SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
P ^ {\theta} \left(i _ {l} (t - \epsilon) \mid i (t)\right) = \left\{ \begin{array}{l l} \epsilon R _ {i _ {1} (t), \dots , i _ {l} (t - \epsilon), \dots , i _ {k} (t)} ^ {i (t)}, & i _ {l} (t - \epsilon) \neq i _ {l} (t), \\ 1 + \epsilon R _ {i (t)} ^ {i (t)} (t), & i _ {l} (t - \epsilon) = i _ {l} (t). \end{array} \right. \tag {31}
$$
When  $\epsilon$  is small, the reverse conditional distribution has the factorized probability:
$$
P ^ {\theta} (i (t - \epsilon) | i (t)) = \Pi_ {l = 1} ^ {K} P ^ {\theta} (i _ {l} (t - \epsilon) | i (t)) \tag {32}
$$
In this way, it becomes possible to generate samples by sequentially sampling from the reverse conditional distribution 32.
Transition to higher probability states The reverse process of the continuous probability flow, as described in Equation (20), causes particles to move towards areas with higher logarithmic probability densities. As the logarithm function is monotonically increasing, this reverse flow pushes particles to higher probability density states. This phenomenon is also observed in the discrete probability flow. By examining the reverse generator, as shown in Equation (30), it can be determined that the transition rate  $R_{j}^{i}(t) > 0$  only when the destination state  $j$  has a higher probability than the source state  $i$ . This implies that transitions only occur in higher probability states. In contrast, the original continuous reverse process (2) and the discrete reverse process from (10) allow any transitions.
Reduction of Standard Deviation We measure the certainty of the sampling method by the expectation of the Conditional Standard Deviation (CSD):
$$
C S D _ {s, t} (X) = \mathbb {E} _ {X _ {t}} \left[ \operatorname {S t d} \left(X _ {s} \mid X _ {t}\right) \right], \tag {33}
$$
where  $\mathrm{Std}(X_s|X_t) = \mathrm{Var}^{\frac{1}{2}}(X_s|X_t) = \mathbb{E}_{X_s}^{\frac{1}{2}}[X_s - \mathbb{E}_{X_s}[X_s|X_t]|X_t]$ .  $CSD_{s,t}(X)$  is 0 when the process is deterministic, such as the continuous probability flow. In the discrete situation, there does not exist any deterministic map. However, our discrete probability flow significantly reduces  $CSD_{s,t}(X)$ . Table 2 presents numerical evidence of this phenomenon. Therefore, we posit that the discrete probability flow enhances the certainty of the sampling outcomes.
The concept of probability flow was initially introduced in [46] as a deterministic alternative to the Itô diffusion. In the work [44], they presented the Denoising Diffusion Implicit Model (DDIM) and demonstrated its equivalence to the probability flow. Subsequently, [29] investigated the relationship between the probability flow and optimal transport. They hypothesized that the probability flow could be considered a Monge optimal transition map and provided a proof for a specific case. Additionally, they conducted numerical experiments that supported their conjecture, showing negligible errors. However, [30] has discovered an initial distribution that renders probability flow not optimal.
The discrete diffusion models were first introduced by [43], who considered a binary model. Following the success of continuous diffusion models, discrete models have garnered more attention. The bulk of research on discrete models has focused primarily on the design of the forward process [24, 23, 4, 5, 26, 16, 9]. Continuous time discrete state models were introduced by [6] and subsequently developed by [47].
We conduct numerical experiments using our novel sampling method by Discrete Probability Flow (DPF) on synthetic data. The primary goal is to demonstrate that our method can generate samples of comparable quality with higher certainty.
Experiments are conducted on synthetic data using the same setup as SDDM [47], with the exception that we replaced the generator  $Q$  with Equation (26). In addition to the binary situation ( $S = 2$ ) studied in [47], we also perform experiments on synthetic data with the state size  $S$  set to 5 and 10. To evaluate the quality of the generated samples, we generated 40,000/4,000 samples for binary data / other type of data using SDDM and DPF, and measured the Maximum Mean Discrepancy (MMD) with the Laplace kernel [15]. The results are shown in Table 1. It can be seen that the MMD value obtained using DPF is slightly higher than that of SDDM, which may be attributed to the structure of the reverse generator 10. Specifically, DPF approximates an additional term,  $Q_{t}(y,x)$ , with the neural network, which potentially introduces additional errors to the sampling process, leading to a higher MMD value compared to SDDM. However, such difference is minimal and does not significantly impact the quality of the generated samples. As evident from the visualization of the
distributions obtained from SDDM and DPF in Figure 1, it is clear that DPF can generate samples that are comparable to those generated by SDDM.
In addition, we also compare the sampling certainty of DPF and SDDM by computing  $CSD_{s,t}$  using a Monte-Carlo based method. Specifically, we set  $s = 0$  and  $t = T$ , and sample  $4,000x_{t}s$  with  $10x_{s}s$  for each  $x_{t}$ . We then estimate  $\mathbb{E}(x_s|x_t)$  and  $\mathrm{Std}(x_s|x_t)$  using the sample mean and sample standard deviation, respectively. The results of certainty are presented in Table 2. Our findings indicate that DPF significantly reduces the  $CSD$ , which suggests a higher certainty. Additionally, we visualize the results of 4,000 generated samples (in blue) from a single initial point (in red) in the binary case in Figure 2. It is apparent that the sampling of SDDM exhibits high uncertainty, as it can sample the entire pattern from a single initial point. In contrast, our method reduces such uncertainty and is only able to sample a limited number of states.
To provide a more intuitive representation of the generated samples originating from various initial points, we select  $20 \times 20$  initial points arranged in the grid, and distinguish them using different colors. Subsequently, we visualize the results by sampling 10 outcomes from each initial point, as shown in Figure 3. We observe that the visualization of SDDM samples appears disorganized, indicating significant uncertainty. In contrast, the visualization of DPF samples exhibits clear regularity, manifesting in two aspects: (1) the generated samples from the same initial point using DPF are clustered by color, demonstrating the better sampling certainty of our DPF. (2) Both of the generated samples and initial points are colored similarly at each position. For example, in the lower
right area, a majority of the generated samples are colored purple, which corresponds to the color assigned to the initial points  $x_{T}$  in that area. This observation demonstrates that most of the sampling results obtained through DPF are closer to their respective initial points, aligning with our design intention of optimal transport. It is worth noting that similar phenomena are observed across different state sizes, and we have provided these results in the Appendix.
Finally, we extended our DPF to the CIFAR-10 dataset, and compare it with the  $\tau$  LDR-0 method proposed in [6]. The visualization results are shown in Figure 4. It can be seen that our method greatly reduces the uncertainty of generating images by sampling from the same initial  $x_{T}$ . Detailed experimental settings and more experimental results are presented in the Appendix.
In this study, we introduce a discrete counterpart of the probability flow and established its connections with the continuous formulations. We began by demonstrating that the continuous probability flow corresponds to a Monge optimal transport map. Subsequently, we proposed a method to modify a discrete diffusion model to achieve a Kantorovich plan, which naturally defines the discrete probability flow. We also discovered shared properties between continuous and discrete probability flows. Finally, we propose a novel sampling method that significantly reduces sampling uncertainty. However, there are still remaining aspects to be explored in the context of the discrete probability flow. For instance, to obtain more general conclusions under a general initial condition, the semi-group method [54] could be employed. Additionally, while we have proven the existence of a Kantorovich plan in a small time interval, it is possible to extend this to a global solution. Moreover, the definition of the probability flow has been limited to a specific type of discrete diffusion model, which also could be extended to a broader range of models. These topics remain open for future studies.
We would like to thank all the reviewers for their constructive comments. Our work was supported in National Natural Science Foundation of China (NSFC) under Grant No.U22A2095 and No.62072482.