SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $\sigma_0(\theta)\in \mathcal{P}(\mathbb{S}^{d - 1})$  is the proposal distribution, and:
$$
w _ {\mu , \nu , \sigma_ {0}, f, p} (\theta) = \frac {f \left(\mathbf {W} _ {p} ^ {p} \left(\theta \sharp \mu , \theta \sharp \nu\right)\right)}{\sigma_ {0} (\theta)}
$$
is the importance weighted function. The detailed derivation is given in Appendix B.2. Let  $\theta_{1},\ldots ,\theta_{L}$  be i.i.d samples from  $\sigma_0(\theta)$ , the importance sampling estimator of the EBSW (IS-EBSW) is:
$$
\widehat {\operatorname {I S - E B S W}} _ {p} (\mu , \nu ; f, L) = \left(\sum_ {l = 1} ^ {L} \left[ \mathrm {W} _ {p} ^ {p} \left(\theta_ {l} \sharp \mu , \theta_ {l} \sharp \nu\right) \hat {w} _ {\mu , \nu , \sigma_ {0}, f, p} \left(\theta_ {l}\right) \right]\right) ^ {\frac {1}{p}}, \tag {8}
$$
where  $\hat{w}_{\mu,\nu,\sigma_0,f,p}(\theta_l) = \frac{w_{\mu,\nu,\sigma_0,f,p}(\theta_l)}{\sum_{l'=1}^{L} w_{\mu,\nu,\sigma_0,f,p}(\theta_{l'})}$  is the normalized importance weights. When  $\sigma_0(\theta) = \mathcal{U}(\mathbb{S}^{d-1}) = \frac{\Gamma(d/2)}{2\pi^{d/2}}$  (a constant of  $\theta$ ), we can replace  $w_{\mu,\nu,\sigma_0}(\theta_l)$  with  $f(W_p^p(\theta_l \sharp \mu, \theta_l \sharp \nu))$ . When we choose the energy function  $f(x) = e^x$ , computing the normalized importance weights
$$
\hat {w} _ {\mu , \nu , \sigma_ {0}, f, p} (\theta_ {l}) = \frac {w _ {\mu , \nu , \sigma_ {0} , f , p} (\theta_ {l})}{\sum_ {l ^ {\prime} = 1} ^ {L} w _ {\mu , \nu , \sigma_ {0} , f} (\theta_ {l ^ {\prime}})}
$$
is equivalent to computing the Softmax function.
Computational algorithms and complexities. The computational algorithm of IS-EBSW can be derived from the algorithm of the SW distance by adding only one to two lines of code for computing the importance weights. For a better comparison, we give algorithms for computing the SW distance and the EBSW distance in Algorithm 1 and Algorithm 4 in Appendix B.1 and Appendix B.2 respectively. When  $\mu$  and  $\nu$  are two discrete measures that have at most  $n$  supports, the time complexity and the space complexity of the IS-EBSW distance are  $\mathcal{O}(Ln\log n + Lnd)$  and  $\mathcal{O}(L(n + d))$  which are the same as the SW.
Unbiasedness. The IS approximation is asymptotically unbiased for  $\mathrm{EBSW}_p^p (\mu ,\nu ;f)$ . However, having a biased estimation is not severe since the unbiasedness cannot be preserved after taking the  $p$ -rooth  $(p > 1)$  like in the case of the SW distance. Therefore, an unbiased estimation of  $\mathrm{EBSW}_p^p (\mu ,\nu ;f)$  is not very vital. Moreover, we can show that  $\widehat{\mathrm{EBSW}}_p^p (\mu ,\nu ;f,L)$  is an unbiased estimation of the power  $p$  of a valid distance. We refer the reader to Appendix B.2 for the detailed definition and properties of the distance. From this insight, it is safe to use the IS-EBSW in practice.
Gradient Estimation. In statistical inference, we might want to estimate the gradient  $\nabla_{\phi}\mathrm{EBSW}_p(\mu_\phi ,\nu ;f)$  for doing minimum distance estimator [45]. Therefore, we derive the gradient estimator of the EBSW with importance sampling in Appendix B.2.
3.3.2 Sampling Importance Resampling and Markov Chain Monte Carlo
The second approach is to somehow sample from the slicing distribution  $\sigma_{\mu,\nu}(\theta;f,p)$ . For example, when we have  $\theta_1,\ldots,\theta_L$  are approximately distributed following  $\sigma_{\mu,\nu}(\theta;f,p)$ , we can take  $\left(\frac{1}{L}\sum_{l=1}^{L}\mathrm{W}_p^p(\theta_l\sharp \mu,\theta_l\sharp \nu)\right)^{\frac{1}{p}}$  as the approximated value of the EBSW. Here, we consider two famous approaches in statistics: Sampling Importance Resampling [14] (SIR) and Markov Chain Monte Carlo (MCMC). For MCMC, we utilize two variants of the Metropolis-Hasting algorithm: independent Metropolis-Hasting (IMH) and random walk Metropolis-Hasting (RMH).
Sampling Importance Resampling. Similar to importance sampling in Section 3.3.1, the SIR uses a proposal distribution  $\sigma_0(\theta)$  to obtain  $L$  samples  $\theta_1', \ldots, \theta_L'$  and the corresponding normalized
importance weights:
$$
\hat {w} _ {\mu , \nu , \sigma_ {0}, f, p} (\theta_ {l} ^ {\prime}) = \frac {w _ {\mu , \nu , \sigma_ {0} , f , p} (\theta_ {l} ^ {\prime})}{\sum_ {i = 1} ^ {L} w _ {\mu , \nu , \sigma_ {0} , f , p} (\theta_ {i} ^ {\prime})}.
$$
After that, the SIR creates the resampling distribution which is a Categorical distribution  $\hat{q}(\theta) = \sum_{l=1}^{L} \hat{w}_{\mu,\nu,\sigma_0,f,p}(\theta_l) \delta_{\theta'}$ . Finally, the SIR draws  $L$  samples  $\theta_1, \ldots, \theta_L \stackrel{i.i.d.}{\sim} \hat{q}(\theta)$ . We denote the SIR estimation of the EBSW distance as SIR-EBSW.
Markov Chain Monte Carlo. MCMC creates a Markov chain that has the stationary distribution as the target distribution. The most famous way to construct such a Markov chain is through the Metropolis-Hastings algorithm. Let the starting sample follow a prior distribution  $\theta_{1} \sim \sigma_{0}(\theta) \in \mathcal{P}(\mathbb{S}^{d-1})$ , a transition distribution  $\sigma_{t}(\theta_{t} | \theta_{t-1}) \in \mathcal{P}(\mathbb{S}^{d-1})$  for any timestep  $t > 1$  is used to sample a candidate  $\theta_{t}'$ . After that, the new sample  $\theta_{t}$  is set to  $\theta_{t}'$  with the probability  $\alpha$  and is set to  $\theta_{t-1}$  with the probability  $1 - \alpha$  with
$$
\alpha = \min \left(1, \frac {\sigma_ {\mu , \nu} (\theta_ {t} ^ {\prime} ; f)}{\sigma_ {\mu , \nu} (\theta_ {t - 1} ; f)} \frac {\sigma_ {t} (\theta_ {t - 1} | \theta_ {t} ^ {\prime})}{\sigma_ {t} (\theta_ {t} ^ {\prime} | \theta_ {t - 1})}\right) = \min \left(1, \frac {f (\mathbf {W} _ {p} ^ {p} (\theta_ {t} ^ {\prime} \sharp \mu , \theta_ {t} ^ {\prime} \sharp \nu)))}{f (\mathbf {W} _ {p} ^ {p} (\theta_ {t - 1} \sharp \mu , \theta_ {t - 1} \sharp \nu)))} \frac {\sigma_ {t} (\theta_ {t - 1} | \theta_ {t} ^ {\prime})}{\sigma_ {t} (\theta_ {t} ^ {\prime} | \theta_ {t - 1})}\right).
$$
In theory,  $T$  should be large enough to help the Markov chain to mix to the stationary distribution and the first  $M < T$  samples are often dropped as burn-in samples. However, to keep the computational complexity the same as the previous computational methods, we set  $T = L$  and  $M = 0$ . The first choice of transition distribution is  $\sigma_t(\theta_t|\theta_{t - 1}) = \mathcal{U}(\mathbb{S}^{d - 1})$  which leads to independent Metropolis-Hasting (IMH). The second choice is  $\sigma_t(\theta_t|\theta_{t - 1}) = \mathrm{vMF}(\theta_t|\theta_{t - 1},\kappa)$  (the von Mises-Fisher distribution [17] with location  $\theta_{t - 1}$ ) which leads to random walk Metropolis-Hasting (RMH). Since both above transition distributions are symmetric  $\sigma_t(\theta_t|\theta_{t - 1}) = \sigma_t(\theta_{t - 1}|\theta_t)$ , the acceptance probability turns into:
$$
\alpha = \min  \left(1, \frac {f \left(\mathbf {W} _ {p} ^ {p} \left(\theta_ {t} ^ {\prime} \sharp \mu , \theta_ {t} ^ {\prime} \sharp \nu\right)\right)}{f \left(\mathbf {W} _ {p} ^ {p} \left(\theta_ {t - 1} \sharp \mu , \theta_ {t - 1} \sharp \nu\right)\right)}\right)
$$
which means that the acceptance probability equals 1 and  $\theta_t^\prime$  is always accepted as  $\theta_{t}$  if it can increase the energy function. We refer to the IMH estimation and the RMH estimation of the EBSW distance as IMH-EBSW and RMH-EBSW in turn.
Computational algorithms and complexities. We refer the reader to Algorithm 5, Algorithm 6, and Algorithm 7 in Appendix B.3 for the detailed algorithms of the SIR-EBSW, the IMH-EBSW, and the RMH-EBSW. Without counting the complexity of the sampling algorithm, the time complexity and the space complexity of both the SIR and the MCMC estimation of EBSW are  $\mathcal{O}(Ln\log n + Lnd)$  and  $\mathcal{O}(L(n + d))$  which are the same as the IS-EBSW and the SW distance. However, the practical computational time and memory of the SIR and the MCMC estimation depend on the efficiency of implementing the sampling algorithm e.g., resampling and acceptance-rejection.
Unbiasedness. The SIR and the MCMC sampling do not give an unbiased estimation for  $\mathrm{EBSW}_p^p (\mu ,\nu ;f)$ . However, they are also unbiased estimations of the power  $p$  of a valid distance. We refer to Appendix B.3 for detailed definitions and properties of the distance. Therefore, it is safe to use the approximation from the SIR and the MCMC.
Gradient Estimation: In the IS estimation, the expectation is with respect to the proposal distribution  $\sigma_0(\theta)$  that does not depend on two input measures  $\mu_{\phi}$ . In the SIR estimation and the MCMC estimation, the expectation is with respect to the slicing distribution  $\sigma_{\mu_{\phi},\nu}(\theta ,f)$  that depends on  $\mu_{\phi}$ . Therefore, the log-derivative trick (Reinforce) should be used to derive the gradient estimation. We give the detailed derivation in Appendix B.3. However, the log-derivative trick is often unstable in practice. A simpler solution is to create the slicing distribution from an independent copy of  $\mu_{\phi}$ . In particular, we denote  $\mu_{\phi^{\prime}}$  is the independent copy of  $\mu_{\phi}$  with  $\phi^\prime$  equals  $\phi$  in terms of value. Therefore, we can obtain the slicing distribution  $\sigma_{\mu_{\phi^{\prime}},\nu}(\theta ;f)$  that does not depend on  $\mu_{\phi}$ . This approach still gives the same value of distance, we refer to Appendix B.3 for a more careful discussion.
In this section, we first visualize the shape of the energy-based slicing distribution in a simple case. After that, we focus on showing the favorable performance of the EBSW compared to the other sliced Wasserstein variants in point-cloud gradient flows, color transfer, and deep point-cloud reconstruction.
In experiments, we denote EBSW-e for the exponential energy function i.e.,  $f(x) = e^{x}$ , and EBSW-1 for the identity energy function i.e.,  $f(x) = x$ . We use  $p = 2$  for all sliced Wasserstein variants.
4.1 Visualization of energy-based slicing distribution
We visualize the shape of the energy-based slicing distribution in two dimensions in Figure 1. In particular, we consider comparing two empirical distributions in the left-most figure (taken from [11]). We utilize the SIR, the IMH, and the RMH to obtain  $10^{4}$  Monte Carlo samples from the energy-based slicing distribution. For the IMH and the RHM, we burn in the  $10^{4}$  samples. After that, we use the von Mises kernel density estimation to obtain the density function. We also present the ground-truth density of the energy-based slicing distribution by uniform discretizing the unit-sphere. Moreover, we also show the optimal vMF distribution from the DSW (tuning  $\kappa \in \{1,5,10,50,100\}$ ) and the "max" projecting direction from the Max-SW ( $T = 100$ , step size 0.1). The middle figure is according to the energy function  $f_{1}(x) = x$  and the right-most figure is according to the energy function  $f_{e}(x) = e^{x}$ . From the figures, we observe that all sampling methods can approximate well the true slicing distribution. In contrast, the vMF distribution from v-DSW is misspecified to approximate the energy distribution, and the "max" projecting direction from the Max-SW can capture only one mode. We also observe that the exponential energy function makes the density more concentrated around the modes than the identity energy function (polynomial of degree 1).
4.2 Point-Cloud Gradient Flows
We model a distribution  $\mu(t)$  flowing with time  $t$  along the gradient flow of a loss functional  $\mu(t) \to \mathcal{D}(\mu(t), \nu)$  that drives it towards a target distribution  $\nu$  [40] where  $\mathcal{D}$  is our sliced Wasserstein variants. We consider discrete flows, namely. we set  $\nu = \frac{1}{n} \sum_{i=1}^{n} \delta_{Y_i}$  as a fixed empirical target distribution and the model distribution  $\mu(t) = \frac{1}{n} \sum_{i=1}^{n} \delta_{X_i(t)}$ . Here, the model distribution is parameterized by a time-varying point cloud  $X(t) = (X_i(t))_{i=1}^n \in (\mathbb{R}^d)^n$ . Starting from an initial condition at time  $t = 0$ , we integrate the ordinary differential equation  $\dot{X}(t) = -n \nabla_{X(t)} \left[ \mathcal{D} \left( \frac{1}{n} \sum_{i=1}^{n} \delta_{X_i(t)}, \nu \right) \right]$  for each iteration. We choose  $\mu(0)$  and  $\nu$  are two point-cloud shapes in ShapeNet Core-55 dataset [4]. After that, we solve the flows by using the Euler scheme with 500 iterations and step size 0.0001.
Quantitative Results. We show the Wasserstein-2 distance between  $\mu(t)$  and  $\nu$  ( $t \in \{0, 100, 200, 300, 400, 500\}$ ) and the computational time of the SW variants in Table 1. Here, we set  $L = 100$  for SW, and EBSW variants. For the Max-SW we set  $T = 100$ , and report the best result for the step size for finding the max projecting direction in  $\{0.001, 0.01, 0.1\}$ . For the v-DSW, we report the best result for  $(L, T) \in \{(10, 10), (50, 2), (2, 50)\}$ ,  $\kappa \in \{1, 10, 50\}$ , and the learning rate for finding the location in  $\{0.001, 0.01, 0.1\}$ . We observe that the IS-EBSW-e helps to drive the flow to converge faster than baselines. More importantly, the computational time of the IS-EBSW-e is approximately the same as the SW and is faster than both the Max-SW and the v-DSW. We report more detailed experiments with other EBSW variants and different settings of hyperparameters  $(L, T)$  in Table 3 in Appendix C.1. From the additional experiments, we see that the EBSW-e variants give lower Wasserstein-2 distances than the baseline with the same scaling of complexity (same  $L / T$ ). Despite having comparable performance, the SIR-EBSW-e, the IMH-EBSW-e, and the RMH-EBSW-e
$(\kappa = 10)$  are slower than the IS-EBSW-e variant. Also, we see that the EBSW-e variants are better than the EBSW-1 variants. We refer to Table 4 for comparing gradient estimators of the EBSW.
Qualitative Results. We show the point-cloud flows of the SW, the Max-SW, the v-DSW, and the IS-EBSW-e, in Figure 2. The flows from the SIR-EBSW-e, the IMG-EBSW-e, and the RMH-EBSW-e are added in Figure 4 in the Appendix C.1. From the figure, the transitions of the flows from the EBSW-e variants are smoother to the target than other baselines.
4.3 Color Transfer
We build a gradient flow that starts from the empirical distribution over the normalized color palette (RGB) of the source image to the empirical distribution over the normalized color palette (RGB) of the target image. Since the value of the color palette is in the set  $\{0, \dots, 255\}^3$ , we must do an additional rounding step at the final step of the Euler scheme with 2000 steps and step size 0.0001.
Results. We use the same setting for the SW variants as in the previous section. We show both transferred images, corresponding computational times, and Wasserstein-2 distances in Figure 3. We observe the same phenomenon as in the previous section, namely, the IS-EBSW-e variants perform the best in terms of changing the color of the source image to the target in the Wasserstein-2 metric while the computational time is only slightly higher. Moreover, the transferred image from the IS-EBSW-e is visually more similar to the target image in color than other baselines, namely, it has a less orange color. We refer to Figure 5 in Appendix C.2 for additional experiments including the results for the SIR-EBSW-e, the IMH-EBSW-e, the RMH-EBSW-e, the results for the identity energy function, the results for changing hyperparameters  $(L,T)$ , and the results for comparing gradient estimators. Overall, we observe similar phenomenons as in the previous gradient flow section.
4.4 Deep Point-Cloud Reconstruction
We follow [33] to train point-cloud autoencoders with sliced Wasserstein distances on the ShapeNet Core-55 dataset [4]. In short, we aim to estimate an autoencoder that contains an encoder  $f_{\phi}$  that maps a point cloud  $X \in \mathbb{R}^{nd}$  to a latent code  $z \in \mathbb{R}^h$ , and a decoder  $g_{\psi}$  that maps the latent
code  $z$  to a reconstructed point cloud  $\tilde{X} \in \mathbb{R}^{nd}$ . We want to have the pair  $f_{\phi}$  and  $g_{\psi}$  such that  $\tilde{X} = g_{\psi}(f_{\phi}(X)) \approx X$  for all  $X \sim p(X)$  which is our data distribution. To do that, we solve the following optimization problem:  $\min_{\phi, \gamma} \mathbb{E}_{X \sim \mu(X)}[S(P_X, P_{g_{\gamma}(f_{\phi}(X)))]$ , where  $S$  is a sliced Wasserstein variant, and  $P_X$  denotes the empirical distribution over the point cloud  $X$ . The backbone for the autoencoder is a variant of Point-Net [37] with an embedding size of 256. We train the autoencoder for 200 epochs using an SGD optimizer with a learning rate of 1e-3, a batch size of 128, a momentum of 0.9, and a weight decay of 5e-4. We give more detail in Appendix C.3
Quantitative Results. We evaluate the trained autoencoders on a different dataset: ModelNet40 dataset [46] using two distances: sliced Wasserstein distance ( $L = 1000$ ), and the Wasserstein distance. We follow the same hyper-parameters settings as the previous sections and show the reconstruction errors at epochs 20, 100, and 200 from the SW, the Max-SW, the DSW, and the IS-EBSW-e in Table 2. The reconstruction errors are the average of corresponding distances on all point-clouds. From the table, we observe that the IS-EBSW-e can help to train the autoencoder faster in terms of the Wasserstein distance and the sliced Wasserstein distance. We refer to Table 5 in Appendix C.3 for similar ablation studies as in the previous sections including the results for the SIR-EBSW-e, the IMH-EBSW-e, the RMH-EBSW-e, the results for the identity energy function, the results for changing hyperparameters ( $L, T$ ), and the results for comparing gradient estimators.
Qualitative Results. We show some ground-truth point-clouds ModelNet40 and their corresponding reconstructed point-clouds from different models ( $L = 100$ ) at epochs 200 and 20 in Figure 6-7 respectively. Overall, the qualitative results are visually consistent with the quantitative results.
Limitations. The first limitation of EBSW is that its MCMC variants are not directly parallelizable, resulting in slow computation. Additionally, a universal effective choice of the energy-based function is an open question. In the paper, we use simple and computationally effective energy-based functions, such as the exponential function and the polynomial function. Finally, proving the triangle inequality of EBSW is challenging due to the expressiveness of the energy-based slicing distribution.
Conclusion. We have presented a new variant of sliced Wasserstein distance, named energy-based sliced Wasserstein (EBSW) distance. The key ingredient of the EBSW is the energy-based slicing distribution which has a density at a projecting direction proportional to an increasing function of the one-dimensional Wasserstein value of that direction. We provide theoretical properties of the EBSW including the topological properties, and statistical properties. Moreover, we propose to compute the EBSW with three different techniques including importance sampling, sampling importance resampling, and Markov Chain Monte Carlo. Also, we discuss the computational properties of different techniques. Finally, we demonstrate the favorable performance of the EBSW compared to existing projecting directions selection sliced Wasserstein variants by conducting experiments on point-cloud gradient flows, color transfer, and deep point-cloud reconstruction.
Source SW 85.04(s),  $W_{2} = 412.76$  Max-SW 126.06(s),  $W_{2} = 460.82$  v-DSW 92.41(s),  $W_{2} = 340.92$  IS-EBSW-e 86.7(s),  $W_{2} = 264.95$  Target
We would like to thank Peter Mueller for his insightful discussion during the course of this project. NH acknowledges support from the NSF IFML 2019844 and the NSF AI Institute for Foundations of Machine Learning.
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Supplement to "Energy-Based Sliced Wasserstein Distance"
We first provide skipped proofs in the main text in Appendix A. We then provide additional materials including additional background, detailed algorithms, and discussion in Appendix B. Additional experimental results in point-cloud gradient flows, color transfer, and deep point-cloud reconstruction in Appendix C. Finally, we report the computational infrastructure in Appendix D.
Non-negativity and Symmetry. the non-negativity and symmetry properties of the EBSW follow directly by the non-negativity and symmetry of the Wasserstein distance since it is an expectation of the one-dimensional Wasserstein distance.
Identity. We need to show that  $\mathrm{EBSW}_p(\mu, \nu; f) = 0$  if and only if  $\mu = \nu$ . First, from the definition of EBSW, we obtain directly  $\mu = \nu$  implies  $\mathrm{EBSW}_p(\mu, \nu; f) = 0$ . For the reverse direction, we use the same proof technique in [3]. If  $\mathrm{EBSW}_p(\mu, \nu; f) = 0$ , we have  $\int_{\mathbb{S}^{d-1}} \mathrm{W}_p(\theta \sharp \mu, \theta \sharp \nu) \, \mathrm{d}\sigma_{\mu, \nu}(\theta; f, p) = 0$ . Hence, we have  $W_p(\theta \sharp \mu, \theta \sharp \nu) = 0$  for  $\sigma_{\mu, \nu}(\theta; f, p)$ -almost surely  $\theta \in \mathbb{S}^{d-1}$ . Since  $\sigma_{\mu, \nu}(\theta; f, p)$  is continuous, we have  $W_p(\theta \sharp \mu, \theta \sharp \nu) = 0$  for all  $\theta \in \mathbb{S}^{d-1}$ . From the identity property of the Wasserstein distance, we obtain  $\theta \sharp \mu = \theta \sharp \nu$  for  $\sigma_{\mu, \nu}(\theta; f, p)$ -a.e  $\theta \in \mathbb{S}^{d-1}$ . Therefore, for any  $t \in \mathbb{R}$  and  $\theta \in \mathbb{S}^{d-1}$ , we have:
$$
\begin{array}{l} \mathcal {F} [ \mu ] (t \theta) = \int_ {\mathbb {R} ^ {d}} e ^ {- i t \langle \theta , x \rangle} d \mu (x) = \int_ {\mathbb {R}} e ^ {- i t z} d \theta \sharp \mu (z) = \mathcal {F} [ \theta \sharp \mu ] (t) \\ = \mathcal {F} [ \theta \sharp \nu ] (t) = \int_ {\mathbb {R}} e ^ {- i t z} d \theta \sharp \nu (z) = \int_ {\mathbb {R} ^ {d}} e ^ {- i t \langle \theta , x \rangle} d \nu (x) = \mathcal {F} [ \nu ] (t \theta), \\ \end{array}
$$