Geometry and analytic properties of the sliced Wasserstein space

For $\Omega=\mathbb{R}^{d},\mathbb{S}^{d-1}\times\mathbb{R}$ we denote by $\mathscr{P}(\Omega)$ the space of all Borel probability measures on $\Omega$. Let $1\leq p<\infty$. For probability measures $\mu,\nu\in\mathscr{P}_{p}(\mathbb{R}^{d})\mathrel{\mathop{\mathchar 58\relax}}=\{\mu\in\mathscr{P}(\mathbb{R}^{d})\>\mathrel{\mathop{\mathchar 58\relax}}\>\int_{\mathbb{R}^{d}}|x|^{p}d\mu(x)<\infty\}$, the $p$-Wasserstein distance, $W_{p}$, is defined as follows:

(1.1)

$$\begin{split}W_{p}(\mu,\nu)\mathrel{\mathop{\mathchar 58\relax}}=&\inf_{\gamma\in\Gamma(\mu,\nu)}\left(\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{p}\,d\gamma(x,y)\right)^{1/p}\\ &\text{ where }\Gamma(\mu,\nu)=\left\{\gamma\in\mathscr{P}(\mathbb{R}^{d}\times\mathbb{R}^{d})\mathrel{\mathop{\mathchar 58\relax}}\,\pi^{1}_{\#}\gamma=\mu,\,\pi^{2}_{\#}\gamma=\nu\right\}.\end{split}$$

To define the sliced Wasserstein distance, we introduce the following notation: for each $\theta\in\mathbb{S}^{d-1}$, let $\mathbb{R}\theta\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Span}{\{\theta\}}\subset\mathbb{R}^{d}$, and define the projection $\pi^{\theta}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}\theta$ by

(1.2)

$$\pi^{\theta}(x)=(\theta\cdot x)\theta.$$

The $p$-sliced Wasserstein distance $SW_{p}$ is defined by

(1.3)

$$SW_{p}(\mu,\sigma)=\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{\mathbb{S}^{d-1}}W_{p}^{p}(\pi^{\theta}_{\#}\mu,\pi^{\theta}_{\#}\sigma)\,d\theta\right)^{\frac{1}{p}}.$$

The Radon transform provides a natural language to describe objects relating to the sliced Wasserstein distance. Consider an integrable function $f\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}$, for $d\geq 2$. We use both $Rf$ and $\widehat{f}$ to denote its Radon transform: For $\theta\in\mathbb{S}^{d-1}$ and $r\in\mathbb{R}$

(1.4)

$$R_{\theta}f(r)=Rf(\theta,r)=\widehat{f}(\theta,r)\mathrel{\mathop{\mathchar 58\relax}}=\int_{\theta^{\perp}}f(r\theta+y^{\theta})\,dy^{\theta},$$

where $\theta^{\perp}\mathrel{\mathop{\mathchar 58\relax}}=\{y\in\mathbb{R}^{d}\mathrel{\mathop{\mathchar 58\relax}}\;y\cdot\theta=0\}$ and $dy^{\theta}$ is the $(d-1)$-dimensional Lebesgue measure on $\theta^{\perp}$. By Fubini’s theorem, $R_{\theta}f(r)$ exists for $\mathscr{L}^{1}$ a.e. $r\in\mathbb{R}$ for each $\theta\in\mathbb{S}^{d-1}$ when $f\in L^{1}(\mathbb{R}^{d})$.

Note that $Rf$ is even on $\mathbb{S}^{d-1}\times\mathbb{R}$, meaning that $Rf(-\theta,-r)=Rf(\theta,r)$. This motivates defining the $d$-dimensional “Radon domain” $\mathbb{P}_{d}$ by

(1.5)

$$\mathbb{P}_{d}\mathrel{\mathop{\mathchar 58\relax}}=(\mathbb{S}^{d-1}\times\mathbb{R})/\sim\text{, where the equivalence relation is given by }(-\theta,-r)\sim(\theta,r).$$

The Radon transform can be extended to distributions (see [25, Chapter 1.5] and [57]); in particular, when $\mu$ is a bounded measure, the distributional extension is consistent with the definition of $R_{\theta}\mu$ as a pushforward of $\mu$ by the projection map $x\mapsto x\cdot\theta$ (see Remark A.1). Thus, for $\mu\in\mathscr{P}_{p}(\mathbb{R}^{d})$ we have $\widehat{\mu}^{\theta}=R_{\theta}\mu\in\mathscr{P}_{p}(\mathbb{R})$, and

$$SW_{p}(\mu,\sigma)=\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{\mathbb{S}^{d-1}}W_{p}^{p}(\widehat{\mu}^{\theta},\widehat{\sigma}^{\theta})\,d\theta\right)^{\frac{1}{p}}.$$

Note that $\widehat{\mu}^{\theta}\in\mathscr{P}_{p}(\mathbb{R})$ whereas $\pi^{\theta}_{\#}\mu\in\mathscr{P}_{p}(\mathbb{R}\theta)$; we will sometimes use the latter when it is more convenient to consider measures on subspaces of $\mathbb{R}^{d}$ than on $\mathbb{R}$.

Henceforth we will focus on the case $p=2$, and write $SW=SW_{2}$ and $W=W_{2}$.

We obtain a number of geometric and analytic properties of the $SW$ distance and of the associated length space, and investigate their implications on the sliced Wasserstein gradient flows and statistical estimation rates in the metrics. Throughout this paper, we use $X\lesssim_{\alpha_{1},\cdots,\alpha_{k}}Y$ as a shorthand for the inequality $X\leq C(\alpha_{1},\cdots,\alpha_{k})Y$, where $C(\alpha_{1},\cdots,\alpha_{k})>0$ is a finite positive constant depending on $(\alpha_{i})_{i=1}^{k}$. When summarizing results or making remarks, we sometimes omit the dependence on the parameters and write $\lesssim$ for the sake of simplicity; however, all rigorous statements contain clear characterizations of the constants.

Basic properties. In Section 2 we establish some basic properties of the $SW$ metric. In particular in Proposition 2.4 we show that $(\mathscr{P}_{2}(\mathbb{R}^{d}),SW)$ is a complete metric space, which we refer to as the $SW$ space. We then turn to the intrinsic geometry of the $SW$ space. Example 2.5 shows that, unlike the Wasserstein space, the $SW$ space is not a geodesic space. That is, one cannot in general find a continuous curve connecting two measures in the SW space with its length equal to the distance between the measures.

Tangential structure of sliced Wasserstein space. We show in Section 3 that the $SW$ space has a tangent structure which resembles the tangent structure of the Wasserstein space. Recall that in the Wasserstein space, each absolutely continuous curve $(\mu_{t})_{t\in I}$ defined on an interval $I\subset\mathbb{R}$ corresponds to a measure-valued distributional solution of the continuity equation

$$\partial_{t}\mu_{t}+\nabla\cdot(v_{t}\mu_{t})=0,\;\text{ with }\|v_{t}\|_{L^{2}(\mu_{t})}=|\mu^{\prime}|_{W}(t)\text{ for a.e. }t\in I,$$

where $|\mu^{\prime}|_{W}$ is the metric derivative w.r.t $W$; see [1, Theorem 8.3.1]. Theorem 3.9 establishes an analogous result for the sliced Wasserstein space – for each absolutely continuous curve $(\mu_{t})_{t\in I}$ in the sliced Wasserstein space, there exists a vector-valued flux $(J_{t})_{t\in I}$ such that $J_{t}$ is in a suitable subspace of $\mathcal{S}^{\prime}(\mathbb{R}^{d};\mathbb{R}^{d})$ and

$$\partial_{t}\mu_{t}+\nabla\cdot J_{t}=0,\;\text{ with }\mathinner{\!\left\lVert\tfrac{d\widehat{J}_{t}}{d\widehat{\mu}_{t}}\right\rVert}_{L^{2}(\widehat{\mu}_{t})}=|\mu^{\prime}|_{SW}(t)\text{ for a.e. }t\in I.$$

We note two key differences: the metric derivative is characterized by the weighted $L^{2}$ norm in the Radon domain, finiteness of which does not imply $J_{t}\ll\mu_{t}$ in general (see Remark 3.10). Moreover, Sharafutdinov’s results of Radon transform on Sobolev spaces [57] imply that $\|Rf\|_{L^{2}(\mathbb{P}_{d})}=\|f\|_{\dot{H}^{-(d-1)/2}(\mathbb{R}^{d})}$, thus we can formally understand $|\mu^{\prime}|_{SW}(t)$ as corresponding to a weighted high order negative Sobolev norm of the flux $J_{t}$, in contrast to the weighted $L^{2}$-norm in the Wasserstein case. Hence, at least for absolutely continuous measures, the formal Riemannian metric measuring infinitesimal length in the sliced Wasserstein space corresponds to a weaker space than the one for the Wasserstein metric. Furthermore in Section 3.2, we characterize its tangent space which has the following key property analogous to its Wasserstein counterpart: Each absolutely continuous curve $(\mu_{t})_{t\in I}$ is associated to a unique (up to a $\mathscr{L}^{1}\,\raisebox{-0.5468pt}{\reflectbox{\rotatebox[origin={br}]{-90.0}{$\lnot$}}}\,_{I}$-null set) family of tangent vectors $(J_{t})_{t\in I}$, which moreover attain the metric derivative through the quadratic form $J_{t}\mapsto\|d\widehat{J}_{t}/d\widehat{\mu}_{t}\|_{L^{2}(\widehat{\mu}_{t})}$.

Intrinsic sliced Wasserstein length space. In Section 4, motivated by the general of lack geodesics in $(\mathscr{P}_{2}(\mathbb{R}^{d}),SW)$, we introduce the sliced Wasserstein length metric $\ell_{SW}$ defined as the infimum of the lengths of curves between measures in the SW space. We establish the basic properties of the metric space $(\mathscr{P}_{2}(\mathbb{R}^{d}),\ell_{SW})$; in particular we prove in Proposition 4.5 that geodesics exist, which further implies that in general $\ell_{SW}\neq SW$.

Comparison of sliced Wasserstein metric with negative Sobolev norms and Wasserstein metric. In Section 5 we establish some of the key results of this paper, namely the comparison theorems of $SW$ metric with negative Sobolev norms near absolutely continuous measures and comparisons of $SW$ with the Wasserstein metric near discrete measures. In particular, consider an absolutely continuous measure $\mu$ bounded away from zero and infinity on some bounded open convex domain $\Omega$. Theorem 5.2 establishes that

$$\|\mu-\nu\|_{\dot{H}^{-(d+1)/2}(\mathbb{R}^{d})}\lesssim SW(\mu,\nu)\leq\ell_{SW}(\mu,\nu)\lesssim SW(\mu,\nu)\lesssim\|\mu-\nu\|_{\dot{H}^{-(d+1)/2}(\mathbb{R}^{d})},$$

for all measures $\nu$ which are bounded above and below by constant multiples of $\mu$ and coincide with $\mu$ near the boundary of $\Omega$. In other words we show that near $\mu$, $SW$ is equivalent to $\dot{H}^{-(d+1)/2}$.

On the other hand, Theorem 5.5 states

$$SW(\mu^{n},\nu)\leq\ell_{SW}(\mu^{n},\nu)\leq\frac{1}{d}W(\mu^{n},\nu)\leq(1+o(1))SW(\mu^{n},\nu)$$

for $\nu$ near discrete measures of the form $\mu^{n}=\sum_{i=1}^{n}m_{i}\delta_{x_{i}}$.

These two results provide interesting insights about the $SW$ metric. Near smooth measures it behaves like a highly negative Sobolev space, in contrast to the Wasserstein metric which for such measures behaves like the $\dot{H}^{-1}$ norm as noted by Peyre [49], while near discrete measures $SW$ behaves like the Wasserstein distance.

Approximation by discrete measures in sliced Wasserstein length. Manole, Balakrishnan, and Wasserman [37, Proposition 4] have shown that a finite random sample (i.e. the empirical measure of the set of $n$ random points) of a probability measure on $\mathbb{R}^{d}$ estimates the measure in the sliced Wasserstein distance at a parametric rate $O(n^{-1/2})$ for a large class of measures; see also [41]. This is in stark contrast with the Wasserstein distance where the approximation error is poor in high dimensions and scales like $n^{-1/d}$. We start by pointing out a connection between the results on the parametric finite-sample estimation in the sliced Wasserstein distance and the results in statistical literature, that our results in Section 5 identify. Namely it is known that finite-sample estimation of measures with respect to maximum mean discrepancy (MMD) also enjoys parametric rate  [61, Theorem 3.3]. MMD distance is nothing but the norm in the dual of a reproducing kernel Hilbert space (RKHS). In particular the results of [61] apply to the dual of the Sobolev space $H^{s}$ with $s>\frac{d}{2}$ (when the spaces embeds in the spaces of Hölder continuous functions and are RKHS). Our Theorem 5.2 says that near absolutely continuous measures, SW behaves like the $\dot{H}^{-(d+1)/2}$-norm; as the associated norm $\|\cdot\|_{H^{-(d+1)/2}(\mathbb{R}^{d})}$ is an MMD, we can formally understand $SW$ to exhibit behaviors like an MMD. Thus the MMD parametric estimation can be seen as a tangential or a linearized analogue of the finite sample estimation rates in SW distance.

Here we investigate the finite sample estimation rates in the SW intrinsic length metric $\ell_{SW}$. The goal is to gain a better understanding of the extent to which $SW$ and $\ell_{SW}$ share properties. In Theorem 6.3, we establish that the finite sample approximation in $\ell_{SW}$ happens at the parametric rate up to a logarithmic correction, namely that

$$\ell_{SW}(\mu,\mu^{n})\lesssim\sqrt{\frac{\log n}{n}}\;\text{ with high probability, where }\mu^{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}\text{ with }X_{i}\overset{i.i.d.}{\sim}\mu.$$

While this is consistent with the geometric view of $(\mathscr{P}_{2}(\mathbb{R}^{d}),\ell_{SW})$ as a curved or nonlinear dual of Reproducing Kernel Hilbert Space (see beginning of Section 6 for discussion), the statement and the proof requires dealing with discrete measures where such heuristic view does not hold.

Implications on gradient flows. Section 7 applies the comparison results on $\ell_{SW}$, $SW$ to obtain comparisons for the metric slopes. Given a metric space $(X,m)$, recall that metric slope $|\partial\mathcal{E}|_{m}$ of a functional $\mathcal{E}\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow\mathbb{R}$ is defined by

(1.6)

$$|\partial\mathcal{E}|_{m}(u)=\limsup_{v\xrightarrow[]{d}u}\frac{[\mathcal{E}(u)-\mathcal{E}(v)]_{+}}{m(u,v)}.$$

Let $V\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}$ and consider the potential energy $\mathcal{V}(\mu)\mathrel{\mathop{\mathchar 58\relax}}=\int_{\mathbb{R}^{d}}V(x)\,d\mu(x)$. Proposition 7.2 states that when $V$ is smooth and compactly supported, for suitable absolutely continuous $\mu\in\mathscr{P}_{2}(\mathbb{R}^{d})$ it holds that

$$|\partial\mathcal{V}|_{\dot{H}^{(d+1)/2}(\mathbb{R}^{d})}(\mu)\lesssim|\partial\mathcal{V}|_{\ell_{SW}}(\mu)\leq|\partial\mathcal{V}|_{SW}(\mu)\lesssim|\partial\mathcal{V}|_{\dot{H}^{(d+1)/2}(\mathbb{R}^{d})}(\mu)$$

whereas Proposition 7.5 shows that the slope behaves quite differently at discrete measures, $\mu^{n}=\sum_{i=1}^{n}m_{i}\delta_{x_{i}}$, namely that

$$|\partial\mathcal{V}|_{SW}(\mu^{n})=|\partial\mathcal{V}|_{\ell_{SW}}(\mu^{n})=\sqrt{d}\,|\partial\mathcal{V}|_{W}(\mu^{n}).$$

By considering a sequence of discrete measures $\mu^{n}$ converging to an absolutely continuous measure $\mu$, we may deduce that $|\partial\mathcal{V}|_{SW}$ (resp. $|\partial\mathcal{V}|_{\ell_{SW}}$) is not lower semicontinuous in $SW$ (resp. $\ell_{SW}$) in general, even when $V\in C_{c}^{\infty}(\mathbb{R}^{d})$; see Corollary 7.7. This implies that the potential energy is not $\lambda$-geodesically convex in $(\mathscr{P}_{2}(\mathbb{R}^{d}),\ell_{SW})$. Furthermore we observe in Remark 7.6 that starting from discrete measures with finite number of particles, the curves of maximal slope in the Wasserstein space, after a constant rescaling of time, are the curves of maximal slope in the SW space.

On the other hand, for smooth measures, the curves of maximal slope with respect to the Wasserstein metric are not curves of maximal slope in the SW space. We formally show that SW gradient flows of the potential energy satisfy a higher order equation given by a pseudodifferential operator of order $d$, which is consistent with the rigorous results of Proposition 7.2. We conclude that the framework of gradient flows in metric spaces would not be the right tool to study such equations; PDE based approaches may provide an avenue for creating a well-posedness theory, which remains an open problem.

Since the introduction of the sliced Wasserstein distance [51], numerous variants have been considered. Deshpande et al. [20] proposed the max-sliced Wasserstein distance (max-SW distance), which is the maximum of the 1D Wasserstein distances, instead of the average as in the $SW$ case. Niles-Weed and Rigollet [45] and Paty and Cuturi [48] independently proposed the $k$-dimensional generalization (max-$k$-SW distance) for $1\leq k\leq d$. Generalizations to spherical [9] and other nonlinear projections [28] have also been considered to more effectively capture the geometric structure of data. Based on the ideas of partial optimal transportation [23], Bai, Schmitzer, Thorpe, and Kolouri [4] introduced sliced optimal partial transport to compare of measures with different masses. Further projection-based transport metrics include the distributional sliced Wasserstein distance introduced by Nguyen, Ho, Pham, and Bui [43] and the convolution sliced Wasserstein distance proposed by Nguyen and Ho [42].

The sliced Wasserstein distances have found numerous applications in image processing. In fact, utility of the sliced Wasserstein barycenter for tasks such as image synthesis, color transfer, and texture mixing served as a motivation behind the introduction of sliced Wasserstein distance [51]. Bonneel, Rabin, Peyré, and Pfister [11] further studied efficient numerical methods to compute sliced Wasserstein and related barycenters, and their applications. Kolouri, Park, and Rhode proposed the Radon cumulative distribution transport (Radon CDT) [29] for image classification; Radon CDT effectively computes the sliced Wasserstein ‘geodesic’, by taking the Radon inverse of the displacement interpolation between the Radon transform of the measures. However, we note that such inverse will in general fail to be a curve in the space of probability measures, as the Radon inverse of nonnegative functions need not be nonnegative.

Gradient flows related to the sliced Wasserstein distance have been applied to various machine learning and image processing tasks. Bonnotte noticed [12] that the continuous analogue of the the isotropic Iterative Distribution Transfer (IDT) algorithm, introduced by [50] to transfer the color palette of a reference picture to a target picture, is the Wasserstein gradient flow of $\mu\mapsto\frac{1}{2}SW^{2}(\mu,\sigma)$. Liutkus et al. [35] utilizes the Wasserstein gradient flows of the entropy-regularized version of the same energy functional for generative modelling. Gradient flow of the same energy in the sliced Wasserstein space have been considered by Bonet et al. [10] also for generative modelling; they also study the JKO scheme with respect to $SW$, and establish existence and uniqueness of minimizers of the scheme when the optimization is restricted to probability measures supported on a common compact set [10, Section 3.2]. Sliced Iterative Normalizing Flows (SINF) [18], useful for sampling and density evaluation, can be seen as a max-SW variant of the isotropic IDT algorithm.

Other applications in machine learning include: sliced Wasserstein generative adversarial nets by Deshpande, Zhang, Schwing [21]; max-SW generative adversarial nets [20] for generative modelling; sliced Wasserstein autoencoder by [30]; and use of $SW_{1}$ distance for unsupervised domain adaptation [31].

On the statistical side, Manole, Balakrishnan, and Wasserman [37] established, based on the 1-dimensional results by Bobkov and Ledoux [8], the parametric estimation rate $\mathbb{E}SW(\mu^{n},\mu)\lesssim n^{-1/2}$ for the empirical measure $\mu^{n}$ of $n$ i.i.d samples of $\mu$, and further investigated statistical properties of the trimmed sliced Wasserstein distances. Nietert, Goldfeld, Sadhu, and Kato [44] established empirical estimation rate in $SW$ and max-$SW$ for log-concave distributions with explicit constants dependent on the intrinsic dimension, and showed robustness to data contamination and explored efficient computational methods. Lin, Zheng, Chen, Cuturi, and Jordan [34] investigated the max-$k$-sliced distances and their corresponding integral variants integral projection robust Wasserstein (IPRW) distance, also known as the $k$-sliced Wasserstein distances ($k$-SW distances), and established several statistical properties including sample complexity $O(n^{-1/k})$. More recently, Olea, Rush, Velez, and Wiesel [46] explored the connection between a certain linear predictor problems and distributionally robust optimization based on a modified max-SW. For applications in Approximate Bayesian Computation, we refer the readers to [40, Chapter 4] and the references therein.

Regarding analytic and topological properties, Bonnotte [12] showed that $SW_{p}$ is indeed a distance on $\mathscr{P}_{p}(\mathbb{R}^{d})$ for $1\leq p<\infty$, and established that, for measures supported on $\Omega\subset\subset\mathbb{R}^{d}$ – i.e. $\Omega$ compactly contained in $\mathbb{R}^{d}$ – we have

$$SW_{p}\leq W_{p}\lesssim_{\Omega}SW_{p}^{\frac{1}{2p(d+1)}}.$$

More recently, Bayraktar and Guo [5] showed that $SW_{p}$, $W_{p}$, and the $p$-max-sliced Wasserstein distance induce the same topology on $\mathscr{P}_{p}(\mathbb{R}^{d})$ for $p\geq 1$. We note here that this does not directly imply completeness of $(\mathscr{P}_{2}(\mathbb{R}^{d}),SW)$, as not all Cauchy sequences in $SW$ need be Cauchy in $W$.

Bonnotte also showed the existence of the Wasserstein gradient flow of the energy functional $\mu\mapsto\frac{1}{2}SW^{2}(\mu,\sigma)$ for the target measure $\sigma$, despite the lack of geodesic convexity of the energy functional, and derived the corresponding PDE [12, Chapter 5], the continuous-time version of the previously mentioned isotropic IDT algorithm. Due to the lack of convexity of the energy functional, even the asymptotic convergence of the gradient flow remains open. Nevertheless, Li and Moosmüller [33] recently established almost sure convergence of the discrete isotropic IDT algorithm with step-sizes satisfying certain summability conditions. More recently, Cozzi and Santambrogio [17] established the convergence rate $t\mapsto SW^{2}(\mu_{t},\sigma)=O(t^{-1})$ when the target measure $\sigma$ is any isotropic Gaussian.

As our work was nearing completion, we became aware of the independent work by Kitagawa and Takatsu [26] on the sliced Wasserstein spaces. In their work, Kitagawa and Takatsu establish in the metric completeness of sliced optimal-transportation-based spaces, which generalizes our Proposition 2.4, and also demonstrate that the SW spaces are not geodesic spaces, generalizing our Example 2.5. Their work focuses on isometrically embedding the SW type spaces into larger spaces and the barycenter problem. See also [27] for their more recent work on disintegrated optimal transport for metric fiber bundles.

Here we provide a brief overview of the key properties of the Radon transform. We refer the readers to Appendix A for precise statements and to the book by Helgason [25] for a more thorough introduction.

The dual Radon transform. Given an integrable function $\mathfrak{g}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{P}_{d}\rightarrow\mathbb{R}$ we define its dual Radon transform, which we write $R^{\ast}\mathfrak{g}$ or $\widecheck{\mathfrak{g}}$, by

(2.1)

$$R^{\ast}\mathfrak{g}(x)=\widecheck{\mathfrak{g}}(x)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{\mathbb{S}^{d-1}}\mathfrak{g}(\theta,x\cdot\theta)\,d\theta.$$

As

$$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}}g(\theta,r)\,d\mathscr{L}^{1}(r)\,d\theta=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}^{d}}g(\theta,x\cdot\theta)\,d\mathscr{L}^{d}(x)\,d\theta=\int_{\mathbb{R}^{d}}R^{\ast}g(x)\,d\mathscr{L}^{d}(x),$$

by Fubini’s theorem $R^{\ast}\mathfrak{g}$ is well-defined for $\mathscr{L}^{d}$-a.e. $x\in\mathbb{R}^{d}$ whenever $\mathfrak{g}\in L^{1}(\mathbb{P}_{d})$. Furthermore, the dual transform $R^{\ast}$ satisfies

(2.2)

$$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}}Rf(\theta,r)\mathfrak{g}(\theta,r)\,dr\,d\theta=\int_{\mathbb{R}^{d}}f(x)R^{\ast}\mathfrak{g}(x)dx$$

whenever either $Rf\mathfrak{g}$ or $fR^{\ast}\mathfrak{g}$ are absolutely integrable; see [25, Lemma 5.1] for further details. In particular, the extension of the Radon transform to finite measures $\mu$ as the pushforward of $\mu$ under the map $x\mapsto x\cdot\theta$ is consistent with (2.2) (see Remark A.1). Consequently, we will often use the duality formula for bounded measures in the form

(2.3)

$$\int_{\mathbb{R}^{d}}R^{\ast}\mathfrak{g}\,d\mu=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}}\mathfrak{g}\,d\widehat{\mu}\;\;\text{ for }\mu\in\mathcal{M}_{b}(\mathbb{R}^{d})\text{ and }\mathfrak{g}\in C_{0}(\mathbb{P}_{d}).$$

Spaces related to the Radon transform. To add clarity, we denote the functions defined on $\mathbb{P}_{d}$ by a different set of symbols – e.g. $\mathfrak{f},\mathfrak{g},\mathfrak{u},\mathfrak{v},\mathfrak{J}$. We denote by $\mathcal{M}(\Omega)$ the space of locally finite signed Borel measures on $\Omega$. We note that $\mathcal{M}(\mathbb{P}_{d})$ can be identified with

(2.4)

$$\{\mathfrak{J}\in\mathcal{M}(\mathbb{S}^{d-1}\times\mathbb{R})\mathrel{\mathop{\mathchar 58\relax}}d\mathfrak{J}(-\theta,-r)=d\mathfrak{J}(\theta,r)\}.$$

We write $\mathcal{M}_{b}(\Omega)$ for the space of bounded Borel measures on $\Omega$. Any $\Omega=\mathbb{R}^{d},\mathbb{P}_{d},I\times\mathbb{R}^{d},I\times\mathbb{P}_{d}$ is a Polish space, hence $\mathcal{M}(\Omega)$ can be equivalently understood as a space of signed Radon measures. Finally, we denote by $\mathcal{M}(\Omega;\mathbb{R}^{d})\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{M}(\Omega)^{d}$ the space of vector valued Radon measures.

We will mostly treat $\theta\in\mathbb{S}^{d-1}$ as a parameter and $r\in\mathbb{R}$ as the variable, which is reflected in our notation. For instance, for a function $\mathfrak{g}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{P}_{d}\rightarrow\mathbb{R}$ we write $\mathfrak{g}^{\theta}(r)=\mathfrak{g}(\theta,r)$. Denoting by $\operatorname{Vol}_{\mathbb{S}^{d-1}}$ the normalized volume measure on $\mathbb{S}^{d-1}$ satisfying $\int_{\mathbb{S}^{d-1}}d\operatorname{Vol}_{\mathbb{S}^{d-1}}=1$, for each $\mathfrak{J}\in\mathcal{M}_{b}(\mathbb{P}_{d})$ we write $\mathfrak{J}^{\theta}\in\mathcal{M}_{b}(\mathbb{R})$ for its disintegration with respect to $\operatorname{Vol}_{\mathbb{S}^{d-1}}$ – i.e. $\mathfrak{J}=\mathfrak{J}^{\theta}\,d\operatorname{Vol}_{\mathbb{S}^{d-1}}(\theta)$; for precise statement of the disintegration theorem, see [19, III-70] or [1, Theorem 5.3.1]. We will always consider $\mathfrak{J}\in\mathcal{M}_{b}(\mathbb{P}_{d})$ with its first marginal equal to $\operatorname{Vol}_{\mathbb{S}^{d-1}}$.

We denote by $\mathcal{S}(\Omega)$ with $\Omega=\mathbb{R}^{d},\mathbb{P}_{d}$ the Schwartz-Bruhat space of smooth rapidly decreasing functions [15, 47]. We note that $\mathcal{S}(\mathbb{R}^{d})$ is the usual Schwartz class, whereas $\mathcal{S}(\mathbb{P}_{d})$ can be identified with the subspace of $\mathcal{S}(\mathbb{S}^{d-1}\times\mathbb{R})$ of even functions, namely the set

(2.5)

$$\{\mathfrak{g}\in\mathcal{S}(\mathbb{S}^{d-1}\times\mathbb{R})\mathrel{\mathop{\mathchar 58\relax}}\,\mathfrak{g}(-\theta,-r)=\mathfrak{g}(\theta,r)\}$$

We write $\mathcal{S}^{\prime}(\Omega)$ with $\Omega=\mathbb{R}^{d},\mathbb{P}_{d}$ to denote the space of continuous linear functionals on $\mathcal{S}(\Omega)$ –i.e. the space of tempered distributions on $\Omega$.

The $L^{2}$-Sobolev theory of Radon transforms will be crucial in understanding the differential structure of the sliced Wasserstein space. For this purpose, we use Sobolev spaces $H_{t}^{s}$ with attenuated ($t>0$) or amplified ($t<0$) low frequencies, introduced by Sharafutdinov [57]. For each $1\leq k\leq d$ let us denote by $\mathcal{F}_{k}$ the $k$-dimensional Fourier transform

$$\mathcal{F}_{k}f(\xi)=(2\pi)^{-k/2}\int_{\mathbb{R}^{k}}f(x)e^{-ix\cdot\xi}\,dx.$$

For $s\in\mathbb{R}$ and $t>-\frac{d}{2}$, the Hilbert space $H_{t}^{s}(\mathbb{R}^{d})$ is defined as the completion of $\mathcal{S}(\mathbb{R}^{d})$ under the norm

(2.6)

$$\|f\|_{H_{t}^{s}(\mathbb{R}^{d})}^{2}=\int_{\mathbb{R}^{d}}|\xi|^{2t}(1+|\xi|^{2})^{s-t}|\mathcal{F}_{d}f(\xi)|^{2}\,d\xi.$$

Similarly we define the analogous space $H_{t}^{s}(\mathbb{P}_{d})$ for $s\in\mathbb{R}$ and $t>-\frac{1}{2}$ in the Radon domain as the completion of Schwartz functions $\mathcal{S}(\mathbb{P}_{d})$ on the Radon domain under the norm

(2.7)

$$\|\mathfrak{g}\|_{H_{t}^{s}(\mathbb{P}_{d})}^{2}=\frac{1}{2(2\pi)^{d-1}}\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}}|\zeta|^{2t}(1+\zeta^{2})^{s-t}|\mathcal{F}_{1}\mathfrak{g}(\theta,\zeta)|^{2}\,d\zeta\,d\theta.$$

Here and in the sequel the one-dimensional Fourier transform $\mathcal{F}_{1}$ always applies to the scalar variable when applied to functions defined on $\mathbb{P}_{d}$. We only use the norms $H_{t}^{s}$ or $H_{-t}^{-s}$ for $0\leq t\leq s$. In the first case, we can view the $H_{t}^{s}$ norm as counting derivatives of order between $t$ and $s$, as

$$|\xi|^{2t}+|\xi|^{2s}\lesssim_{t-s}|\xi|^{2t}(1+|\xi|^{2})^{t-s}\lesssim_{t-s}|\xi|^{2t}+|\xi|^{2s}.$$

Thus when $0=t\leq s$ we see $H_{t}^{s}$ coincides with the standard Sobolev space of order $s$. On the other hand, the space $H_{-t}^{-s}$ can be understood as the dual of $H_{t}^{s}$. Indeed, for any $f,g\in\mathcal{S}(\mathbb{R}^{d})$

(2.8)

$$\left|\int_{\mathbb{R}^{d}}f(x)g(x)\,dx\right|\leq\|f\|_{H_{t}^{s}(\mathbb{R}^{d})}\|g\|_{H_{-t}^{-s}(\mathbb{R}^{d})};$$

for details, see [57, Theorem 5.3] and its proof. We provide further information on the relationship between the Radon transform and Sobolev spaces in Appendix A.

Outside of Section 2 we will mostly be interested in the case $t=s$, where $\dot{H}^{s}(\Omega)\mathrel{\mathop{\mathchar 58\relax}}=H_{s}^{s}(\Omega)$ with $\Omega=\mathbb{R}^{d}$ or $\mathbb{P}_{d}$ is equivalent to the more familiar homogeneous Sobolev space.

We only consider the space $H_{t}^{s}(\mathbb{R}^{d})$ where $-\frac{d}{2}<t<\frac{d}{2}$, and $H_{t}^{s}(\mathbb{P}_{d})$ for $-\frac{1}{2}<t<\frac{1}{2}$ which ensures that the identity map continuously embeds $H_{t}^{s}(\mathbb{R}^{d})$ to $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and the same holds for $H_{t}^{s}(\mathbb{P}_{d})$ and $\mathcal{S}^{\prime}(\mathbb{P}_{d})$; see [57, Theorem 5.3], which we have included in the appendix (Theorem A.11) for completeness. Thus, for $\Omega=\mathbb{R}^{d},\mathbb{P}_{d}$, the spaces $H_{t}^{s}(\Omega)$ can be seen as a complete normed subspace of $\mathcal{S}^{\prime}(\Omega)$. We stress that, while we use the norm $\|\cdot\|_{\dot{H}^{(d+1)/2}(\mathbb{R}^{d})}$ in comparison to $SW$, generic elements of spaces $H_{t}^{s}(\mathbb{R}^{d})$ with $|t|>\frac{d}{2}$ are not considered in this paper.

Furthermore, when $0<s<\frac{d}{2}$, $\dot{H}^{s}(\mathbb{R}^{d})$ continuously embeds to $L^{\frac{2d}{d-2s}}(\mathbb{R}^{d})$ by Gagliardo-Nirenberg-Sobolev inequality for fractional Sobolev spaces; see for instance [32, Theorem 11.31]. Thus, we can consider $\dot{H}^{s}(\mathbb{R}^{d})$ as a space of functions in this case.

We note that in some works the definition of homogeneous Sobolev spaces for $s<d/2$ differs from the one we use. Namely $\dot{H}^{s}(\Omega)$ is defined as the subset of $\mathcal{S}^{\prime}(\Omega)$ for which the seminorm (2.6) for $t=s$ is bounded; in this case, elements in $\dot{H}^{s}$ are uniquely defined in $\mathcal{S}^{\prime}$ modulo polynomials; see for instance [62, Remark 3, Section 5.1].

Sharafutdinov showed [57, Theorem 2.1] that the Radon transform can be extended as a bijective isometry between $H_{t}^{s}(\mathbb{R}^{d})$ and $H^{s+(d-1)/2}_{t+(d-1)/2}(\mathbb{P}_{d})$ – i.e. when $t>-\frac{d}{2}$

(2.9)

$$\|f\|_{H_{t}^{s}(\mathbb{R}^{d})}=\|Rf\|_{H^{s+(d-1)/2}_{t+(d-1)/2}(\mathbb{P}_{d})}.$$

The special case $t=s=0$ was observed by Reshetnyak, recorded in [24, Section 1.1.5] and also in [25, Chapter 1, Theorem 4.1]. Whenever $t\in(-d/2,-d/2+1)$, the $H_{s+(d-1)/2}^{t+(d-1)/2}(\mathbb{P}_{d})$-norm is stronger than the topology of $\mathcal{S}^{\prime}(\mathbb{P}_{d})$, thus the continuous extension of the Radon transform applied to any function $f\in H_{t}^{s}(\mathbb{R}^{d})$ is unambiguously defined as an element of $\mathcal{S}^{\prime}(\mathbb{P}_{d})$ independently of $t\in(-d/2,-d/2+1)$ and $s\in\mathbb{R}$. Therefore in the remainder of this paper we refer to this extension simply as the Radon transform.

In Sections 5 and 6 we will make use of the weighted homogeneous Sobolev norm of order $-1$ in dimension 1. Given $\sigma,\mu,\nu\in\mathscr{P}_{2}(\mathbb{R})$, the $\dot{H}^{-1}(\sigma)$-norm of $\mu-\nu$ is defined by

(2.10)

$$\|\mu-\nu\|_{\dot{H}^{-1}(\sigma)}\mathrel{\mathop{\mathchar 58\relax}}=\sup\left\{\int_{\mathbb{R}}\varphi\,d(\mu-\nu)\mathrel{\mathop{\mathchar 58\relax}}\;\varphi\in\mathcal{S}(\mathbb{R})\text{ and }\|\varphi\|_{\dot{H}^{1}(\sigma)}\leq 1\right\}.$$

Operators related to the Radon transform. Calculus using the Radon transform often involves $\Lambda_{d}$, which is defined via

(2.11)

$$\Lambda_{d}=(-\partial_{r}^{2})^{\frac{d-1}{2}}.$$

When $d$ is even, the fractional power of the 1-dimensional Laplace operator is defined using the Hilbert transform; for precise definitions, see Definition A.7 in the appendix. Observe that $\Lambda_{d}$ is well-defined as an operator from $\mathcal{S}(\mathbb{P}_{d})$ to itself, and can be extended as a bounded operator from $H^{r}_{t}(\mathbb{P}_{d})$ to $H^{r-(d-1)}_{t-(d-1)}(\mathbb{P}_{d})$ for $t>d-3/2$; see Remark A.8.

The operators $\Lambda_{d}$ can be understood by their interaction with the Fourier transform, namely

(2.12)

$$\begin{split}(\mathcal{F}_{1}\Lambda_{d}\mathfrak{g})(\theta,\zeta)&=|\zeta|^{d-1}\mathcal{F}_{1}\mathfrak{g}(\theta,\zeta).\end{split}$$

We also use fractional powers of the Laplace operator in $d$-dimensions, which can be defined via the Fourier transform by

$$(\mathcal{F}_{d}(-\Delta)^{s}f)(\xi)=|\xi|^{2s}\mathcal{F}_{d}f(\xi).$$

Again, observe that $(-\Delta)^{s}$ is well-defined as an operator from $\mathcal{S}(\mathbb{R}^{d})$ to itself, and can be extended as a bounded operator from $H^{r}_{t}(\mathbb{R}^{d})$ to $H^{r-2s}_{t-2s}(\mathbb{R}^{d})$ when $t-2s>-\frac{d}{2}$. Rigorous definition of $(-\Delta)^{s}$ for fractional powers $s$ without relying on the Fourier transform can be found in [25, Chapter 7.6], which we have included in Proposition A.5 in the appendix for completeness. The inversion formulae are expressed using these operators: Setting $c_{d}=(4\pi)^{(d-1)/2}\Gamma(d/2)/\Gamma(1/2)$, for each $f\in\mathcal{S}(\mathbb{R}^{d})$ and $\mathfrak{g}\in\mathcal{S}(\mathbb{P}_{d})$ we have

$$c_{d}f=R^{\ast}\Lambda_{d}Rf=(-\Delta)^{(d-1)/2}R^{\ast}Rf\quad\text{ and }\quad c_{d}\mathfrak{g}(\theta,r)=RR^{\ast}(\Lambda_{d}\mathfrak{g}).$$

See Proposition A.9 for further details.

Whenever $t\in(-\frac{d}{2},-\frac{d}{2}+1)$, we have $H_{t}^{s}(\mathbb{R}^{d})\subset\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and $H_{t+(d-1)/2}^{s+(d-1)/2}(\mathbb{P}_{d})\subset\mathcal{S}^{\prime}(\mathbb{P}_{d})$. In this case, straightforward calculations using the inversion formula and the Fourier transform imply

(2.13)

$$J(c_{d}\varphi)=RJ(\Lambda_{d}R\varphi)\text{ for }J\in H_{t}^{s}(\mathbb{R}^{d})\text{ and }\varphi\in\mathcal{S}(\mathbb{R}^{d}).$$

For each of the domains $\Omega=\mathbb{R}^{d},\mathbb{P}_{d},\mathbb{R}$, we will write

(2.14)

$$\langle J,\varphi\rangle_{\Omega}\mathrel{\mathop{\mathchar 58\relax}}=J(\varphi)\text{ for }J\in\mathcal{S}^{\prime}(\Omega)\text{ and }\varphi\in\mathcal{S}(\Omega).$$

In this section we establish some basic properties of the SW distance and the SW space.

Let $\mathscr{P}_{2}(\mathbb{P}_{d})$ be the set of Borel probability measures in the Radon domain with bounded second moment

(2.15)

$$\begin{split}\mathscr{P}_{2}(\mathbb{P}_{d})\mathrel{\mathop{\mathchar 58\relax}}=&\left\{\widehat{\mu}\in\mathscr{P}_{2}(\mathbb{S}^{d-1}\times\mathbb{R})\mathrel{\mathop{\mathchar 58\relax}}\pi^{1}_{\#}\widehat{\mu}=\operatorname{Vol}_{\mathbb{S}^{d-1}},\;\;\widehat{\mu}^{\theta}\in\mathscr{P}_{2}(\mathbb{R}),\phantom{\int_{\mathbb{R}}}\right.\\ &\left.\;\;d\widehat{\mu}^{-\theta}(-r)=d\widehat{\mu}^{\theta}(r),\quad\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}}r^{2}\,d\widehat{\mu}^{\theta}(r)\,d\theta<\infty\right\}\end{split}$$

where $\pi^{1}$ is the projection in the first variable. In other words, $(\widehat{\mu}^{\theta})_{\theta\in\mathbb{S}^{d-1}}$ is a family of measures in $\mathscr{P}_{2}(\mathbb{R})$ parametrized by $\theta\in\mathbb{S}^{d-1}$ additionally satisfying the evenness condition. Observe that for each $\theta\in\mathbb{S}^{d-1}$, we can choose an orthonormal frame $\{\theta^{1},\cdots,\theta^{d}\}$ with $\theta^{1}=\theta$ and $\theta^{i}\in\mathbb{S}^{d-1}$ for $i=1,\cdots,d$. As $|x|^{2}=\sum_{i=1}^{d}|x\cdot\theta^{i}|^{2}$, we have

$$\int_{\mathbb{S}^{d-1}}|x|^{2}\,d\operatorname{Vol}_{\mathbb{S}^{d-1}}(\theta)=\sum_{i=1}^{d}\int_{\mathbb{S}^{d-1}}|x\cdot\theta^{i}|^{2}\,d\operatorname{Vol}_{\mathbb{S}^{d-1}}(\theta^{i})=d\int_{\mathbb{S}^{d-1}}|x\cdot\theta|^{2}\,d\operatorname{Vol}_{\mathbb{S}^{d-1}}(\theta).$$

Thus

$$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}}r^{2}d\widehat{\mu}^{\theta}(r)\,d\theta=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}^{d}}|x\cdot\theta|^{2}\,d\mu(x)\,d\theta=\frac{1}{d}\int_{\mathbb{R}^{d}}|x|^{2}\,d\mu(x).$$

Equivalently,

(2.16)

$$SW^{2}(\mu,\delta_{0})=\frac{1}{d}W^{2}(\mu,\delta_{0})\text{ for any }\mu\in\mathscr{P}_{2}(\mathbb{R}^{d}).$$

Thus the finite second moment condition in the Euclidean and Radon domain coincide. Hence, $\mu\in\mathscr{P}_{2}(\mathbb{R}^{d})$ if and only if $\widehat{\mu}\in\mathscr{P}_{2}(\mathbb{P}_{d})$.