Sliced-Wasserstein Distance
The sliced Wasserstein distance \({\displaystyle SW_{p}}\) is an alternative distance metric between probability measures that integrates Wasserstein distances over 1-dimensional projections of measures in \(\mathbb{R}^d\). It shares many properties with the Wasserstein distance while being computationally simpler. For further reading see (Santambrogio 2015; Peyré and Cuturi 2020)
Motivation
One situation in which the Wasserstein distance is easier to compute is the 1D case (Peyré and Cuturi 2020). In particular, if the the measures are of the form \(\alpha ={\tfrac {1}{n}}\textstyle \sum _{i=1}^{n}\delta _{x_{i}}\) and \(\beta ={\tfrac {1}{n}}\textstyle \sum _{i=1}^{n}\delta _{y_{i}}\) where \(x_{1}\leq \ldots \leq x_{n}\) and \(y_{1}\leq \ldots \leq y_{n}\) then the Wasserstein distance is given by \(W_{p}(\alpha ,\beta )^{p}={\tfrac {1}{n}}\textstyle \sum _{i=1}^{n}|x_{i}-y_{i}|^{p}\).
More generally, if \(\alpha \in \mathcal{P}_p(\mathbb{R})\), we can define the cumulative distribution function (CDF) as a function \(\mathcal{C}_{\alpha}: \mathbb{R}\rightarrow [0,1]\) defined by \[\mathcal{C}_{\alpha} = \int_{-\infty}^x d\alpha\] as well as the pseudoinverse-CDF \(\mathcal{C}_{\alpha}^{\dagger}: [0,1] \rightarrow \mathbb{R}\cup \{-\infty\}\) defined by \[\mathcal{C}_{\alpha}^{\dagger} (r) = \min \mathcal{C}_\alpha^{-1}([r,\infty]).\] Using these notions, we can write down an explicit formula for the Wasserstein distance between probability distributions on \(\mathbb{R}\): \[\mathcal{W}_p(\alpha,\beta)^p = ||\mathcal{C}_\alpha^{\dagger} -\mathcal{C}_{\beta}^\dagger||_{L^p([0,1])}^p = \int_{0}^1 |\mathcal{C}_\alpha^{\dagger}(r) -\mathcal{C}_{\beta}^\dagger(r)|^pdr\]
The simplicity of the 1D case provokes one to consider whether a Wasserstein-like distance over \(\mathbb {R} ^{d}\) could be built from knowledge of the Wasserstein distance along projections onto 1D axes. The sliced Wasserstein distance provides an affirmative answer to this question.
Definition
Let \(P_{\theta }:\mathbb {R} ^{d}\to \mathbb {R}\) be the projection onto a unit vector \(\theta \in \mathbb {S} ^{d-1}\) i.e. \(P_{\theta }(x)=x\cdot \theta\). The sliced Wasserstein distance \(SW_p\) on \({\mathcal {P}}_p(\mathbb {R} ^{d})\) is given by \[SW_p(\mu ,\nu )=\left(\int _{\mathbb {S} ^{d-1}}W_p(P_{\theta \#}\mu ,P_{\theta \#}\nu )^pd\theta \right)^{1/p}.\]
Metric Structure of Sliced-Wasserstein Space
The sliced Wasserstein distance satisfies all the axioms of a true metric on \(\mathcal{P}_p(\mathbb{R}^d)\) (Peyré and Cuturi 2020). The metric space \((\mathcal{P}_p(\Omega), SW_p)\) is both complete and separable (Kitagawa and Takatsu 2024). In addition, it has a Hilbert space structure, which we discuss below, as well as the fact that \({\displaystyle SW_p}\) and \({\displaystyle W_p}\) are homeomorphic when \(\Omega\) is compact.
We sketch how to show that the sliced Wasserstein distance is a proper metric. The triangle inequality is inherited from \(\displaystyle \mathcal{P}_p(\mathbb{R}^d)\) and \(\displaystyle L^p(\mathbb{S}^{d-1})\). To see this, if \(\mu, \nu, \sigma\) are three measures in \({\mathcal {P}}_p(\mathbb {R} ^{d})\) then \[\begin{align} SW_p(\mu ,\sigma )+SW_p(\sigma ,\nu )&=\left(\int _{\mathbb {S} ^{d-1}}W_p(P_{\theta \#}\mu ,P_{\theta \#}\sigma )^pd\theta \right)^{1/p} + \left(\int _{\mathbb {S} ^{d-1}}W_p(P_{\theta \#}\sigma ,P_{\theta \#}\nu )^pd\theta \right)^{1/p}\\ &\leq \left(\int _{\mathbb {S} ^{d-1}}W_p(P_{\theta \#}\mu ,P_{\theta \#}\sigma )^pd\theta + \int _{\mathbb {S} ^{d-1}}W_p(P_{\theta \#}\sigma ,P_{\theta \#}\nu )^pd\theta \right)^{1/p}\\ &= \left(\int _{\mathbb {S} ^{d-1}}W_p(P_{\theta \#}\mu ,P_{\theta \#}\sigma )^p+W_p(P_{\theta \#}\sigma ,P_{\theta \#}\nu )^p d\theta \right)^{1/p}\\ &\leq \left(\int _{\mathbb {S} ^{d-1}}\Big(W_p(P_{\theta \#}\mu ,P_{\theta \#}\sigma )+W_p(P_{\theta \#}\sigma ,P_{\theta \#}\nu )\Big)^p d\theta \right)^{1/p}\\ &\leq \left(\int _{\mathbb {S} ^{d-1}} W_p(P_{\theta \#}\mu,P_{\theta \#}\nu )^p d\theta \right)^{1/p}\\ &= SW_p(\mu ,\nu ). \end{align}\] Similarly, the positivity and symmetry of \(\displaystyle SW_p\) can be shown from the positivity and symmetry of \(\displaystyle W_p\). To show that \(\displaystyle SW_p(\mu ,\nu )=0\) implies \(\displaystyle \mu =\nu\), one first observes that if \(\displaystyle SW_p(\mu ,\nu )=0\) then \(\displaystyle P_{\theta \#}\mu =P_{\theta \#}\nu\) for any choice of \(\theta\). This observation can be used to show that \(\mu = \nu\) by way of Radon transforms (Santambrogio 2015).
Hilbertian Property of \({\displaystyle SW_{p}}\)
There is an isometric embedding of \((\mathcal{P}_p(\mathbb{R}^d),SW_p)\) into a Hilbert space which demonstrates that \(SW_p\) is a Hilbertian metric. This is a special property of \(SW_p\) which is not satisfied by \(\mathcal{W}_p\), which is generally not Hilbertian on \(\mathbb{R}^d\) with \(d > 1\) (Peyré and Cuturi 2020).
We now demonstrate the Hilbertian nature of \(SW_p\) via an isometric embedding of \((\mathcal{P}_p(\mathbb{R}^d),SW_p)\) into an \(L^p\) space. Recall the pseudoinverse-CDF formulation of Wasserstein distance in \(\mathbb{R}\), which tells us that \[\mathcal{W}_p(P_{\theta \#}\mu, P_{\theta \#}\nu)^p = \int_{0}^1 |\mathcal{C}_{P_{\theta \#}\mu}^{\dagger}(r) -\mathcal{C}_{P_{\theta \#}\nu}^\dagger(r)|^pdr\] which implies that \[SW_p(P_{\theta \#}\mu, P_{\theta \#}\nu)^p = \int_{\mathbb{S}^{d-1}}\int_{0}^1 |\mathcal{C}_{P_{\theta \#}\mu}^{\dagger}(r) -\mathcal{C}_{P_{\theta \#}\nu}^\dagger(r)|^pdrd\theta.\] This tells us that the map \(F: \mathcal{P}_p(\mathbb{R}^d)\rightarrow L^p(\mathbb{S}^{d-1} \times [0,1])\) defined by \(F(\mu): (\theta,r) \mapsto \mathcal{C}_{P_{\theta \#}\mu}^{\dagger}(r)\) is an isometric embedding for \(\mathcal{P}_p(\mathbb{R}^d)\) into \(L^p(\mathbb{S}^{d-1} \times [0,1])\), which means that \(SW_p\) is a Hilbertian metric.
This embedding allows one to construct an analogue of displacement interpolation between measures \(\mu\) and \(\nu\). This is done via a gradient flow on the function \(\sigma \mapsto SW_p(\nu,\sigma)^p\) with initial condition \(\mu_0 = \mu\), as long as such a flow converges to \(\nu\). In the case of \(SW_2\), numerical simulations have shown that the flow indeed converges to \(\nu\) (Peyré and Cuturi 2020).
Topological Equivalence of \({\displaystyle SW_p}\) and \({\displaystyle W_p}\) on compact domains
On compact domains \(\Omega \subset \mathbb{R}\), the topologies on \((\mathcal{P}_p(\Omega), SW_p)\) and \((\mathcal{P}_p(\Omega), W_p)\) are the same. To see this, first notice that \({\displaystyle SW_p(\mu ,\nu )\leq W_p(\mu ,\nu )}\) from the property that \(\displaystyle W_p(P_{\theta \#}\mu ,P_{\theta \#}\nu )\leq W_p(\mu ,\nu ).\) This implies that convergence of a sequence in \({\displaystyle W_p}\) implies convergence in \({\displaystyle SW_p}\), i.e the identity map on \({\displaystyle {\mathcal {P}}_p(\mathbb {R} ^{d})}\) is \({\displaystyle W_p}\)-to-\({\displaystyle SW_p}\)-continuous. Moreover, since \(\Omega\) is compact it follows that \({\displaystyle ({\mathcal {P}}_p(\Omega ),W_p)}\) is compact, and since \((\mathcal{P}_p(\Omega), SW_p)\) is a metric space it is Hausdorff. We conclude that the identity map from \((\mathcal{P}_p(\Omega), W_p)\) to \((\mathcal{P}_p(\Omega), SW_p)\) is a continuous bijection from a compact space to a Hausdorff space, and such a map must be a homeomorphism. This shows that \((\mathcal{P}_p(\Omega), W_p)\) and \((\mathcal{P}_p(\Omega), SW_p)\) are equivalent from a topological perspective.
This result does not easily generalize beyond compact domains. In fact, it is known that \((\mathcal{P}_p(\mathbb{R}^d), W_p)\) and \((\mathcal{P}_p(\mathbb{R}^d), SW_p)\) have distinct topologies (Bayraktar and Guo 2021).
Comparison of \((\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d), W_2)\) and \((\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d), SW_2)\)
There is a large body of work in Optimal Transport Theory devoted to the study of \((\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d),W_2)\) as a result of its highly geometric, Riemannian-esque structure. Recent work (Park and Slepčev 2024) has studied the properties of the 2-Sliced-Wasserstein metric in comparison to the 2-Wasserstein metric. They find for a discrete measure \(\mu\) that \(SW_2(\mu, \cdot)\) behaves similarly to \(W_2(\mu,\cdot)\), and for certain classes of absolutely continuous measures \(SW_2(\mu, \cdot)\) behaves similarly to a negative Sobolev norm.
In terms of geodesic structure is known that \((\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d), SW_2)\) is not a geodesic space, i.e there is not always a continuous curve between measures \(\mu,\nu\) whose length is the distance \(SW_2(\mu,\nu)\). This is in contrast with the fact that \((\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d), W_2)\) gains much of its structure from being a geodesic space. However, it is still possible to put a tangential structure on \((\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d), SW_2)\) in a fashion similar to the tangential structure on \((\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d), W_2)\). All absolutely continuous curves \((\mu_t)_{t \in [0,1]}\) in \((\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d), W_2)\) are solutions of the continuity equation with a flux \(v_t \mu_t\) that is absolutely continuous with respect to \(\mu_t\), and similarly curves in \((\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d), SW_2)\) are solutions of a continuity equation, although the corresponding flux is not necessarily absolutely continuous with respect to \(\mu_t\). Owing to the lack of geodesics in Sliced Wasserstein space, (Park and Slepčev 2024) studied a length metric \(l_{SW_2}\) on \(\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d)\) with respect to \(SW_2\), where \(l_{SW_2}(\mu,\nu)\) is defined as the infimum of the \(SW_2\)-lengths of all curves between \(\mu\) and \(\nu\). Under this metric, geodesics do in fact exist between any two measures in \(\mathcal{P}_{2,\text{ac}}(\mathbb{R}^d)\).
Dual Representation of the Sliced Wasserstein metric
The dual representation of the Wasserstein metric is useful for computational problems. Analogously, there is a dual representation for the Sliced Wasserstein metric. The domain of test functions will be \(\mathcal{A}_p\) and \(\mathcal{Z}_{s}\), where \[\mathcal{A}_p := \left\{ (\Phi_{(\boldsymbol{\cdot})}, \Psi_{(\boldsymbol{\cdot})} \in C(\mathbb{S}^{d-1};C_b(\mathbb{R})))^2 \,\bigg|\, \begin{array}{ c } -\Phi_{\omega}(t) - \Psi_{\omega}(s) \leq |t-s|^p \\ \text{for all } t,s\in \mathbb{R}\text{ and } \omega \in \mathbb{S}^{d-1} \end{array} \right\}\] and \[\mathcal{Z} := \left\{ \zeta \in C(\mathbb{S}^{d-1}) \, \Big| \, |\zeta| < 1, \zeta > 0 \right\}.\] Then the dual representation of the Sliced Wasserstein metric is \[SW_p(\mu,\nu)^p = \sup \left\{ -\int_{\mathbb{S}^{d-1}} \zeta \left( \int_{\mathbb{R}} \Phi_{\theta} dP_{\theta \#} \mu + \int_{\mathbb{R}} \Psi_{\theta} dP_{\theta \#} \nu \right) d\theta \, \bigg| \, \begin{array}{ c }(\Phi,\Psi) \in \mathcal{A}_p, \\ \zeta \in \mathcal{Z} \end{array} \right\}.\]