df5e5241b2adb1e55c58c9a405a0c253ce5022f845f4.jpg)  
+Figure 6: Drift network magnitude of DGFS and PIS.
+
+![](images/14e8dd59ebf6c08484948a3915e2461d3ff96520be0620b9abd828c4d7a8d970.jpg)  
+Figure 7: Gradient variance of DGFS, PIS, and DGFS trained with only full paths.
+
+![](images/88b4d956970d00f10036cf4acdffbfe29ba507a49e9a7eb5377fa0d3ceb4da66.jpg)  
+Figure 8: Manywell results.
+
+![](images/dccf476f0f57b61ea55c976dfcdc20a06b384ff3ccd3dfdbb3968371733ee0bb.jpg)
+
+# Algorithm 1 DGFS training algorithm
+
+Require: DGFS model with parameters  $\theta$ , reward function  $\mu(\cdot)$ , variance coefficient  $\tilde{\sigma}$ . repeat Sample  $\tau$  with control  $\mathbf{f}(\cdot, \cdot; \theta)$  and  $\tilde{\sigma}$ ;  $\triangle \theta \gets \nabla_{\theta} \mathcal{L}(\tau; \theta)$  (as per Equation 15); Update  $\theta$  with Adam optimzier; until some convergence condition
+
+![](images/a162bb249abf7c7385a7f8d1f4ec04308378aec16bfd7b7b461275182fc5e4ac.jpg)  
+Figure 9: Demonstration of DGFS's off-policy exploration ability in a modified MoG+ task.
+
+![](images/3b7a2e835992b968aaf1126b75511fee9efc36f482a318f68cf5da941093ea3b.jpg)
+
+# C.4 DGFS DETAILS
+
+Here we describe details about DGFS algorithm. As shown in Algorithm 1, DGFS first interact with the environment to collect training data. Next, it updates its parameters  $\theta$  with methods described in Section 3. Notice that the training of DGFS is very general and does not have to use on-policy trajectories. Therefore, we could use a different variance coefficient  $\tilde{\sigma}$  to rollout. This could potentially bring better exploration ability and thus better mode capturing performance. In this work, we assume DGFS does on-policy rollout unless otherwise specified.
+
+We use PyTorch to implement DGFS algorithm. For PIS implementation, we take its original PyTorch code base. For DDS, we build our own implementation based on PIS code base. For other methods, we take the DeepMind jax code base.
+
+DGFS contains two sets of deep network parameters: one is the same as the neural network in PIS / DDS, the other is the flow function to amortize the intermediate marginal densities. The former one is for the control network, and express it in the way of  $\mathbf{f}(\cdot ,n) / \sigma = \mathrm{NN}_1(\cdot ,n), + \mathrm{NN}_2(n)\cdot \nabla \log \mu (\cdot)$ . These use Fourier features to embed the step index input. We also ablate diffusion-based sampling methods without gradient information in network parametrization in C.6. The second one is a scalar output network  $\mathrm{NN}(\cdot ,n)$  with similar structure. Same to PIS, we use two 64-dimensional hidden layers after the embedding layer. We set  $\lambda$  to be 2. For all experiments, we train with a batch size of 256 and Adam optimizer. We have not tuned too hard on the learning rate, but simply use  $1\times 10^{-4}$  and  $1\times 10^{-3}$  for the policy network (i.e., drift network) and the flow function network. The training keeps for 5000 optimization steps although almost all experiments converge in the first 1000 steps. Similar to PIS, we set the number of diffusion steps  $N$  to 100 for all experiments. We set the diffusion step size  $h$  to 0.05 for the MoG and Cox task and 0.01 for other tasks for all diffusion based methods. We set  $\sigma$  to 1. Zhang & Chen (2022) set evaluation particle number  $B = 6000$  for the funnel task and  $B = 2000$  for other tasks for unknown reasons. In this work, we set the number of evaluation particles  $B$  to be 2000 for all tasks for consistency.
+
+# C.5 MORE EXPERIMENTS
+
+![](images/867dc2f4af4b2b5d04b2ffd18eb2a154478f0bf6060b05dfa11ddb8a3fff2806.jpg)  
+Figure 10: Ablation on DGFS design choices on the VAE task.
+
+Table 2: Experimental results on benchmarking target densities, where the diffusion-based samplers are not allowed to directly access gradient information of the target log density in the drift network design. Note the PIS-NN and DDS-NN here still require to know the gradient information due to their KL formulation, while the DGFS-NN here is a real black-box algorithm and only need to know zeroth order information from the target.  
+
+<table><tr><td></td><td>MOG</td><td>FUNNEL</td><td>MANYWELL</td><td>VAE</td><td>COX</td></tr><tr><td>PIS-NN (w/ GRADIENT)</td><td>1.943±0.259</td><td>0.374±0.024</td><td>2.712±0.017</td><td>0.368±0.064</td><td>85.36±3.342</td></tr><tr><td>DDS-NN (w/ GRADIENT)</td><td>2.183±0.018</td><td>0.351±0.022</td><td>4.056±1.195</td><td>1.740±1.158</td><td>N/A</td></tr><tr><td>DGFS-NN</td><td>0.023±0.001</td><td>0.354±0.072</td><td>2.659±0.057</td><td>0.496±0.196</td><td>109.0±4.344</td></tr></table>
+
+We conduct ablation studies on a series of design choices of DGFS. In Figure 10, we compare the performance of DGFS and a few its variants on the VAE task. In the VAE task, the unnormalized target density  $\log \mu (\cdot)$  is defined by the log latent prior  $\log p(\cdot)$  plus the log decoder likelihood  $\log p(\mathbf{x}_{\mathrm{obs}}|\cdot)$ , which equals the latent posterior  $\log p(\cdot|\mathbf{x}_{\mathrm{obs}})$  plus a constant value. The VAE is trained on the MNIST dataset, and  $\mathbf{x}_{\mathrm{obs}}$  is a fixed image data. We first remove the usage of the intermediate signals (i.e., the parameterization of flow function in Equation 16), which leads to slightly worse performance. We then try to train DGFS with VP modeling in Appendix B.2. For this VP modeling, we also take the form of Equation 16 for getting intermediate signals. This gives a bias of 0.86. Notice that this is still better than both PIS and DDS, demonstrating the validity of the proposed DGFS training method. We also ablate the weighting coefficient  $\lambda$  and number of diffusion steps  $N$ . We can see their changes lead to small variations but the overall convergence remains good.
+
+Regarding the running time of the proposed algorithm, DGFS has the same inference procedure as PIS, and thus having the same inference speed as PIS. The training overhead of DGFS is slightly  $(20\%)$  higher than PIS due to the introduction of the additional flow function network. In the Funnel task, the training time of DGFS and PIS for one batch are 0.54s and 0.65s respectively.
+
+We then show some missing results from the main text. We put the visualization of the joint of the second and third dimension in Manywell task in Figure 8. We can see that PIS misses one important mode and DDS could not handle the mode separation well enough.
+
+We tried to use a MNIST-pretrained normalizing flow as the target density. However, we find a well-trained normalizing flow almost always assign  $-\infty$  to the log likelihood for out-of-distribution samples, which makes it impossible for any sampler to success. This actually makes sense as the normalizing flow is only trained on a finite size image dataset and never gets out-of-domain inputs.
+
+# C.6 REMOVE GRADIENT INFORMATION FROM PARAMETRIZATION
+
+In the network design, PIS explicitly use the score function—gradient of the log unnormalized density function—as one term in the drift network parametrization. Our work and DDS follow its modeling. In this section, we show performance where the score function are removed from the architecture of all three diffusion-base samplers in Table 2. We attach “-NN” postfix to their names to distinguish between their original algorithms. Notice PIS-NN and DDS-NN still require the knowledge of the score function implicitly based on their KL formulation. On the other hand, DGFS-NN
+
+Table 3: Experimental results on benchmarking target densities. DGFS+ denotes DGFS with additional off-policy exploration trick in the initial stage of training.  
+
+<table><tr><td></td><td>MoG</td><td>FUNNEL</td><td>MANYWELL</td><td>VAE</td><td>Cox</td></tr><tr><td>DGFS</td><td>0.019±0.008</td><td>0.274±0.014</td><td>0.904±0.067</td><td>0.180±0.083</td><td>8.974±1.169</td></tr><tr><td>DGFS+</td><td>0.012±0.002</td><td>0.322±0.029</td><td>0.932±0.123</td><td>0.162±0.015</td><td>10.23±1.237</td></tr></table>
+
+Table 4: Experimental results on benchmarking target densities for methods in Lahlou et al. (2023), with or without off-policy exploration technique. We also present performance of TB and its off-policy variant; see details in Section C.8.  
+
+<table><tr><td></td><td>MOG</td><td>FUNNEL</td><td>MANYWELL</td><td>VAE</td><td>Cox</td></tr><tr><td>LAHLOU ET AL. (2023)</td><td>1.891±0.042</td><td>0.398±0.061</td><td>3.669±0.653</td><td>4.563±1.029</td><td>2728.±51.54</td></tr><tr><td rowspan="2">LAHLOU ET AL. (2023)+TB</td><td>0.024±0.189</td><td>0.373±0.020</td><td>6.257±2.449</td><td>2.529±0.737</td><td>2722.±31.64</td></tr><tr><td>0.018±0.002</td><td>0.309±0.023</td><td>1.218±0.194</td><td>1.097±0.239</td><td>138.6±7.850</td></tr><tr><td>TB+</td><td>0.009±0.003</td><td>0.373±0.031</td><td>1.210±0.192</td><td>0.842±0.248</td><td>240.4±0.998</td></tr></table>
+
+is a real black-box method, i.e., only requires zeroth order information of the unnormalized density function. Despite using less information, DGFS-NN is still very competitive in most of the tasks.
+
+# C.7 OFF-POLICY EXPLORATION ABILITIES DGFS
+
+The usage of the KL divergence in previous works restricts the training to be on-policy, as the training trajectory has to come from the model itself. Using off-policy samples is possible only with importance sampling, which typically results in high variance. This limits the exploration ability of these methods. On the other hand, it is first shown in Malkin et al. (2023) that GFlowNets can potentially benefit from the use of off-policy exploration. To be more specific, as has been shown by a similar analysis in prior works Malkin et al. (2023); Lahlou et al. (2023), the objectives in Equation 9 or Equation 14 do not require the training samples to follow any particular distribution (only to have full support), which means our method supports off-policy training without importance sampling. Therefore, it is possible to employ off-policy exploration by using a larger variance coefficient in the forward process during the rollout stage. In theory, this could help DGFS to better capture diversity in complex multimodal target distributions.
+
+In an algorithmic perspective, we could use a slightly larger variance coefficient  $\tilde{\sigma} >\sigma$  in the rollout stage of Algorithm 1 to ensure that information of more modes fall into the training data, which would drive the training of DGFS to put effort on these distant modes. To demonstrate this property, we carefully create a more difficult Mixture of Gaussian task and name it as  $MoG+$ . We intentionally put a few very distant modes in this MoG+ task, as shown by the contours in Figure 9. We set the exploration variance coefficient  $\tilde{\sigma} = 2$  contrary to  $\sigma = 1$ , which achieves better mode coverage and lower normalizing factor estimation bias with the same neural network capacity.
+
+We conduct an initial experiment to investigate this possibility in DGFS. We test a strategy that linearly anneals the  $\tilde{\sigma}$  from 1.1 to 1 during the first 1000 steps of training, and denote it as DGFS+ in Table 3. Our experiment results show that this simple strategy can improve the performance under some scenarios, and can be treated as a potential hyperparameter to extend the application scope of DGFS.
+
+# C.8 OTHER ABLATION STUDY
+
+We run the method in Lahlou et al. (2023) in this section. As shown in Table 4, this previous method and its off-policy variant are far from comparable to the baselines in Table 1. We also run the trajectory balance method with gradient information (denoted as TB) and its variant with off-policy exploration (denoted as  $\mathrm{TB + }$ ) and present in the same table. These two methods achieve relatively better performance, but still worse than that of DGFS.

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2024/Diffusion in Diffusion_ Cyclic One-Way Diffusion for Text-Vision-Conditioned Generation/full.md

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+# DIFFUSION IN DIFFUSION: CYCLIC ONE-WAY DIFFUSION FOR TEXT-VISION-CONDITIONED GENERATION
+
+Ruoyu Wang $^{1*}$ , Yongqi Yang $^{1*}$ , Zhihao Qian $^{1}$ , Ye Zhu $^{2}$ , Yu Wu $^{1\dagger}$
+
+$^{1}$  School of Computer Science, Wuhan University  
+$^{2}$  Department of Computer Science, Princeton University
+
+{wangruoyu, yongqiyang, qianzhihao, wuyucs}@whu.edu.cn  
+yezhu@princeton.edu
+
+# ABSTRACT
+
+Originating from the diffusion phenomenon in physics that describes particle movement, the diffusion generative models inherit the characteristics of stochastic random walk in the data space along the denoising trajectory. However, the intrinsic mutual interference among image regions contradicts the need for practical downstream application scenarios where the preservation of low-level pixel information from given conditioning is desired (e.g., customization tasks like personalized generation and inpainting based on a user-provided single image). In this work, we investigate the diffusion (physics) in diffusion (machine learning) properties and propose our Cyclic One-Way Diffusion (COW) method to control the direction of diffusion phenomenon given a pre-trained frozen diffusion model for versatile customization application scenarios, where the low-level pixel information from the conditioning needs to be preserved. Notably, unlike most current methods that incorporate additional conditions by fine-tuning the base text-to-image diffusion model or learning auxiliary networks, our method provides a novel perspective to understand the task needs and is applicable to a wider range of customization scenarios in a learning-free manner. Extensive experiment results show that our proposed  $COW$  can achieve more flexible customization based on strict visual conditions in different application settings. Project page: https://wangruoyu02.github.io/cow.github.io/
+
+# 1 INTRODUCTION
+
+In physics, the diffusion phenomenon describes the movement of particles from an area of higher concentration to a lower concentration area till an equilibrium is reached (Philibert, 2006). It represents a stochastic random walk of molecules to explore the space, from which originates the state-of-the-art diffusion generative models (Sohl-Dickstein et al., 2015). Compared to the physical diffusion process, it is widely acknowledged that the diffusion generative model in machine learning also stimulates a random walk in the data space (Song et al., 2020b), however, it is less obvious how the diffusion models stimulate the information diffusion for real-world data along its walking trajectory. In this work, we start by investigating the diffusion phenomenon in diffusion models for image synthesis, namely "diffusion in diffusion", during which the pixels within a single image from different data distributions exchange and interact with each other, ultimately achieving a harmonious state in the data space (see Sec. 3.2).
+
+The diffusion phenomenon is intrinsically stochastic, both in physics and in current machine learning frameworks, given no explicit constraints on the regional concentrations. In other words, regions within an image interfere with each other along the generation process as the sample goes from a noisy Gaussian space to the data space. However, we note that this property of bidirectional interference is not always desired when applying diffusion generative models in practical downstream tasks. For instance, tasks like image inpainting can be viewed as strictly unidirectional information diffusion, propagating information from a known portion (the existing image portion)
+
+![](images/d8ee7c13751783f4a97a03cb31893b75c0228075f21238244e6f97d24b4a3a8a.jpg)  
+Figure 1: Comparison with existing SOTA methods for maintaining the fidelity of text and visual conditions in different application scenarios. We consistently achieve superior fidelity to both text and visual conditions across all three settings. In contrast, other learning-based approaches struggle to attain the same level of performance across diverse scenarios.
+
+to an unknown portion (the missing image portion) while preserving the pixel-level integrity of the known portion. Most existing methods (Ruiz et al., 2022; Gal et al., 2022; Dong et al., 2022; Zhang & Agrawala, 2023) tackle the problem by brute-force learning, where they incorporate the visual condition into pre-trained text2image models by introducing an additional finetuning process to minimize the reconstruction error. However, these tuning-based methods, despite explicitly minimizing reconstruction errors, can not always achieve pleasant fidelity to both the text and the visual conditions, especially when faced with contradictions between conditions like style transfer and mismatched attributes scenarios, as shown in Fig. 1. In addition, they introduce additional learning costs dependent on the pre-trained model and hinder the original distribution modeling ability of the base model. To this end, the ability to control the direction of information diffusion opens up the potential for a new branch of methodological paradigms to achieve versatile customization applications without the need to change the parameters of existing pre-trained diffusion models or learn any auxiliary neural networks.
+
+Following our analysis on the diffusion in diffusion and its connection to practical task needs, we propose Cyclic One-Way Diffusion (COW), a training-free framework that achieves unidirectional diffusion for versatile customization scenarios, ranging from conventional visual-conditioned inpainting to visual-text-conditioned style transformation. From the methodological point of view, we re-inject the semantics (inverted latents) into the generation process and repeatedly "disturb" and "reconstruct" the image in a cyclic way to maximize and encourage information flow from the visual condition to the whole image. From the application point of view, the powerful knowledge of the pre-trained diffusion model empowers us to conduct meaningful editing or stylizing operations while maintaining the fidelity of the visual condition. As a result, even in its unidirectional design, COW can achieve more flexible customization based on strict visual conditions. Extensive experiments and human studies, involving 3,000 responses for 600 image groups, demonstrate that COW consistently outperforms its counterparts in terms of condition consistency and overall fidelity. Besides, COW generates an image in just 6 seconds, far faster than other customization methods like DreamBooth (Ruiz et al., 2022), which takes 732 seconds.
+
+# 2 RELATED WORK
+
+Diffusion Models. The recent state-of-the-art diffusion generative methods (Song et al., 2020a; Dhariwal & Nichol, 2021; Nichol & Dhariwal, 2021; Song et al., 2020b; Hoogeboom et al., 2021; Wu et al., 2022; Zhu et al., 2023b) originate from the non-equilibrium statistical physics (Sohl-
+
+Dickstein et al., 2015), which simulate the physical diffusion process by slowly destroying the data structure through an iterative forward diffusion process and restore it by a reverse annealed sampling process. DDPM (Ho et al., 2020) shows the connection between stochastic gradient Langevin dynamics and denoising diffusion probabilistic models. DDIM (Song et al., 2020a) generalizes the Markovian diffusion process to a non-Markovian diffusion process, and rewrites DDPM in an ODE form, introducing a method to inverse raw data into latent space with low information loss. In our work, we utilize DDIM inversion to access the latent space and find an information diffusion phenomenon in the sampling process towards the higher concentration data manifold driven by the pre-trained gradient estimator.
+
+Downstream Customization Generation. Given a few images of a specific subject, the customization generation task aims to generate new images according to the text descriptions while keeping the subject's identity unchanged. Early approaches mainly relied on GAN-based architectures (Reed et al., 2016; Zhang et al., 2017; Dong et al., 2017; Zhang et al., 2018; Xu et al., 2018; Lin et al., 2018; Wu et al., 2019; Karras et al., 2019; Li et al., 2019; Karras et al., 2020) for customization generation. In recent years, the diffusion methods under text condition image generation task (T2I) have made a great development (Nichol et al., 2021; Ramesh et al., 2022; Oppenlaender, 2022; Ramesh et al., 2021; Ho & Salimans, 2022; Sahara et al., 2022). However, appointing a specific visual condition at a certain location on the results of a generation remains to be further explored. Choi et al. (2021) is a learning-free visual conditioned method that generates a random new face similar to the given face. There are recent customized methods of learning-based visual-text conditioned generation like Ruiz et al. (2022); Gal et al. (2022); Zhang & Agrawala (2023); Rombach et al. (2022); Dong et al. (2022); Brooks et al. (2023); Wu et al. (2023); Wei et al. (2023); Guo et al. (2023); Choi et al. (2021). Methods like DreamBooth (Ruiz et al., 2022) and Textual Inversion (Gal et al., 2022) learn the concept of the visual condition into a certain word embedding by additional training on the pretrained T2I model (Kumari et al., 2022; Dong et al., 2022; Ruiz et al., 2023; Chen et al., 2023). It is worth noting that it is still hard for DreamBooth or Textual Inversion (TI) to keep the id-preservation even given 5 images as discussed in Shi et al. (2023), and DreamBooth tends to overfit the limited fine-tuning data and incorrectly entangles object identity and spatial information, as discussed in Li et al. (2023). ControlNet (Zhang & Agrawala, 2023) trains an additional network for a specific kind of visual condition (e.g., canny edges, pose, and segmentation map). More generally, an inpainting task is also a customization generation with a forced strong visual condition. Compared to these methods, our proposed method can explicitly preserve the pixel-level information of the visual conditions while achieving versatile application scenarios like style transfer and attribute editing.
+
+# 3 METHOD
+
+In this section, we present our methodology design for leveraging the diffusion in diffusion phenomenon to achieve versatile downstream applications without additional learning.
+
+# 3.1 PRELIMINARIES
+
+Denoising Diffusion Probabilistic Models (DDPMs) (Ho et al., 2020) define a Markov chain to model the stochastic random walk between the noisy Gaussian space and the data space, with the diffusion direction written as,
+
+$$
+q \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {t - 1}\right) = \mathcal {N} \left(\sqrt {1 - \beta_ {t}} \mathbf {x} _ {t - 1}, \beta_ {t} I\right), \tag {1}
+$$
+
+where  $t$  represents diffusion step,  $\{\beta_t\}_t^T$  are usually scheduled variance, and  $\mathcal{N}$  represents Gaussian distribution. Then, a special property brought by Eq. 1 is that:
+
+$$
+q \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {t - k}\right) = \mathcal {N} \left(\sqrt {\alpha_ {t} / \alpha_ {t - k}} \mathbf {x} _ {t - k}, \sqrt {1 - \alpha_ {t} / \alpha_ {t - k}} I\right), \tag {2}
+$$
+
+where  $\alpha_{t} = \prod_{i=0}^{t}(1 - \beta_{i})$ . So we can bring  $\mathbf{x}_{\mathbf{t}}$  to any  $\mathbf{x}_{t+k}$  in a one-step non-Markov way in our proposed cyclic one-way diffusion (Sec. 3.3) by adding certain random Gaussian noise.
+
+DDIM (Song et al., 2020a) generalizes DDPM to a non-Markovian diffusion process and connects the sampling process to the neural ODE:
+
+$$
+d \bar {x} (t) = \epsilon^ {(t)} \left(\frac {\bar {x} (t)}{\sqrt {\sigma^ {2} + 1}}\right) d \sigma (t). \tag {3}
+$$
+
+![](images/c856e464040531c651d2dc750e5e438e3a66c41deb63a98e9fcaa0e9c872470f.jpg)  
+Figure 2: Illustration of "diffusion in diffusion". We inverse pictures of pure gray and white back to  $\mathbf{x}_t$ , merge them together with different layouts, and then regenerate them back to  $x_0$  via deterministic denoising. Different columns indicate different replacement steps  $t$ . The resulting images show how regions within an image diffuse and interfere with each other during denoising.
+
+By solving this ODE using Euler integration, we can inverse the real image  $x_0$  (the visual condition) to  $\mathbf{x_t}$  in any corresponding latent space  $\epsilon_t$  (Zhu et al., 2023a; Mokady et al., 2023; Asperi et al., 2023) while preserving its information. The symbols  $\sigma$  and  $\overline{x}$  are the reparameterizations of  $(\sqrt{1 - \alpha} / \sqrt{\alpha})$  and  $(\mathbf{x} / \sqrt{\alpha})$  respectively.
+
+# 3.2 DIFFUSION IN DIFFUSION
+
+Internal Interference in Diffusion Generation. Diffusion in physics is a phenomenon caused by random movements and collisions between particles. The diffusion model, drawing inspiration from non-equilibrium thermodynamics, establishes a Markov chain between a target data distribution and the Gaussian distribution. Subsequently, it learns to reverse this diffusion process, thereby constructing the desired data samples from the Gaussian noise. This inherently simulates a gradual, evolving process that can be viewed as a random walk through a large number of possible data distributions, which eventually gradually approaches the real data distribution. Therefore, the diffusion models share a similar interference phenomenon as the physical diffusion, characterized by continuous information exchange within the data, ultimately achieving harmonious generation results.
+
+We design a toy experiment to reveal this phenomenon more intuitively. To start with, we apply DDIM Inversion to convert gray and white images into various latent codes along the diffusion timeline, spanning from the start  $(t = T)$  to the final state  $(t = 0)$ . Existing literatures (Song et al., 2020a; Zhu et al., 2023a) demonstrate that those intermediate latent codes can well reconstruct the raw image via deterministic denoising. In other words, both latent codes contain information inherited from their respective raw images, i.e., pure gray and white colors. Consequently, at each selected time step  $t$ , we merge half of latent codes from the two images into one and denoise the resulting combination. This allowed us to observe how different pieces of information interact throughout the generation process and thus influence the final image. The results in Fig. 2 show that as the merging time step  $t$  (at which step we put two latent codes together) approaches  $T$  (the Gaussian noise end), the corresponding denoised image  $x_0$  exhibits the spatial diffusion phenomenon, resulting in stronger color blending. Conversely, as  $t$  approaches 0 (the raw image end), the image showcases robust reconstruction ability, with minimal interference between the two colors.
+
+Varying Interference Intensity in Diffusion Generation. Based on the observations above and additional supplementary experiments in Appendix A.4, we can roughly divide the denoising process into three stages. Throughout the reverse diffusion process, the model attends to different levels of information at each stage, essentially embodying a progression from extreme noise to semantic formation to refinement. Introducing guidance too early hinders its reflection in the final image due to the uncontrollable interference caused by excessive noise, while introducing it too late does not allow for the desired high-level semantic modifications. The best to inject visual condition information is the middle stage, where the model gains the capacity to comprehend and generate basic semantic content, striking a balance between controllable inner mutual influence and responsiveness to text conditions. Ultimately, proper refinement at the last stage ensures that the generated
+
+![](images/17dbccd1f0402057ec90b1748a13ecaec461ac9cb95cea179d778f320fc89d78.jpg)  
+Figure 3: The pipeline of our proposed COW method. Given the input visual condition, we stick it on a predefined background and inverse it as the seed initialization of the starting point of the cycle. In the Cyclic One-Way Diffusion process, we "disturb" and "reconstruct" the image in a cyclic way and ensure a continuous one-way diffusion by consistently replacing it with corresponding  $\mathbf{x}_{\mathrm{t}}$ .
+
+images exhibit intricate details of visual condition. These insights and observations pave the way for integrating new visual condition control paradigms into pre-trained diffusion models.
+
+# 3.3 TRAINING-FREE CYCLIC ONE-WAY DIFFUSION
+
+In this subsection, we illustrate how to take advantage of the diffusion phenomenon in the diffusion generation process, to enable effective both pixel-level and semantic-level visual conditioning without training. Our approach involves three main components: Seed Initialization, Cyclic One-Way Diffusion and Visual Condition Preservation, as shown in Fig. 3.
+
+Seed Initialization. It is common to bring the thread end close to the needle before threading. Using random initialization sampled from the Gaussian distribution may introduce features or structures that conflict with the given visual condition. For instance, if the user specifies that the object should be located on the left side, but the initial noise tends to generate it on the right side, there will be a conflict. The model must expend considerable effort during the generation process to correct this inconsistency. To address this issue, we propose to introduce user-specified visual conditions at the initial stage by placing them onto a predefined background, typically a semantically neutral pure gray background. The objective is to inject stable high-level semantic information early in the denoising process, effectively reducing the layout conflicts with the visual condition.
+
+Cyclic One-Way Diffusion (COW). Theoretically, we invert the visual condition into its latent representation by solving the probabilistic flow ODE (Eq. 3) and embedding it in the initial random Gaussian noise that serves as the starting point, which can provide a good generative prior to maintain consistency with the visual condition. However, the implanted information will be continuously disrupted by inner diffusion from the surrounding Gaussian region at every denoising step. Therefore, we introduced "one-way" and "cyclic" strategies to maximize the flow of information from the visual condition to the whole image and minimize undesired interference from other image regions. To be specific, we store inverted latents of the visual condition at each inversion step in the middle stage (the semantic formation stage), denoted as  $\mathbf{x}_{t_1},\mathbf{x}_{t_1 + 1},\ldots ,\mathbf{x}_{t_2}$  and gradually embed them in the corresponding timesteps during the generation process. Through this step-wise information injection, we can ensure the unidirectional propagation of information, i.e., it only propagates from the visual condition to the other regions without interference from information in the background or other parts of the image. Given the limited generative capacity of the model at each step, noise is injected to regress the generative process to earlier stages, as illustrated in Equation 1. This cyclic utilization of the model's generative capacity enables the continuous perturbation of inconsistent semantic information, facilitating the re-diffusion of conditional guidance in subsequent rounds. The Cyclic One-Way Diffusion approach benefits the model from a one-way guidance from the
+
+visual condition, creates additional space by cycles for semantic "disturb" and "reconstruct", and ultimately achieves harmony among the background, visual condition, and text condition.
+
+Visual Condition Preservation. Conflicts between the visual and the text conditions often exist (such as a smiling face condition and a "sad" text prompt), necessitating a method that can effectively balance these conditions. We observe that the middle stage is still subject to some extent of uncertainty, which in turn leaves enough space for controlling the text condition guidance on the generation of the visual condition region. Meanwhile, in the later stage, the model focuses on refining high-frequency details and textures to enhance image quality while maintaining global structure integrity. Thus we explicitly control the degree of visual condition preservation by replacing the corresponding region at an adjustable step  $\mathbf{x}_{t_0}$  in the early phase of this later stage. This approach can effectively preserve fidelity to both the visual and the text conditions, achieving harmonious results of style transfer and attribute editing without additional training.
+
+# 4 EXPERIMENTS
+
+We show the versatility and superiority of COW by comparisons with four SOTA baselines under three different task settings. Also, we conduct an exhaustive ablation study to demonstrate the effectiveness of COW utilizing the diffusion phenomenon in machine learning diffusion generation.
+
+# 4.1 EXPERIMENTS SETUP
+
+Benchmark. To simulate the visual condition processing in real scenarios, we adopt face image masks from CelebAMask-HQ (Lee et al., 2020) as our visual condition. For the text conditions, we design three kinds of settings: normal prompt, style transfer, and attribute editing. Then we combine them as the dual conditions for our TV2I task.
+
+Implementation Details. We implement COW, SD inpainting, and ControlNet on pre-trained T2I Stable Diffusion model (Rombach et al., 2022) sd-v2-1-base, with default configuration condition scale set 7.5, noise level  $\eta$  set 1, image size set 512, 50 steps generation (10 steps for COW), and negative prompt set to "a bad quality and low-resolution image, extra fingers, deformed hands". Note that we implement DB and TI following their official code and use the highest supported Stable Diffusion version sd-v1-5. During generation, we set the visual condition size to 256 and randomly chose a place above half of the image to add the visual condition. We choose  $\mathbf{x}_{t_1}$  to step 25,  $\mathbf{x}_{t_2}$  to 35, cycle number to 10. We use slightly different settings for the three different tasks. We set  $\mathbf{x}_{t_3}$  to be [4, 3], eta to be 0 in the normal prompts,  $\mathbf{x}_{t_3}$  to be 4, eta to be 0.1 in the attribute editing prompts, and  $\mathbf{x}_{t_3}$  to be 4, eta to be 1 in the style prompts. We perform all comparisons with baselines on 200 images and 300 texts (100 texts for every setting, 1 text to 2 faces in order). We use a single NVIDIA 4090 GPU to run experiments since our proposed COW method is training-free.
+
+Comparisons. We perform a comparison with four SOTA works incorporating different levels of the visual condition into pre-trained T2I models: DreamBooth (Ruiz et al., 2022) based on the code $^{1}$ , TI (Gal et al., 2022), SD inpainting (Rombach et al., 2022), and ControlNet on the canny edge condition (Zhang & Agrawala, 2023). DreamBooth (Ruiz et al., 2022) introduces a rare-token identifier along with a class-prior for a more specific few-shot visual concept by finetuning pre-trained T2I model to obtain the ability to generate specific objects in results. TI (Gal et al., 2022) proposes to convert visual concept to word embedding space by training a new word embedding using a given visual condition, and uses it directly in the generation of results of specific objects. ControlNet (Zhang & Agrawala, 2023) incorporate additional visual conditions (e.g., canny edges) by finetuning a pretrained Stable Diffusion model in a relatively small dataset (less than 50k) in an end-to-end way. SD inpainting (Rombach et al., 2022) preserves the exact pixel of the visual condition when generating an image, and it is the inpainting mode of the pre-trained Stable Diffusion.
+
+**Evaluations.** To evaluate the quality of the generated results for this new task, we consider two aspects of assessment: visual fidelity and text fidelity. Following the same evaluation metrics as in previous works (Gal et al., 2022; Ruiz et al., 2022; Dong et al., 2022), we utilize prestrained CLIP (ViT-B/32) to evaluate the average pairwise cosine similarity between CLIP embeddings of generated images and the prompts, denoted as  $CLIP-T$ . Additionally, we adopt a face detection
+
+![](images/b7b04db1d577fdabb564de4090ea83bc356149c06ec83b4d53e7f3c215052efc.jpg)  
+Figure 4: The adaptation of the visual condition to align with the text condition while maintaining the semantic and pixel-level information of the visual condition. In each pair of images, the smaller image is the given visual condition and the other is the generated result. The bolded parts of the text conditions highlight the conflicts between conditions.
+
+Table 1: Quantitative and qualitative comparison between COW and SOTA methods.  
+
+<table><tr><td>Methodology</td><td>Clip-T ↑</td><td>ID-Distance ↓</td><td>Face Detection Rate ↑</td><td>Time ↓</td><td>Condition Consistency ↑</td><td>General Fidelity ↑</td></tr><tr><td>TI</td><td>0.253</td><td>1.186</td><td>70.66%</td><td>3025s</td><td>02.73%</td><td>12.60%</td></tr><tr><td>DreamBooth</td><td>0.329</td><td>1.361</td><td>70.50%</td><td>732s</td><td>09.60%</td><td>28.73%</td></tr><tr><td>ControlNet</td><td>0.305</td><td>1.194</td><td>45.66%</td><td>4s</td><td>11.60%</td><td>04.73%</td></tr><tr><td>SD inpainting</td><td>0.300</td><td>0.408</td><td>100.00%</td><td>5s</td><td>06.33%</td><td>02.07%</td></tr><tr><td>COW (ours)</td><td>0.306</td><td>0.901</td><td>100.00%</td><td>6s</td><td>69.73%</td><td>51.87%</td></tr></table>
+
+model (MTCNN (Zhang et al., 2016)) to detect the presence of the face, and a face recognition model (FaceNet (Schroff et al., 2015)) to obtain the face feature and thus calculate the face feature distance between the generated face region and the given visual condition. The Face Detection Rate and ID-Distance reflect the fidelity of the visual face condition for those generative models. However, relying solely on model predictions can not fully capture the subtle differences between images and can not reflect the overall quality of the images (e.g., realism, richness), which are critical crucial for human perception. Therefore, we further evaluate our model via human evaluation. We design two base criteria and invite 50 participants to be involved in this human evaluation. The two criteria are: 1. Condition Consistency: whether the generated image well matches the visual and textual conditions; 2. General fidelity: whether the chosen image looks more like a real image in terms of image richness, face naturalness, and overall image fidelity. It's important to note that when assessing the latter criterion, participants are not provided with textual and visual conditions to prevent additional information from potentially interfering with the assessment process.
+
+# 4.2 EXPERIMENTAL RESULTS
+
+COW enables versatile applications under the setting of TV2I tasks, including inpainting, attribute editing, and style transfer. More comparison of generated images are included in Appendix A.1.5.
+
+Quantitative Results. Quantitative results in Tab. 1 show that our method almost perfectly retains faces (the given visual condition) in the synthesized images, and achieves second-best text conditional fidelity in an efficient time cost. These qualitative results prove that our method generally outperforms previous works in terms of fidelity to visual and text conditions. When processing semantically complex visual conditions, such as faces, by explicitly considering and incorporating low-level visual details, our approach can motivate the model to generate results that are highly consistent with the original conditions. In addition, our method is training-free, thus we can generate the prediction using fewer computations.
+
+Human Evaluations. We conduct a preference test where participants select their favorite image among a shuffled set, including ours and four compared methods. Each image is evaluated by both
+
+![](images/da6020e324b4536ac203e3b066048a27ed0395c0047358f545790c3812702822.jpg)
+
+![](images/14ddbdf2aeaa4ede18978c1a3fc8ba427ec353e1c2a726f194c246eef354298f.jpg)
+
+![](images/1130354da08a6afcf2a2ee68af4b03959e29a26ccabb001c5a015449a643d9d7.jpg)  
+The number of cycles increases
+
+![](images/6302c5aafaff9120a190aa7eaadfa953bd104ddf40933344027c256c0067bd4b.jpg)
+
+![](images/4d48245190190095012e5d7c8e91befad0af481442fc4e68d7754839227fa4e1.jpg)
+
+![](images/1e6ac771a1de34c7b9b036dea4ae08f2d0eff9bd86387d19875866a9b66a45fd.jpg)
+
+![](images/1bf32a5296ea60efdcda6ca4bfa96ed1d290b6903e00a2a622f9a7e71dcc3039.jpg)  
+a person in black dress at a vintage car show.  
+a person is cooking in a large kitchen.
+
+![](images/3dfae5623e583d712cb3dee5027186877ab323bb19bfc42c72829664e6241925.jpg)  
+Figure 5: Analysis of the cycling process that diffuses "visual seed" to its surroundings. The left-most figure shows a given face condition. The right shows the images generated with given text conditions. The cycle number increases from the left to the right.
+
+![](images/44932a39320d0a176f408a9271bf2afb75fb5d109950dc39b3faf272792ddfc9.jpg)
+
+![](images/fa43d94fecfe61011d7879a0dc51ce10237d4a1f8aab4a8cf8654dfe3b49df3d.jpg)
+
+criteria and five different participants, resulting in a total of 3,000 responses for 600 image groups in Tab. 1. Our method consistently outperforms the others across all three settings, as detailed in Appendix A.3. These results demonstrate that our method better integrates text and visual conditions while preserving a satisfying image fidelity.
+
+One-Way Diffusion during Cycling Process. To emphasize the role of cyclic one-way diffusion strategy during the generation process, we set the noise level  $\eta$  to 0 to slow down the information diffusion. As shown in Fig. 5, as the cycle proceeds, the background pixels are gradually matched to the text condition and fused with the given face condition. This vividly demonstrates that the information in the visual condition keeps spreading and diffusing to the surrounding region along with cycles. In addition, the model is capable of understanding the semantics of the implanted visual conditions properly. Additionally, we conduct a comprehensive ablation study in Appendix A.2 to validate the effectiveness of COW and explore optimal hyperparameter configurations.
+
+Trade-offs between Conditions for Different Modalities. The TV2I generation task faces many challenges due to the semantic gap between text and images and the complexity of multimodal data. In general, textual information provides a high-level description of the image to be generated, such as the type, color, and other attributes of the object, while visual information contains more low-level details, such as shape, texture, and so on. The model needs to be able to understand the semantic information in the textual description and translate this information into visually specific details. COW strikes a balance between meeting the visual and text conditions as shown in Fig. 4. For example, when given a photo of a young woman but the text is "an old person", our method can make the woman older to meet the text description by adding wrinkles, changing skin elasticity and hair color, etc. while maintaining the facial expression and the identity of the given woman. Additionally, in Fig. 6, we showcase a series of samples containing varying degrees of changes of visual conditions in the generated output: (1) almost unchanged, (2) slightly perturbed (e.g., adding accessories), (3) attribute editing (e.g., from smiling to angry), (4) style transfer (e.g., from a photo to a comic picture), and (5) cross-domain transformation (e.g., from a human face to a lion). In particular, even in cases involving significant conflicts, such as transitioning between different species, our method can adeptly preserve certain individual characteristics of the given individual while seamlessly integrating them with the attributes of the target species as shown in Fig.8. These results demonstrate the ability of COW to effectively understand and balance the information of different modalities and adaptively adjust to produce high-quality images under a wide range of conditions, showcasing its versatility and effectiveness in handling diverse customization scenarios.
+
+More Applications. We directly apply COW to other applications to demonstrate its generalization ability. As shown in Fig. 7 (a), we present results involving a generalized visual condition (cats) paired with different text conditions. Our method harmoniously grows a whole body out of the visual condition under the guidance of the text condition. Furthermore, we explore extreme cases where the visual condition size equals the whole image size as illustrated in Fig. 7 (b). The results indicate that our method can still produce pleasant outcomes and maintain its style transfer and attribute editing capabilities even in the context of whole-image generation. Additionally, we include results under multiple visual conditions (e.g., two faces) in Appendix A.1.2.
+
+![](images/f54429ceb18622592f79c89e031e37ef52947220eb27d01a3890443a876dcb46.jpg)  
+Figure 6: Different degrees of changes in visual conditions. The small image on the left shows the given visual condition and the corresponding generated result is on the right.
+
+![](images/1a5babadf1c4badb839dc050b13c600a340edfc95104eea5fc8b3007b8581e8c.jpg)  
+Figure 7: More applications: (a) generalized condition and (b) whole image generation/editing. The small images represent the visual conditions, while the texts serve as prompts.
+
+# 5 CONCLUSION AND DISCUSSION
+
+In this paper, we investigate the diffusion (physics) in diffusion (machine learning) properties and propose our Cyclic One-Way Diffusion (COW) method to strict the bidirectional diffusion into a unidirectional diffusion from the given visual condition via a pre-trained frozen diffusion model, fertilizing a versatility of customization application scenarios. Our method novelly explores the potential of utilizing the intrinsic diffusion property for specific task needs. All the experiments and evaluations demonstrate our method can generate images with high fidelity to both semantic-text and pixel-visual conditions in a training-free, efficient, and effective manner.
+
+Limitations. The pre-trained diffusion model is sometimes not robust enough to handle extremely strong conflicts between the visual and the text conditions. For example, when given the text "a profile of a person", and a front face condition, it is very hard to generate a harmonious result that fits both conditions. In this case, the model would follow the guidance of the text generally.
+
+Social Impact. Image generation and manipulation have been greatly used in art, entertainment, aesthetics, and other common use cases in people's daily lives. However, it can be abused in telling lies for harassment, distortion, and other malicious behavior. Too much abuse of generated images will decrease the credibility of the image. Our work doesn't surpass the capabilities of professional image editors, which adds to the ease of use of this proposed process. Since our model fully builds on the pre-trained T2I model, all the fake detection works distinguishing the authenticity of the image should be able to be directly applied to our results.
+
+# ACKNOWLEDGEMENT
+
+This work was partially supported by the National Natural Science Foundation of China under Grant 62372341. This work is jointly advised by Dr. Ye Zhu and Dr. Yu Wu. Also, Ye and Yu appreciate the support from their postdoc advisor, Dr. Olga Russakovsky from Princeton University, for her help in their early career development stage.
+
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+
+# A APPENDIX
+
+In the supplementary material, Appendix A.1 showcases an array of TV2I generation results along with comprehensive analyses. We delve into the flexible adaptation of visual conditions under text guidance (Appendix A.1.1), presenting additional TV2I outcomes involving scenarios with multiple faces (Appendix A.1.2). We explore the effects of varying sizes of visual conditions (Appendix A.1.3) and examine failure cases (Appendix A.1.4). Additionally, we conduct extensive image comparisons with baseline methods (Appendix A.1.5). Our thorough ablation study in Appendix A.2 validates the effectiveness of Seed Initialization and provides insights into optimal hyperparameter configurations. Detailed human evaluation results are elaborated upon in Appendix A.3. Moreover, we illuminate the varying sensitivity to text conditions during the denoising process (Appendix A.4.1) and elucidate the semantic formation process of diffusion generation (Appendix A.4.2). Concluding the supplemental material, we furnish the pseudo-code of our proposed pipeline in Appendix A.5.
+
+# A.1 TV2I GENERATION RESULTS COMPARISON AND ANALYSIS
+
+# A.1.1 DEGREE OF CHANGES ON VISUAL CONDITIONS
+
+COW offers a flexible change in the visual condition region according to the text guidance. The level of changes that may occur within the seed image depends on the discrepancy between textual and visual conditions as shown in Fig. 6. We show a set of samples containing gradual changes of the generated sample: (1) almost unchanged, (2) slightly perturbed (e.g., add accessories), (3) attribute variation (e.g., from smile to angry), (4) style transfer (e.g., from photo to comic picture), and (5) cross-domain transformation (e.g., from human face to lion). The results demonstrate that the seed images can be well preserved given no explicit conflict between the visual-text conditioning pairs, while appropriate levels of text-guided changes can also be well reflected in the case of attribute variation, style transfer, and cross-domain transformation.
+
+We further provide examples with stronger conflicts (cross-domain) in Fig. 8 to demonstrate the model could even fit in the case where the visual condition has a very strong conflict with other conditions. The samples clearly show our method could fit in different conflict levels of text and visual conditions and can handle strong conflicts between conditions.
+
+![](images/9df8ca52799c0b6e72299f7bb199f3fb79a1fb880fe0b6c84017f367d7cdbb19.jpg)  
+Figure 8: Generated results of cross-domain transformation. Images with green edges are visual conditions.
+
+# A.1.2 MULTIPLE VISUAL CONDITION RESULTS
+
+We notice that the customized generation methods like DreamBooth (Ruiz et al., 2022) or TI (Gal et al., 2022) learn each object concept separately, which makes the direct generation of multi-target visual conditions infeasible. Additionally, other approaches (Kumari et al., 2022; Avrahami et al., 2023; Gu et al., 2024), that focus on multi-concept customization are often limited to combining 2-3 semantically distinct concepts, and when generating subjects with high similarities, they tend to become confused. Therefore, we try a more difficult setting of two human face conditions in one
+
+![](images/ba9b5b4fbb24f001e2a17d6929e69e56b1bc892c688e3d2d92e62427259ee82a.jpg)
+
+![](images/83cbc0e41a210a18645d26241c102c5e820f8dfb42dfd96680b9511891e8614b.jpg)
+
+![](images/362d4fa7c5f18498935c2e9455437b49a1b4a254bf903bb0175fbf69fff90656.jpg)  
+a wedding photo of two people.
+
+![](images/243fefe7c7ae68c503aced38055a2c618e5c9486bac2e3f91767073c442490ac.jpg)  
+two friends dined at a restaurant.
+
+![](images/f0d00a67a5fd9777edd4bbea696b8da2182f7272ffd63295545f99f9841c18fe.jpg)  
+a couple walked hand in hand through the park.
+
+![](images/d3706968116ea9b392d53b594b9ff0c0e01def9790ce3498d37aa73d03d41d27.jpg)  
+two people strolled along the beach during the sunset.
+
+![](images/710cfd934063d3ce8b322aa1bfaee6b93506a25849ec49cd4004b8be821a011a.jpg)  
+two business partners discussed their new project together.
+
+![](images/384252c15043651511e8d4c7b96af7ce7dce742a6b8566480acd69bace02a3d5.jpg)  
+two workout buddies jogged side by side during their morning run.  
+Figure 9: Generated results using two faces as the visual conditions.
+
+diffusion generation and provide a set of results in Fig. 9. As it can be seen from the figure, our method could be easily extended to the multiple visual condition setting.
+
+# A.1.3 DIFFERENT SIZES OF VISUAL CONDITION
+
+![](images/2ee43b00c23afa7f4e9029db4ed4a934383aaefbb830c0af1b45078be3eefc0f.jpg)  
+Figure 10: Generated results from different sizes of visual condition. We experimented with Seed Initialization for different sizes of large, medium, and small, respectively.
+
+We directly change the face size of the visual condition applied in the original process. As shown in Fig. 10, although the smaller visual condition is more susceptible to the background, our method still successfully delivers visual condition information and maintains satisfactory results.
+
+# A.1.4 ANALYSIS OF FAILURE CASES
+
+We find the proposed method may fail in two cases as shown in Fig. 11. First, there is a strong orientation mismatch between the visual condition face and the text prompt (e.g., back view text prompt but with front view face visual condition). Second, the model may fail when there is a clear
+
+![](images/35b36afdbf6a0211f8951012f91574f29be354512786d559c864ef1cf45663b5.jpg)  
+a person with his back to the camera
+
+![](images/12825bded10946b484a3b3513499da59bf864b60eaa2778792969a253c240bf3.jpg)  
+close-up of a hand turning the pages of an old, worn book.
+
+![](images/2742873ea7c804d76090c185510d5006864679d445a4286bb40c545bd9ba0e40.jpg)  
+a person wearing a masquerade mask at a glamorous ball  
+a person wrapped in a scarf and wearing goggles in a snowy mountain landscape
+
+![](images/1af8bf6e5f5915ca21da0f4ce61d52272e2bbe47982739561ae329e72007b6d4.jpg)  
+Figure 11: Some failure cases of COW.
+
+text instruction of face coverings while the visual condition is an unobstructed face. In this case, the model may generate the covering at other random positions in the image.
+
+# A.1.5 MORE IMAGE COMPARISONS WITH BASELINES
+
+In Fig. 12, we include a more intuitive comparison with baseline results. We compare current SOTA work that incorporates visual conditions into pre-trained T2I models: DreamBooth (Ruiz et al., 2022), TI (Gal et al., 2022), ControlNet (Zhang & Agrawala, 2023), and SD inpaint (Rombach et al., 2022)
+
+# A.2 MORE ABLATION STUDIES
+
+We ablate on different settings of Seed Initialization (SI) to explore the best setting for SI and verify the effectiveness of SI, as shown in Fig. 13.
+
+Different Initialization Position. We show the ablation analysis on Seed Initialization (SI) to see its effectiveness in Fig. 13. The first three columns show different initialization positions for SI, which simulate user customization of visual conditions. We observe that our model can freely generate reasonable images given the pre-defined initialization position.
+
+Different Seed Initialization Background. We also try white, black, and random noise backgrounds as the pre-defined background images for SI. The results show using these backgrounds could lightly decrease the quality of generated images compared to the ones in the gray (default) background. We argue this originates from the fact that the average color value (gray) is a better starting point for all possible synthesized images.
+
+Ablation of Seed Initialization. The results w/o Seed Initialization exhibit an unnatural merging between background and visual condition, as another person emerges separately from the visual condition. This clearly shows the effectiveness of our SI. Without SI, we can not pre-generate the image layout during the Chaos Phase, where the model only fits the textual condition without considering the visual condition. For example, the model may already generate a person at other positions rather than the user's pre-defined location, which is nearly impossible for us to meet the strict visual condition restrictions given such a layout.
+
+Ablation of different hyper-parameters. We conduct a quantitative ablation study on different hyper-parameters of COW, including the number of cycles, or positions of the start and end points in Tab. 2. Here we use three quantitative metrics to analyze different models.  $CLIP-T$  is the CLIP-text cosine-similarity, which is the similarity of generated image and text condition calculated by the prestrained CLIP (ViT-B/32) model. We also perform human evaluations on those ablation models with 10 volunteers to better evaluate the visual quality of the generated samples. (Each volunteer carefully goes through 216 images generated by the six model variants and rates the best model variant as 5 and the worst as 0, similar to the Mean of Score test in previous works Zhu et al. (2022). Here we report the averaged ratings for each model.) We also include the time cost per image generation. Since different starting and ending points would involve different inversion steps, the time cost is slightly different according to the diffusion inversion cost. The results show the model
+
+![](images/503bf9ed8875a0cc683f305421e9c7122b1624ca915f9688af09ae77a8c58a1f.jpg)  
+Figure 12: Comparison of our COW-generated images with TV2I baseline.
+
+performance is sensitive to the starting and ending points, which confirms our motivation that we should repeatedly go through the semantic formation stage to have a controllable and directional generation. As we can see, a cycle number of 10 is sufficient and achieves a good balance between performance and speed for the inner diffusion process.
+
+![](images/320ff7030e4988950e2e64f6720401b913b70261819df28f3cb4acb93a70b435.jpg)  
+Figure 13: Ablation study on Seed Initialization. Given the left-most visual and text conditions, we show generated images at different initialization positions (left, middle, and right), with different initial backgrounds (gray, white, black, and random noise), and without Seed Initialization.
+
+<table><tr><td>Cycle</td><td>Position</td><td>Clip-T↑</td><td>Human Rating↑</td><td>Time Cost↓</td></tr><tr><td>10</td><td>700⇒500</td><td>0.31</td><td>4.6</td><td>5.21</td></tr><tr><td>10</td><td>900⇒700</td><td>0.30</td><td>1.5</td><td>5.52</td></tr><tr><td>10</td><td>800⇒400</td><td>0.30</td><td>3.2</td><td>7.96</td></tr><tr><td>10</td><td>500⇒300</td><td>0.27</td><td>0.8</td><td>4.85</td></tr><tr><td>5</td><td>700⇒500</td><td>0.29</td><td>2.5</td><td>3.32</td></tr><tr><td>20</td><td>700⇒500</td><td>0.30</td><td>3.6</td><td>8.83</td></tr></table>
+
+Table 2: Ablation of different hyper-parameters of the cycle number and the cycle position using three metrics including Clip-T, Human Rating, and Time Cost.
+
+# A.3 HUMAN EVALUATION
+
+We conduct a human evaluation based on two criteria: 1) Condition Consistency: whether the generated image well matches both the visual and textual conditions; 2) General Fidelity: whether the chosen image looks more like a real image in terms of image richness, face naturalness, and overall image fidelity. It is worth noting that we inform the participants of the application setting: input the face text condition pairs and then get an output image, and ask them to choose a favorite result if they were the users under the two basic criteria. The voting rate for our results greatly surpasses the other baselines in both criteria across all three settings, as shown in Tab. 3, and this demonstrates the general quality when applied to TV2I generation. We show interface examples of questionnaire systems received by participants regarding these two criteria in Fig. 14 and Fig. 15.
+
+The participants recruited are all undergraduate or graduate students, all over 18 years old, ranging from 18-26 years old. Therefore, there are no ethical concerns for young participants (under 18 years
+
+Table 3: The condition correspondence and general fidelity results of human study for three settings: normal prompt, style transfer, and attribute editing, with condition correspondence to the left of every column and general fidelity to the right. Note that our method outperforms all of the baselines in terms of the general fidelity shown by the results of human users' judgment.  
+
+<table><tr><td></td><td colspan="2">TI</td><td colspan="2">DB</td><td colspan="2">ControlNet</td><td colspan="2">SD inpainting</td><td colspan="2">COW</td></tr><tr><td rowspan="3">normal style editing</td><td>2.00%</td><td>10.60%</td><td>2.80%</td><td>21.40%</td><td>5.40%</td><td>3.00%</td><td>5.20%</td><td>2.00%</td><td>84.60%</td><td>63.00%</td></tr><tr><td>3.60%</td><td>15.40%</td><td>13.40%</td><td>26.80%</td><td>24.20%</td><td>7.20%</td><td>10.60%</td><td>3.00%</td><td>48.20%</td><td>47.60%</td></tr><tr><td>2.60%</td><td>11.80%</td><td>12.80%</td><td>37.80%</td><td>5.40%</td><td>4.00%</td><td>3.20%</td><td>1.20%</td><td>76.00%</td><td>45.20%</td></tr></table>
+
+![](images/0999fadf822ca537f53338d362b25d54467e2b973400291ff0c1fd0339ea35b7.jpg)  
+Figure 14: The interface of our human study on conditional consistency.
+
+![](images/7d042077e77809bef76d6762cebaa38686668581557f18f097a7b8b7e1753b37.jpg)  
+Figure 15: The interface of our human study on general fidelity.
+
+old). Before taking the questionnaire, all participants are informed about the research's purpose, the compensation, and the user study content. All participants consent to participate. We do not provide compensation to volunteers. In addition, all elements of the questionnaire are free from potential psychological or moral hazards.
+
+# A.4 PROPERTIES OF DIFFUSION GENERATION PROCESS
+
+# A.4.1 SENSITIVITY TO TEXT CONDITION DURING DENOISING
+
+![](images/674e7dee2cdeea5276c6a5c8e35affa5a6b89c85ca96486aaf0670c96478fe66.jpg)  
+Figure 16: Text-sensitivity during denoising process. Each line represents a CLIP cosine similarity between the generated image and text with the text condition injected at different steps. The image is generated in 1000 overall unconditional denoising process with 100 steps text conditional guidance starting from  $t$ . Generally, the denoising process is more responsive to the text condition in the beginning and almost stops reacting to the text condition when high-level semantics are settled.
+
+![](images/2ab070179aaacc3d40bfee05dd0978ef8ef0060e850e48354e4b04fb9d8dd7b2.jpg)  
+(a)
+
+![](images/16b55cf69a7cf5c98f1a4de58da4930c0ddfcf3316d4b8e0512cff4ee8be4a13.jpg)  
+(b)  
+Figure 17: Different sizes come with different semantic formation processes. Each red curve represents a face disturb-and-reconstruct process, (a) size is  $256 \times 256$  and (b) size is  $128 \times 128$ . We disturb the reconstruction process by sticking the origin image to a random noise background  $(512 \times 512)$  at different steps. The general semantic is settled earlier when the size is larger.
+
+We study the sensitivity of DMs to text conditions during the denoising process by injecting text conditions at different denoising steps. We replace a few unconditional denoising steps with text-conditioned ones during the generation of image random sampled from Gaussian noise. We calculate the Clip cosine similarity between the final generated image and text condition. As shown in Fig. 16, the model is more sensitive to text conditions in the early denoising stage and less sensitive in the late stage. This also demonstrates our method reasonably utilizes the sensitivity to text conditions through proposed Seed Initialization and Cyclic One-Way Diffusion.
+
+![](images/63dd103b1c863fe06f0bbfc44f2f65f688dad8867e78fe81847f30790c482356.jpg)  
+Figure 18: Illustration of the semantic formation process and the mutual interference. Each red curve represents a face disturb-and-reconstruct process. We only show one face for instance. We disturb the reconstruction process by sticking the origin image to a random noise background at different steps. The cosine similarity between the original image and the reconstructed image increases as the denoising step goes.
+
+# A.4.2 ILLUSTRATION OF SEMANTIC FORMATION PROCESS OF DIFFUSION GENERATION
+
+To quantitatively assess the influence between regions during the denoising process, we inverse an image  $x_0$  to obtain different transformed  $\mathbf{x_t}$ , and plant it in a replacement manner into a sub-region of random Gaussian noise in the same step. We subsequently apply this composite noise map to several denoising steps before extracting the planted image to continue the generation process. As shown in Fig. 18, the influence level weakens with the denoising process.
+
+To see the degree of impact exerted on images of different sizes during the denoising process, we conduct the same experiments with images of two different sizes  $(128\times 128,256\times 256)$ . We inverse images to  $\mathbf{x_t}$ , and reconstruct them with disturbance introduced by sticking them to a random noise background  $(512\times 512)$  at the corresponding step for 100 steps. We calculate the MSE loss of the original image and the final reconstructed image after being disturbed. As shown in Fig. 17, the semantics settle earlier when the size is larger.
+
+# A.5 PSEUDO CODE
+
+We show the pseudo-code for our proposed Cyclic One-Way Diffusion approach in Algo. 1.
+
+Algorithm 1: Cyclic One-Way Diffusion  
+Input: visual condition  $V_{0}$  ,text prompt  $P$  target cycle number  $c$  , ends of the cycle  $t_1,t_2$  , id preserve step  $t_3$    
+Output: Generated Image  $\mathcal{X}$    
+Definition: deterministic sampling  $p s$  , stochastic sampling  $p d$    
+// Step 1: Seed Initialization   
+ $X_0\gets$  paste  $V_{0}$  into the user-specified background by the user-specified size and position;   
+ $x_0,v_0\gets \mathrm{Encoder}(X_0),\mathrm{Encoder}(V_0);$ $x_{t_1},v_t\gets \mathrm{ODEInverse}(x_0,t_1),\mathrm{ODEInverse}(v_0,t);$    
+// Step 2: Cyclic One-Way Diffusion   
+while cycnum  $<  c$  do for  $t = t_1,t_1 - 1,\dots ,t_2$  do  $x_{t}^{\prime}\gets \mathrm{Replace}(x_{t},v_{t})$  .  $x_{t - 1}\gets p_s(x_t',t,P)$  end   
+ $x_{t_1}\gets \mathrm{InjectNoise}(x_{t_2})$    
+end   
+// Step 3: Visual Condition Preserve   
+for  $t = t_2,t_2 - 1,\ldots ,0$  do if  $t = t_3$  then  $x_{t}^{\prime}\gets \mathrm{Replace}(x_{t},v_{t})$  .  $x_{t - 1}\gets p_d(x_t',t,P)$  end   
+ $x_{t - 1}\gets p_d(x_t,t,P)$    
+end

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+# DIFFUSIONSAT: A GENERATIVE FOUNDATION MODEL FOR SATELLITE IMAGERY
+
+Samar Khanna $^{1*}$ , Patrick Liu $^{1}$ , Linqi Zhou $^{1}$ , Chenlin Meng $^{1}$ , Robin Rombach $^{2}$ , Marshall Burke $^{1}$ , David B. Lobell $^{1}$ , Stefano Ermon $^{1,3}$
+
+$^{1}$ Stanford University,  $^{2}$ Stability AI,  $^{3}$ CZ Biohub
+
+*Correspondence to: samarkhanna [at] cs.stanford.edu
+
+# ABSTRACT
+
+Diffusion models have achieved state-of-the-art results on many modalities including images, speech, and video. However, existing models are not tailored to support remote sensing data, which is widely used in important applications including environmental monitoring and crop-yield prediction. Satellite images are significantly different from natural images – they can be multi-spectral, irregularly sampled across time – and existing diffusion models trained on images from the Web do not support them. Furthermore, remote sensing data is inherently spatio-temporal, requiring conditional generation tasks not supported by traditional methods based on captions or images. In this paper, we present DiffusionSat, to date the largest generative foundation model trained on a collection of publicly available large, high-resolution remote sensing datasets. As text-based captions are sparsely available for satellite images, we incorporate the associated metadata such as geolocation as conditioning information. Our method produces realistic samples and can be used to solve multiple generative tasks including temporal generation, superresolution given multi-spectral inputs and in-painting. Our method outperforms previous state-of-the-art methods for satellite image generation and is the first large-scale generative foundation model for satellite imagery. The project website can be found here: https://samar-khanna.github.io/DiffusionSat/
+
+# 1 INTRODUCTION
+
+Diffusion models have achieved state of the art results in image generation (Sohl-Dickstein et al., 2015; Ho et al., 2020; Dhariwal & Nichol, 2021; Kingma et al., 2021; Song & Ermon, 2019; 2020). Large scale models such as Stable Diffusion Rombach et al. (2022) (SD) have been trained on Internet-scale image-text datasets to generate high-resolution images from user-provided captions. These diffusion-based foundation models, used as priors, have led to major improvements in a variety of inverse problems like inpainting, colorization, deblurring (Luo et al., 2023), medical image reconstruction (Khader et al., 2023; Xie & Li, 2022), and video generation (Blattmann et al., 2023).
+
+Similarly, there are a variety of high-impact ML tasks involving the analysis of satellite images, such as disaster response, environmental monitoring, poverty prediction, crop-yield estimation, urban planning and others (Gupta et al., 2019; Burke et al., 2021; Ayush et al., 2021b; 2020; Jean et al., 2016; You et al., 2017; Wang et al., 2018; RuBwurm & Korner, 2020; Martinez et al., 2021; M Rustowicz et al., 2019; Yeh et al., 2021). These tasks consist of important inverse problems, such as super-resolution (from frequent low resolution images to high resolution ones), cloud removal, temporal in-painting and more. However, satellite images fundamentally differ from natural images in terms of perspective, resolutions, additional spectral bands, and temporal regularity. While foundation models have been recently developed for discriminative learning on satellite images Cong et al. (2022); Ayush et al. (2021a); Bastani et al. (2022), they are not designed to and cannot solve the inverse problems (eg: super-resolution) described above.
+
+To fill this gap, we propose DiffusionSat, a generative foundation model for satellite imagery inspired from SD. Using commonly associated metadata with satellite images including latitude, longitude, timestamp, and ground-sampling distance (GSD), we train our model for single-image generation on a collection of publicly available satellite image data sets. Further, inspired from
+
+![](images/c0c222a7c31546fa92e559b900f50f7e4e4c6678d9c020aebd88a86a8a5fcca7.jpg)  
+Figure 1: Conditioning on freely available metadata and using large, publicly available satellite imagery datasets shows DiffusionSat is a powerful generative foundation model for remote sensing data.
+
+ControlNets Zhang & Agrawala (2023), we design conditioning models that can easily be trained for specific generative tasks or inverse problems including super-resolution, in-painting, and temporal generation. Specifically, our contributions include:
+
+1. We propose a novel generative foundation model for satellite image data with the ability to generate high-resolution satellite imagery from numerical metadata as well as text.  
+2. We design a novel 3D-conditioning extension which enables DiffusionSat to demonstrate state-of-the-art performance on super-resolution, temporal generation, and in-painting  
+3. We collect and compile a global generative pre-training dataset from large, publicly available satellite image datasets (see section 3.1).
+
+# 2 BACKGROUND
+
+Diffusion Models Diffusion models are generative models that aim to learn a data distribution  $p_{\mathrm{Data}}$  from samples (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song & Ermon, 2019; Song et al., 2020b; Song & Ermon, 2020). Given an input image  $x \sim p_{\mathrm{Data}}$ , we add noise to create a noisy input  $x_{t} = \alpha_{t}x + \sigma_{t}\epsilon$ , where  $\epsilon \sim \mathcal{N}(\mathbf{0},\mathbf{I})$  is Gaussian noise.  $\alpha_{t}$  and  $\sigma_{t}$  denote a noise schedule parameterized by diffusion time  $t$  (higher  $t$  leads to more added noise). The diffusion model  $\epsilon_{\theta}$  then aims to denoise  $x_{t}$ , and is optimized using the score-matching objective:
+
+$$
+\mathbb {E} _ {x \sim p _ {\mathrm {D a t a}}, \epsilon \sim \mathcal {N} (\mathbf {0}, \mathbf {I})} \left[ | | y - \epsilon_ {\theta} (x _ {t}; t, c) | | _ {2} ^ {2} \right] \tag {1}
+$$
+
+where the target  $y$  can be the input noise  $\epsilon$ , the input image  $x$  or the "velocity"  $v = \alpha_{t}\epsilon - \sigma_{t}x$ . We can additionally condition the denoising model with side information  $c \in \mathbb{R}^{D}$ , which can be a class embedding, text, or other images etc.
+
+Latent diffusion models (LDMs) (Vahdat et al., 2021; Sinha et al., 2021; Rombach et al., 2022) first downsample the input  $x$  using a VAE with an encoder  $\mathcal{E}$  and a decoder  $\mathcal{D}$ , such that  $\tilde{x} = \mathcal{D}(\mathcal{E}(x))$  is a reconstructed image. Instead of denoising the input image  $x$ , the diffusion process is used on a downsampled latent representation  $z = \mathcal{E}(x)$ . This approach reduces computational and memory cost and has formed the basis for the popularly used StableDiffusion (SD) model (Rombach et al., 2022).
+
+# 3 METHOD
+
+First, we describe our method for the following tasks of interest: single-image generation, conditioned on text and metadata, multi-spectral superresolution, temporal prediction, and temporal inpainting.
+
+# 3.1 SINGLE IMAGE GENERATION
+
+Our first goal is to pre-train DiffusionSat to be able to generate single images given an input text prompt and/or metadata. Concretely, we begin by considering datasets such that each image
+
+$\mathbf{x} \in \mathbb{R}^{C \times H \times W}$  is paired with an associated text-caption  $\tau$ . Our goal is to learn the conditional data distribution  $p(\mathbf{x}|\tau)$  such that we can sample new images  $\tilde{\mathbf{x}} \sim p(\cdot|\tau)$ .
+
+LDMs are popularly used for text-to-image generation primarily for their strong ability to use text prompts. An associated text prompt  $\tau$  is tokenized, encoded via CLIP (Radford et al., 2021), and then passed to the DM  $\epsilon_{\theta}(\mathbf{x}_t; t, \tau)$  via cross-attention (Vaswani et al., 2017) at each layer. However, while text prompts are widely available for image datasets such as LAION-5B (Schuhmann et al., 2022), satellite images typically either do not have such captions, or are accompanied by object-detection boxes, segmentation masks, or classification labels. Moreover, requiring such labels precludes the use of vast amounts of unlabeled satellite imagery. Ideally, we would like to pretrain DiffusionSat on existing labelled and unlabeled datasets, without necessarily curating expensive labels.
+
+To solve this challenge, we note that satellite images are commonly associated with metadata including their timestamp, latitude, longitude, and various other numerical information that are correlated with the image (Christie et al., 2018). We thus consider datasets where each image  $\mathbf{x} \in \mathbb{R}^{C \times H \times W}$  is paired with a text-caption  $\tau$ , as well as cheaply available numerical metadata  $\mathbf{k} \in \mathbb{R}^{M}$ , where  $M$  is the number of metadata items. We thus wish to learn the data distribution  $p(\mathbf{x}|\tau, \mathbf{k})$ . With good enough metadata  $\mathbf{k}$ , we want to still sample an image of high quality even if  $\tau$  is poor or missing.
+
+We now turn to conditioning on  $\mathbf{k}$ . One option is to naively incorporate each numerical metadata item  $k_{j}, j\in \{1,\dots ,M\}$ , into the text caption with a short description. However, this approach unnecessarily discretizes continuous-valued covariates and can suffer from text-encoders' known shortcomings related to encoding numerical information (Radford et al., 2021). Instead, we choose to encode the metadata using the same sinusoidal timestep embedding eq. (2) used in diffusion models:
+
+$$
+\operatorname {P r o j e c t} (k, 2 i) = \sin \left(k \Omega^ {- \frac {2 i}{d}}\right), \operatorname {P r o j e c t} (k, 2 i + 1) = \cos \left(k \Omega^ {- \frac {2 i}{d}}\right) \tag {2}
+$$
+
+where  $k$  is the metadata or timestep value,  $i$  is the index of feature dimension in the encoding,  $d$  is the dimension, and  $\Omega = 10000$  is a large constant. Each metadata value  $k_{j}$  is first normalized to a value between 0 and 1000 (since the diffusion timestep  $t \in \{0, \dots, 1000\}$ ), and is then projected via the sinusoidal encoding. A different MLP for each metadataum encodes the projected metadata value identically to the diffusion timestep  $t$  (Ho et al., 2020) as follows eq. (3):
+
+$$
+f _ {\theta_ {j}} \left(k _ {j}\right) = \operatorname {M L P} \left(\left[ \text {P r o j e c t} \left(k _ {j}, 0\right), \dots , \text {P r o j e c t} \left(k _ {j}, d\right) \right]\right) \tag {3}
+$$
+
+where  $f_{\theta_j}$  represents the learned MLP embedding for metadata value  $k_j$ , corresponding to metadata type  $j$  (eg: longitude). Our embedded is then  $f_{\theta_j}(k_j) \in \mathbb{R}^D$ , where  $D$  is the embedding dimension. The  $M$  metadata vectors are then added together  $\boldsymbol{m} = f_{\theta_1}(k_1) + \dots + f_{\theta_M}(k_M)$ , where  $\boldsymbol{m} \in \mathbb{R}^D$ , which is then also added with the embedded timestep  $t = f_{\theta}(t) \in \mathbb{R}^D$ , so that the final conditioning vector is  $\boldsymbol{c} = \boldsymbol{m} + t$ .
+
+To summarize, we first we first encode an image  $\mathbf{x} \in \mathbb{R}^{C \times H \times W}$  using the SD variational autoencoder (VAE) (Rombach et al., 2022; Esser et al., 2021) to a latent representation  $\mathbf{z} = \mathcal{E}(\mathbf{x}) \in \mathbb{R}^{C' \times H' \times W'}$ . Gaussian noise is then added to the latent image features to give us  $\mathbf{z}_t = \alpha_t \mathbf{z} + \sigma_t \epsilon$  (see section 2). The conditioning vector  $c$  created from embedding metadata and the diffusion timestep, as well as the CLIP-embedded text caption  $\tau' = \mathcal{T}_{\theta}(\tau)$ , are passed through a DM  $\epsilon_{\theta}(\mathbf{z}_t; \tau', c)$  to predict the added noise. Finally, the VAE decoder  $\mathcal{D}$  upsamples the denoised latents to full resolution (fig. 1).
+
+Lastly, we initialize the encoder  $\mathcal{E}$ , the decoder  $\mathcal{D}$ , the CLIP text encoder  $\mathcal{T}_{\theta}$ , and the denoising unet  $\epsilon_{\theta}$  all with SD 2.1's weights. We only update the denoising UNet  $\epsilon_{\theta}$  and the metadata and timestep embeddings  $f_{\theta_j}$  during training to speed up convergence using the rich semantic information in the pretrained SD weights. During training, we also randomly zero out the metadata vector  $\pmb{m}$  with a probability of 0.1 to allow the model to generate images when metadata might be unavailable or inaccurate. A similar strategy is employed to learn unconditional generation by Ho et al. (2020).
+
+Single Image-Text-Metadata Datasets There is no equivalent of a large, text-image dataset (eg: LAION (Schuhmann et al., 2022)) for satellite images. Instead, we compile publicly available annotated satellite data and contribute a large, high-resolution generative dataset for satellite images. Detailed descriptions on how the caption is generated for each dataset are in the appendix. (i) fMoW: Function Map of the World (fMoW) Christie et al. (2018) consists of global, high-resolution (GSD  $0.3\mathrm{m}-1.5\mathrm{m}$ ) MAXAR satellite images, each belonging to one of 62 categories. We crop each image to 512x512 pixels. The metadata we consider include longitude, latitude, GSD (in meters), cloud
+
+![](images/96274b6d655721fee3e9f2c19eea820ed9cbcfd73074af8467821cd218225a47.jpg)  
+Figure 2: DiffusionSat flexibly extends to a variety of conditional generation tasks. We design a 3D version of a ControlNet (Zhang & Agrawala, 2023) which can accept a sequence of images. Like regular ControlNets, our 3D ControlNet keeps a trainable copy of SD weights for the downsampling and middle blocks. Latent image features are reshaped to combine the batch and temporal dimensions before being input to these layers. The output of each SD block is then passed through a temporal layer (top right), which re-expands the temporal dimension before passing the latent features though a 3D convolution (initialized with zeros) and a temporal, pixel-wise transformer. The metadata associated with each input image is projected as in fig. 1.
+
+cover (as a fraction), year, month, and day. To generate a caption, we consider the semantic class and the country code. (ii) Satlas: Satlas Bastani et al. (2022) is a large-scale, multi-task dataset of NAIP and Sentinel-2 satellite images. For our dataset, we use the NAIP imagery in Satlas-small, roughly of the same size as fMoW. We use the same metadata as in item (i). (iii) SpaceNet: Spacenet Van Etten et al. (2018; 2021) is a collection of satellite image datasets for tasks including object detection, semantic segmentation and road network mapping. We consider a subset of Spacenet datasets, namely Spacenet v1, Spacenet v2, and Spacenet v5. We use the same metadata as earlier.
+
+# 3.2 CONTROL SIGNAL CONDITIONAL GENERATION
+
+Single-image DiffusionSat can generate a high-resolution satellite image given associated prompt and metadata, but it cannot yet solve the inverse problems described in section 1. To leverage its pretrained weights, we can use it as a prior for conditional generation tasks which do encompass inverse problems such as super-resolution and in-painting. Thus, we now consider generative tasks where we can additionally condition on control signals (eg: sequences of satellite images)  $\mathbf{s} \in \mathbb{R}^{T \times C' \times H' \times W'}$  with associated metadata  $\mathbf{k}_s \in \mathbb{R}^{T \times M}$ , a single caption  $\tau$  and target metadata  $\mathbf{k} \in \mathbb{R}^M$ . Here,  $C'$ ,  $H'$ , and  $W'$  reflect the possible difference in the number of channels, height, and width, respectively, between the conditioning images and the target image. The goal is to sample  $\tilde{\mathbf{x}} \sim p(\cdot | \mathbf{s}; \mathbf{k}_s; \tau; \mathbf{k})$ , where  $\tilde{\mathbf{x}}$  is a sample conditioned on the control signal  $\mathbf{s}$  for a given caption  $\tau$  and given metadata  $\mathbf{k}$ .
+
+Temporal Generation Recent works for video diffusion have proposed using 3D convolutions and temporal attention (Blattmann et al., 2023; Wu et al., 2022; Zhou et al., 2022), while others propose using existing 2D UNets and concatenating temporal frames in the channel dimension (Voleti et al., 2022; An et al., 2023). However, sequences of satellite images differ in a few key ways from frames of images in a video: (i) there is high variance in the length of time separating images in the sequence, while frames in video data are usually separated by a fixed amount of time (fixed frame rate) (ii) the length of time between images can be on the order of months or years, therefore capturing a wider range of semantic information than consecutively placed frames in video (eg: season, human development, land cover). (iii) there is a sense of "global time" across locations. Even if one compares satellite images across different countries or terrains, patterns may be similar if the
+
+![](images/c764db59755c7b3037f7bae9ad064bf5e7f10dc0e7fcbd8536c31b13c85cce60.jpg)  
+Figure 3: Here we generate samples from single-image DiffusionSat. We see that changing the coordinates from a location in Paris to one in USA changes the type of stadium generated, with American football and baseball more likely to appear in the latter location. Additionally, for locations that receive snow, DiffusionSat accurately captures the correlation between location and season. However, naively incorporating the metadata into the text caption results in poorer conditioning flexibility across geography and season, (eg: with winter and summer time images produced for both August and January, or a lack of “zooming in” when lowering the GSD).
+
+year is known to be 2012 as opposed to 2020 (especially for urban landscapes). This is not the case for video data, where "local" time across frames is sufficient to provide semantic meaning.
+
+Usually, sequences of satellite images have fewer images than frames in natural image videos. As such, generating long sequences of images is less useful than conditioning on existing satellite imagery to predict the future or interpolate in the past (He et al., 2021; Bastani et al., 2023). Thus, we introduce our novel conditioning framework shown in fig. 2 to solve the inverse problem of frame-by-frame conditional temporal prediction. Unlike 2D ControlNet, we use 3D zero-convolutions between each StableDiffusion block (Zhang & Agrawala, 2023). Our temporal attention layers, similar to VideoLDM (Blattmann et al., 2023), further enable the model to condition on temporal control signals. We introduce a learned parameter  $\alpha_{i}$  for each block  $i$  to "mix" in the output of the temporal attention layer to prevent noise from early stages in training from affecting our pre-trained weights (fig. 2).
+
+A key advantage of our approach is the ability to provide each item in the control sequence  $s$  with its own associated metadata, which is done similarly as in fig. 1, by projecting each metadata individually and embedding it with an MLP. The embedded metadata for each image is then concatenated with its image and passed through the 2D layers of the ControlNet. DiffusionSat is thus invariant to the ordering of images in the control sequence  $s$ , since each image's temporal position is solely determined by the timestamp in its metadata. A single DiffusionSat model can then be trained to predict images in the past and future, or interpolate within the temporal range of the sequence.
+
+Super-resolution with multi-spectral input Unlike in section 3.2, our input is a sequence s of lower resolution (GSD) images than the target image and can contain a differing number of channels. The output of the model is still a high-resolution RGB image, as before.
+
+Temporal Inpainting The task is functionally equivalent to 3.2, except the goal is to in-paint corrupted pixels (eg: from cloud cover, flooding, fire-damage) rather than predict a new frame in s.
+
+# 4 EXPERIMENTS
+
+We describe the experiments for the tasks in section 3. Implementation details are in appendix A.1.
+
+<table><tr><td>Method</td><td>FID↓</td><td>IS↑</td><td>CLIP↑</td></tr><tr><td>SD 2.1</td><td>117.74</td><td>6.42</td><td>17.23</td></tr><tr><td>SD 2.1†</td><td>37.99</td><td>7.42</td><td>16.59</td></tr><tr><td>SD 2.1‡</td><td>24.23</td><td>7.60</td><td>18.62</td></tr><tr><td>Ours</td><td>15.80</td><td>6.69</td><td>17.20</td></tr></table>
+
+Table 1: Single-image  $512 \times 512$  generation on the validation set of fMoW. † refers to finetuned SD 2.1 without any metadata information. ‡ refers to incorporating the metadata in the text caption.  
+
+<table><tr><td>Method</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>MSE↓</td></tr><tr><td>Pix2Pix</td><td>0.1374</td><td>8.2722</td><td>0.6895</td><td>0.1492</td></tr><tr><td>DBPN</td><td>0.1518</td><td>11.8568</td><td>0.6826</td><td>0.0680</td></tr><tr><td>SD</td><td>0.1671</td><td>10.2417</td><td>0.6403</td><td>0.0962</td></tr><tr><td>SD + CN</td><td>0.1626</td><td>10.0098</td><td>0.6506</td><td>0.1009</td></tr><tr><td>Ours</td><td>0.1703</td><td>10.3924</td><td>0.6221</td><td>0.0928</td></tr></table>
+
+Table 2: Image sample quality quantitative results on fMoW superresolution. DBPN refers to Haris et al. (2018), Pix2Pix is from Isola et al. (2017). Our method beats other super-resolution models for multi-spectral data.
+
+For single image generation, we report standard visual-quality metrics such as FID (Heusel et al., 2017), Inception Score (IS), and CLIP-score (Radford et al., 2021). For conditional generation, given a reference ground-truth image, we report pixel-quality metrics including SSIM (Wang et al., 2004), PSNR, LPIPS (Zhang et al., 2018) with VGG (Simonyan & Zisserman, 2014) features. As noted in Gong et al. (2021) and He et al. (2021), LPIPs is a more relevant perceptual quality metric used in evaluating satellite images. Our metrics are reported on a sample size of 10,000 images.
+
+# 4.1 SINGLE IMAGE GENERATION
+
+We first consider single-image generation, as the task that DiffusionSat is pre-trained on. We compare against a pre-trained SD 2.1 model Rombach et al. (2022), a SD 2.1 model finetuned on our dataset with our captions, but without metadata, and finally a SD 2.1 model finetuned on our dataset with the metadata included in the caption (see table 1). We find that including the metadata, even within the caption, is better than a caption formed from just the labels of satellite images. This is reflected in better FID scores, which measure visual quality. We expect that the text-metadata model  $\ddagger$  does better in terms of CLIP score given its more highly descriptive caption. However, treating metadata numerically, as in DiffusionSat, further improves generation quality and control, as seen in fig. 3.
+
+# 4.2 CONTROL SIGNAL CONDITIONAL GENERATION
+
+We now use single-image DiffusionSat as an effective prior for the conditional generation tasks of super-resolution, temporal generation/prediction, and in-painting. We describe the dataset for each task and demonstrate results using our 3D conditioning approach on Texas-housing super-resolution, fMoW super-resolution using fMoW-Sentinel multispectral inputs, temporal generation on the fMoW-temporal dataset, and temporal inpainting on the xBD natural disaster dataset. DiffusionSat achieves state of the art LIPs and close to optimal performance on the SSIM and PSNR metrics as well.
+
+fMoW Superresolution Using the dataset provided in Cong et al. (2022), we create a fMoW-Sentinel-fMoW-RGB dataset with paired Sentinel-2 (10m-60m GSD) and fMoW (0.3-1.5m GSD) images at each of the original fMoW-RGB locations. Given all 13 multi-spectral bands of the Sentinel-2 image (here  $T = 1$ ), we aim to reconstruct the corresponding high resolution RGB image. Super-resolution (section 4.2) given low-resolution (10m-60m), multi-spectral input is especially difficult, since most fMoW-RGB images are  $< 1\mathrm{m}$  GSD. We find that DiffusionSat once again outperforms strong super-resolution baselines, such as SD (which has shown to (table 2, fig. 4). We further note that while methods such as DBPN (Haris et al., 2018) yield strong PSNR/SSIM, these metrics don't reflect human perception and favor blurriness over sharp detail (Zhang et al., 2018; Sahara et al., 2022b).
+
+Texas Housing Superresolution The dataset for this task is introduced by Spatial Temporal Superresolution (STSR) (He et al., 2021) and contains 286717 houses built between 2014 and 2017 in Texas. Each location consists of two high-resolution images from NAIP (GSD  $1\mathrm{m}$ ) and 2 low-resolution images from Sentinel-2 (GSD  $10\mathrm{m}$ ). A high resolution image at a time  $t$  and corresponding low-resolution images at times  $t$  and  $t'$  form the control signal  $\mathbf{s}$ , and the task is to reconstruct the other high resolution image  $\mathbf{x}$  at time  $t'$ .
+
+![](images/f97e020b3678d1b5a0ea328f58ba3d773337cafec38468db62aaa1ce24786646.jpg)  
+Figure 4: Generated samples from fMoW-Sentinel superresolution validation set. The conditioning image is the Sentinel-2 multispectral (MS) image represented here as SWIR, NIR, RGB. The desired output is the high-resolution (HR) fMoW-RGB image. Our method is able to capture fine-grained details better than other baselines, even when the low-resolution MS image lacks detail. SD tends to "hallucinate" details.
+
+![](images/c0ba758c0b1e11997a307ad294f28bbce49a7c81c88d078083c83b78b9d7d162.jpg)
+
+We also perform an ablation on the efficacy of pretraining on our single image datasets against finetuning directly on SD weights. We find a significant improvement from DiffusionSat pretraining, and from using the 3D ControlNet (across all metrics) over simply stacking the images in the channel dimension and using a 2D ControlNet (table 3).
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">t&#x27; &gt; t</td><td colspan="3">t&#x27; &lt; t</td></tr><tr><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td></tr><tr><td>Pix2Pix</td><td>0.5432</td><td>20.8420</td><td>0.4243</td><td>0.3909</td><td>17.9528</td><td>0.4909</td></tr><tr><td>cGAN Fusion</td><td>0.5976</td><td>21.5226</td><td>0.3936</td><td>0.4220</td><td>17.8763</td><td>0.4726</td></tr><tr><td>DBPN</td><td>0.5781</td><td>21.4716</td><td>0.5101</td><td>0.4572</td><td>18.9330</td><td>0.5910</td></tr><tr><td>SRGAN</td><td>0.5361</td><td>21.1968</td><td>0.5261</td><td>0.4221</td><td>18.9772</td><td>0.5694</td></tr><tr><td>STSR (EAD)</td><td>0.6470</td><td>22.4906</td><td>0.3695</td><td>0.5225</td><td>19.7675</td><td>0.4275</td></tr><tr><td>STSR (EA64)</td><td>0.6570</td><td>22.5552</td><td>0.3764</td><td>0.5338</td><td>19.8547</td><td>0.4342</td></tr><tr><td>SD + 3D ControlNet</td><td>0.4747</td><td>17.8023</td><td>0.4166</td><td>0.3458</td><td>16.1467</td><td>0.4351</td></tr><tr><td>Ours + ControlNet</td><td>0.5403</td><td>20.3982</td><td>0.3874</td><td>0.4657</td><td>18.1007</td><td>0.3652</td></tr><tr><td>Ours + 3D ControlNet</td><td>0.5982</td><td>21.0299</td><td>0.3247</td><td>0.4825</td><td>18.4604</td><td>0.3534</td></tr></table>
+
+Table 3: Sample quality results on Texas housing validation data.  $t' > t$  represents generating an image in the past given a future HR image, and  $t' < t$  is the task for generating a future image given a past HR image.
+
+fMoW Temporal Generation Many locations in fMoW (Christie et al., 2018) contain multiple images across time. For our experiments, if  $T < 4$ , we add copies of the latest image to pad the sequence s to 4 images. Given a sequence s of conditioning images, DiffusionSat can predict another image at any desired target time by appropriately adjusting the target metadata  $\mathbf{k}_s$  (section 3.2) Since prior works aren't designed to predict an image at any given target time, we consider tasks where the target image is chronologically prior to or later than the first image in s.
+
+Our experiments show that DiffusionSat outperforms STSR and MCVD (Voleti et al., 2022), as well as regular SD with our 3D ControlNet. Quantitative and qualitative results are in table 4 and fig. 5, respectively. These reveal DiffusionSat's improved ability over the baselines to capture the target date's season (eg: snow, terrain color, crop maturity) as well as development of roads and buildings. Other models, lacking the ability to reason about metadata covariates, often simply copy an input image in the conditioning sequence as their generated output. In A.3.1, we showcase DiffusionSat's novel ability to generate sequences of satellite images without prior conditioning images s.
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">t&#x27; &gt; t</td><td colspan="3">t&#x27; &lt; t</td></tr><tr><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td><td>SSIM↑</td><td>PSNR↑</td><td>LPIPS↓</td></tr><tr><td>STSR (EAD) (He et al., 2021)</td><td>0.3657</td><td>13.5191</td><td>0.4898</td><td>0.3654</td><td>13.7425</td><td>0.4940</td></tr><tr><td>MCVD (Voleti et al., 2022)</td><td>0.3110</td><td>9.6330</td><td>0.6058</td><td>0.2721</td><td>9.5559</td><td>0.6124</td></tr><tr><td>SD + 3D CN</td><td>0.2027</td><td>11.0536</td><td>0.5523</td><td>0.2218</td><td>11.3094</td><td>0.5342</td></tr><tr><td>DiffusionSat + CN</td><td>0.3297</td><td>13.6938</td><td>0.5062</td><td>0.2862</td><td>12.4990</td><td>0.5307</td></tr><tr><td>DiffusionSat + 3D CN</td><td>0.3983</td><td>13.7886</td><td>0.4304</td><td>0.4293</td><td>14.8699</td><td>0.3937</td></tr></table>
+
+Table 4: Sample quality quantitative results on fMoW-temporal validation data.  $t' > t$  represents generating an image in the past given a future image, and  $t' < t$  is the task for generating a future image given a past image.
+
+![](images/18a7b39c7dd547523f5ad1c527d0ccd33a324083e2d6953de28787207297fa44.jpg)  
+Figure 5: Generated samples from the fMoW-temporal validation set, for temporal prediction. The 4 columns in the center are ground-truth images from the temporal sequence. To the right, we see generated samples for the future-prediction task. The goal is to generate the image marked by the date in red, given the 3 other images (to its left) as conditioning signals. Similarly, for the past-prediction task on the left, the goal is to predict the image marked by the date in blue given the 3 images to its right. DiffusionSat leverages pretrained weights to capture seasonal changes and predict human development better than the baselines. Images are best viewed zoomed in.
+
+In-painting Rather than artificially corrupt input images, we use the xBD dataset (Gupta et al., 2019) which is a subset of the xView-2 (Lam et al., 2018) challenge to assess damage caused by natural disasters. Since each location carries a pre- and post-disaster satellite image, we consider the in-painting task of reconstructing damaged areas in the post-disaster image, or introducing destruction to the pre-disaster image, where  $T = 1$ . We demonstrate qualitative results in fig. 6. DiffusionSat's capability to reconstruct damaged roads and houses for a variety of disasters including floods, wind, fire, earthquakes etc will be important for disaster response teams to identify access routes and assess damage. We also show that DiffusionSat can add damage from different natural disasters, which can be useful for forecasting or preparing areas for evacuation.
+
+# 5 RELATED WORK
+
+Diffusion Models Diffusion models (Ho et al., 2020; Song et al., 2020b; Kingma et al., 2021) have recently dominated the field of generative modeling, including application areas such as speech (Kong et al., 2020; Popov et al., 2021), 3D geometry (Xu et al., 2022; Luo & Hu, 2021; Zhou et al., 2021), and graphics (Chan et al., 2023; Poole et al., 2022; Shue et al., 2023). Besides advancements in the theoretical foundation, large-scale variants built on latent space (Rombach et al., 2022; Sahara et al., 2022a; Ho et al., 2022) have arguably been the most influential. With these foundation models came a slew of novel applications such as subject customization (Ruiz et al., 2023; Liu et al., 2023; Kumari et al., 2023) and text-to-3D generation (Poole et al., 2022; Lin et al., 2023; Wang et al., 2023). Many works have also demonstrated these models' impressive adaptation capabilities through
+
+![](images/50af7b43ef78f1611b082e02c79edc97523d874e3419d6fd258dea2129fc3cad.jpg)  
+Figure 6: Inpainting results. The two columns marked "before" and "after" represent ground truth images. The "pred past" column is generated by conditioning on the "after" image, and the "pred future" column likewise by conditioning on the "before" image. DiffusionSat successfully reconstructs damaged roads and houses from floods, fires, and wind, even when large portions of the conditioning image are masked by clouds or damage.
+
+finetuning. For example, ControlNet (Zhang & Agrawala, 2023), T2IAAdapter (Mou et al., 2023), and InstructPix2Pix (Brooks et al., 2023), which add additional trainable parameters, have proven to be highly successful in adding control signals to the pre-trained diffusion networks.
+
+Generative Models for Remote Sensing Image super-resolution is well studied for natural image datasets (Dong et al., 2015; Ledig et al., 2017; Sahara et al., 2022b; Haris et al., 2018; Rombach et al., 2022). Generative Adversarial Networks (GANs) Goodfellow et al. (2014) such as SR-GAN Ledig et al. (2017) are among the most popular remote-sensing super-resolution methods (Wang et al., 2020; Ma et al., 2019; Gong et al., 2021; Cornebise et al., 2022; Bastani et al., 2023; Rabbi et al., 2020). Other methods have tailor-made convolutional architectures for Sentinel-2 image-input superresolution (Razzak et al., 2023; Tarasiewicz et al., 2023). More recently, Spatial-Temporal Super Resolution (STSR) He et al. (2021) uses a conditional-pixel synthesis approach to condition on a combination of high and low resolution images to generate a high-resolution image at an earlier or later date. In general, these models lack the flexibility and generality of latent-diffusion models across a variety of tasks and datasets, and can suffer from unstable training (Kodali et al., 2017). Our work aims to address these shortcomings by proposing a single approach based on pretrained LDMs that can flexibly translate to downstream generative tasks via our novel conditioning mechanism.
+
+# 6 CONCLUSION
+
+In this work, we provide DiffusionSat, the first generative foundation model for remote sensing data based on the latent-diffusion model architecture of StableDiffusion Rombach et al. (2022). Our approach consists of two components: (i) a single-image generation model that can generate high-resolution satellite data conditioned on numerical metadata and text captions (ii) A novel 3D control signal conditioning module that generalizes to inverse problems such as multi-spectral input super-resolution, temporal prediction, and in-painting.
+
+For future work, we would like to explore expanding DiffusionSat to even larger and more diverse satellite imagery datasets. Testing the feasibility of DiffusionSat on generating synthetic data (Le et al., 2023) might also augment existing discriminative methods to scale to larger datasets. Lastly, investigating faster sampling methods or more efficient architectures will enable easier deployment or use of DiffusionSat in resource-constrained settings.
+
+We hope that DiffusionSat spurs future investigation into solving inverse problems posed by remote-sensing data. Doing so would unlock societal benefits to important applications including object detection given super-resolved Sentinel-2 images (Shermeyer & Van Etten, 2019), crop-phenotyping (Zhang et al., 2020), ecological conservation efforts (Boyle et al., 2014; Johansen et al., 2007), natural disaster (eg: landslide) hazard assessment (Nichol et al., 2006), archaeological prospection (Beck et al., 2007), urban planning Li et al. (2019); Xiao et al. (2006); Piyoosh & Ghosh (2017), and precise agricultural applications (Gevaert et al., 2015).
+
+# 7 ACKNOWLEDGEMENTS
+
+This research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via 2021-2011000004, NSF(#1651565), ARO (W911NF-21-1-0125), ONR (N00014-23-1-2159), CZ Biohub, HAI. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes not-withstanding any copyright annotation therein.
+
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+
+# A APPENDIX
+
+# A.1 TRAINING DETAILS
+
+We list implementation details for our experiments in this section. All models are trained on half-precision and with gradient checkpointing, borrowing from the Diffusers (von Platen et al., 2022) library.
+
+Single-Image DiffusionSat We use 8 NVIDIA A100 GPUs. The text-to-image models are trained with a batch size of 128 for 100000 iterations, which we determined was sufficient for convergence. We choose a constant learning rate of 2e-6 with the AdamW optimizer. We train two variants- one for images of resolution 512x512 pixels, and one for 256x256 pixels.
+
+For sampling, we use the DDIM (Song et al., 2020a) sampler with 100 steps and a guidance scale of 1.0. We generate 10000 samples on the validation sets of fMoW-RGB.
+
+Super-resolution We use the 512 single-image DiffusionSat model as our prior. We train our ControlNet Zhang & Agrawala (2023) by upsampling the conditional multi-spectral image to  $256 \times 256$  pixels, which we found to work better than conditioning on  $64 \times 64$  conditioning images. We use 4 NVIDIA A100 GPUs, and train the model for 50000 iterations with a learning rate of 5e-5 using the AdamW optimizer. We drop Sentinel bands B1, B9, B10, which we find to not be useful, similar to Cong et al. (2022). We use the same sampling configuration as above.
+
+Texas Housing We use the 256 single-image DiffusionSat model as our prior. We train our 3D ControlNet on sequences of the HR image and the two LR Sentinel-2 images. We use 4 NVIDIA A100 GPUs, and train the model for 50000 iterations with a learning rate of 5e-5 using the AdamW optimizer. The sampling configuration is the same as above.
+
+fMoW Temporal We use the 256 single-image DiffusionSat model as our prior. We train our 3D ControlNet on sequences of at-most 3 conditioning images on the fMoW-temporal dataset. If the location has less than 3 images, we pick one of the conditioning images and copy it over until the sequence is padded to length. We avoid samples where there is only 1 image per location. We train using 4 NVIDIA A100 GPUs, for 40000 iterations with a learning rate of 4e-4 using the AdamW optimizer. The sampling configuration matches the ones above.
+
+# A.2 DATASETS
+
+# A.2.1 CAPTIONS AND METADATA
+
+The text captions are dependent on the metadata fields available for each dataset. For the captions below, fields denoted in angle brackets below are filled in using the metadata for each example. Some sections of each caption, denoted by square brackets below, are randomly and independently dropped out of caption instances at a  $10\%$  rate. We label datasets from the same satellite sources with the same image type (e.g., both Texas Housing and Satlas use NAIP images, so both are labelled as "satlas" images).
+
+<table><tr><td>Dataset</td><td>Caption</td></tr><tr><td>fMoW</td><td>&quot;a [fmow] satellite image [of a &lt;object&gt;] [in &lt;country&gt;] &quot;</td></tr><tr><td>SpaceNet</td><td>&quot;a [spacenet] satellite image [of &lt;object&gt;] [in &lt;city&gt;] &quot;</td></tr><tr><td>Satlas</td><td>&quot;a [satlas] satellite image [of &lt;object&gt;] &quot;</td></tr><tr><td>Texas Housing</td><td>&quot;a [satlas] satellite image [of houses] [built in &lt;year_built&gt;] [covering &lt;num_ acres&gt; acres]&quot;</td></tr><tr><td>xBD</td><td>&quot;a [fmow] satellite image [&lt;before/after&gt;] being affected by a &lt;disaster_type&gt; natural disaster&quot;</td></tr></table>
+
+Table 5: Captions created for each dataset type based on available label information.
+
+Besides the captions, we also incorporate numerical metadata from 7 fields. Each field was normalized based on high and low reference values:  $m_{norm} = m / (high - low) \times scale$ , where scale is a scaling
+
+<table><tr><td>field</td><td>description</td><td>min</td><td>max</td></tr><tr><td>lon</td><td>longitude, in degrees</td><td>-180</td><td>180</td></tr><tr><td>lat</td><td>longitude, in degrees</td><td>-90</td><td>90</td></tr><tr><td>gsd</td><td>ground sampling distance</td><td>0</td><td>10</td></tr><tr><td>cloud_cover</td><td>proportion of pixels with cloud cover</td><td>0</td><td>1</td></tr><tr><td>year</td><td>year of the satellite image</td><td>1980</td><td>2100</td></tr><tr><td>month</td><td>month of the year</td><td>0</td><td>12</td></tr><tr><td>day</td><td>day of the month</td><td>0</td><td>31</td></tr></table>
+
+<table><tr><td>Dataset</td><td>Image</td><td>Caption</td><td>Metadata</td></tr><tr><td>fmow</td><td></td><td>a fmow satellite image of a car dealership in United States of America</td><td>lon: -76.781
+lat: 17.98
+gsd: 0.941
+cloud_cover: 0
+year: 2010
+month: 10
+day: 6</td></tr><tr><td>satlas</td><td></td><td>a satlas satellite image of 26 ms buildings</td><td>lon: 78.995
+lat: 85.048
+gsd: 2
+cloud COVER: 0
+year: 2013
+month: 6
+day: 22</td></tr><tr><td>spacenet</td><td></td><td>a spacenet satellite image of 144 buildings covering an area of 9280.166 squared meters in Rio</td><td>lon: -43.636
+lat: -22.892
+gsd: 0.793
+cloud COVER: 0
+year: 1980
+month: 0
+day: 0</td></tr></table>
+
+Figure 8: Sample captions and pre-normalization metadata for the fMoW, Satlas, and SpaceNet datasets.
+
+factor of 1000, such that low maps to 0 and high maps to scale. The fields are summarized in Figure 7. We include examples of both the captions and numerical metadata in Figure 8.
+
+# A.3 TEMPORAL GENERATION
+
+In this section, we provide further results for the temporal generation task, demonstrating the powerful capabilities of DiffusionSat.
+
+# A.3.1 SEQUENCE GENERATION
+
+We first demonstrate how we can generate temporal sequences of satellite images unconditionally i.e. without any prior conditioning image, unlike in section 4.2. To do so, we first generate a satellite image using single-image DiffusionSat given a caption and desired metadata for our image. We then apply our novel 3D-conditioning ControlNet, already trained for temporal generation, on the first image to generate the next image in the sequence, given some desired metadata (eg: how many years/months/days into the future or past). We now re-apply the 3D-conditioning ControlNet on the first 2 generated images to get the third image of the sequence. Repeating this procedure, we are
+
+able to auto-regressively sample sequences of satellite images given desired metadata properties. Generated samples using this procedure are shown in fig. 9.
+
+![](images/86ba83610cb08fc55bd76391413b158eeeb992c821282f8b7eb8ea772535e4bd.jpg)  
+Figure 9: Auto-regressively generated sequences of satellite images given a caption (for the sequence) and desired metadata (per image). The image sampled from single-image DiffusionSat is outlined in red. On the left, we generate sequences backwards in time (i.e.: into the past) given the first generated image. For example, for the park in Japan (2nd column from the left), the metadata, from the bottom to the top image, is: (1.07, 2014, 5, 25), (1.62, 2013, 6, 17), (1.17, 2011, 3, 17), (1.17, 2010, 11, 8). The metadata is in order (GSD, year, month, day). We omit listing the latitude and longitude, since that remains the same for the sequence, but it is inputted as metadata, as described in fig. 1. On the right, we generate images forwards in time (i.e. into the future) given the first generated image outlined in red. For example, for the crop field in Chile, the metadata, from the bottom to the top image, is: (1.03, 2016, 2, 23), (1.03, 2015, 9, 17), (0.97, 2014, 10, 12), (1.17, 2014, 8, 13). As we can see, our model generates realistic sequences that reflect trends in detail and development both forwards and backwards in time.
+
+Our results demonstrate a novel way of generating arbitrarily long sequences of satellite images- our conditioning mechanism can flexibly handle both conditional and unconditional generation. The generated samples reflect season and trends in development (eg: past images have fewer structures, future images usually have more detail).
+
+# A.4 GEOGRAPHICAL BIAS
+
+Concerns about bias for the outputs of machine learning models are natural given the large, potentially biased datasets they are trained on (Huang et al., 2021). We perform an evaluation of the generation quality of single-image DiffusionSat across latitude and longitude around the globe in fig. 10 and fig. 11
+
+Our results show no particular favoritism for location, even though one would expect better generation quality for regions in North America and Europe. We would still like to point out a few caveats:
+
+(i) FID or LIPPs scores may not be the most informative metric towards estimating bias in sample quality. We use it as a measure of generation quality given a lack of better alternatives for the novel problem of estimating geographical bias in generative remote sensing models.  
+(ii) The FID scores are dependent on sample size, and so while the scores might be evenly distributed, it still remains the case that there are far more dataset samples from developed regions of the world, and a dearth of images for large swaths (eg: across Africa). Even so, for a severely biased model we would expect poorer generation quality for data-poor regions of the world.  
+(iii) We estimate only one angle of bias. Bias may still exist along different axes, such as generating types of buildings, roads, trees, crops, and understanding the effects of season. We leave this investigation to future work.
+
+![](images/8615caea9aa38221dbb7a99af7779d197dbac35902637a810d1aa02e80be16c7.jpg)  
+Figure 10: FID scores of single-image DiffusionSat prompted on 10k samples of the fMoW-RGB validation set for coordinates around the world.
+
+![](images/f6029cec73d60ec6954bd8ba2d6756e81a9bee77c37a2f1b9c4fdc13460b6168.jpg)  
+Figure 11: LPIPs scores of super-resolution DiffusionSat prompted on 10k samples of the fMoW-Sentinel-fMoW-RGB validation set for coordinates around the world.

2024/DiffusionSat_ A Generative Foundation Model for Satellite Imagery/images.zip

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