# Stereographic Spherical Sliced Wasserstein Distances

Huy Tran Denotes equal contribution. Department of Computer Science,
Vanderbilt University, Nashville, TN  Yikun Bai11footnotemark: 1 Department of
Computer Science, Vanderbilt University, Nashville, TN  Abihith
Kothapalli11footnotemark: 1 Ashkan Shahbazi  Department of Computer Science,
Vanderbilt University, Nashville, TN  Xinran Liu  Department of Computer
Science, Vanderbilt University, Nashville, TN
Rocio Diaz Martin  Department of Computer Science, Vanderbilt University,
Nashville, TN  Soheil Kolouri  Department of Computer Science, Vanderbilt
University, Nashville, TN

###### Abstract

Comparing spherical probability distributions is of great interest in various
fields, including geology, medical domains, computer vision, and deep
representation learning. The utility of optimal transport-based distances,
such as the Wasserstein distance, for comparing probability measures has
spurred active research in developing computationally efficient variations of
these distances for spherical probability measures. This paper introduces a
high-speed and highly parallelizable distance for comparing spherical measures
using the stereographic projection and the generalized Radon transform, which
we refer to as the Stereographic Spherical Sliced Wasserstein (S3W) distance.
We carefully address the distance distortion caused by the stereographic
projection and provide an extensive theoretical analysis of our proposed
metric and its rotationally invariant variation. Finally, we evaluate the
performance of the proposed metrics and compare them with recent baselines in
terms of both speed and accuracy through a wide range of numerical studies,
including gradient flows and self-supervised learning.

##  1 Introduction

Applications involving distributions defined on a hypersphere are remarkably
diverse, highlighting the importance of spherical geometries across various
disciplines. These applications include: 1) mapping the distribution of
geographic or geological features on celestial bodies, such as stars and
planets [39, 8, 60], 2) magnetoencephalography (MEG) imaging [75] in medical
domains, 3) spherical image representations and $360\degree$ images [13, 38],
such as omnidirectional images in computer vision [40], 4) texture mapping in
computer graphics [24, 21], and more recently, 5) deep representation
learning, where the latent representation is often mapped to a bounded space,
commonly a sphere, where cosine similarity is utilized for effective
representation learning [11, 76].

The analysis of distributions on hyperspheres is traditionally approached
through directional statistics, also referred to as circular/spherical
statistics [37, 52, 50, 61]. This specialized field is dedicated to the
statistical analysis of directions, orientations, and rotations. More
recently, with the growing application of optimal transport theory [74, 62] in
machine learning, due in part to its favorable statistical, geometrical, and
topological properties, there has been an increasing interest in using optimal
transport to compare spherical probability measures [14, 32].

One of the main bottlenecks in optimal transport theory is its high
computational cost, generally of cubic complexity. This has sparked extensive
research to develop faster solvers [15, 71, 10] or computationally superior
equivalent distances [67, 6, 43]. Notably, sliced variations of optimal
transport distances, such as sliced Wasserstein distances [67] and their
various extensions [43, 49, 55], have emerged as effective solutions. These
methods use integral geometry and the Radon transform [34] to represent high-
dimensional distributions via a set of their one-dimensional marginals. By
doing so, they take advantage of more efficient optimal transport solvers
designed for one-dimensional probability measures, offering a pragmatic
approach to comparing probability measures. Owing to their computational
efficiency and implementation simplicity, these methods have been recently
adapted for spherical measures, leading to the development of spherical sliced
optimal transport methods [5, 65].

The main challenge in developing spherical sliced optimal transport methods
lies in extending the classical Radon transform to its spherical counterparts.
In the context of sliced optimal transport, such an extension must: 1) map
probability measures on the hypersphere to a family of probability measures on
one-dimensional domains (e.g., $\mathbb{R}$ or $\mathbb{S}^{1}$), and 2) be
injective so that a sliced distance can be defined (otherwise, one will obtain
only a pseudo-metric). These requirements rule out many of the existing
extensions of the Radon transform to the sphere, e.g., the classic Funk-Radon
transform, which takes integrals along all great circles [34, 64]. Recently,
[65] proposed using two such spherical extensions for the Radon transform,
namely, the vertical slice transform [29] and a normalized version of the
semicircle transform [31], to define sliced optimal transport on the sphere.
Notably, the semicircle transform was also used in [5] to define a spherical
sliced Wasserstein discrepancy for empirical probability measures.

The recent works on sliced spherical optimal transport [5, 65] map a
distribution defined on a hypersphere into its marginal distributions on a
unit circle, thus requiring circular optimal transport to compare these
marginals. Importantly, the calculation of optimal transport between two one-
dimensional measures defined on a circle is more expensive (requiring an
additional binary search) than when the measures are defined on the real line
[53, 36]. Motivated by this observation, in this paper, we explore a spherical
Radon transform aimed at converting a spherical distribution into one-
dimensional marginals along the real line. We utilize the stereographic
projection, which is a smooth bijection, in composition with an injective
(nonlinear) map to transform the hypersphere into a hyperplane, where we then
apply the classic Radon transform. The key advantage of the injective map is
its ability to manage the distortion of distances introduced by the
stereographic projection. We support our proposed method with detailed
theoretical analysis and comprehensive numerical studies.

Contributions. Our specific contributions are as follows:

  * •

Introducing a computationally efficient transport distance, Stereographic
Spherical Sliced Wasserstein distance (S3W), for spherical probability
measures.

  * •

Providing a rotationally invariant variation of the proposed distance,
Rotationally Invariant Stereographic Spherical Sliced Wasserstein distance
(RI-S3W).

  * •

Offering theoretical analysis of the proposed distances.

  * •

Demonstrating the performance, both in terms of speed and accuracy, of the
proposed distances in diverse applications, including gradient flows on the
sphere, representation learning with Sliced-Wasserstein Auto-Encoders (SWAEs),
spherical density estimation via normalizing flows, sliced-Wasserstein
variational inference on the sphere, and self-supervised learning.

##  2 Background

###  2.1 Stereographic Projection

Stereographic projection is a mathematical technique to map points on a sphere
onto a plane. Originating in Greek astronomy, it projects points from a
sphere, typically from one of its poles, onto a plane tangential to the
opposite pole or onto the “equator plane.” This projection is conformal,
preserving angles and shapes locally, making it significant in fields like
cartography, complex analysis, and computer graphics. It elegantly translates
spherical geometry into planar terms, maintaining the intricate relationships
between points and angles found on the sphere’s surface.

Figure 1: Depiction of stereographic projection from
$\mathbb{S}^{2}\backslash\\{s_{n}\\}$ to $\mathbb{R}^{2}$ (a), the
stereographic Radon transform integration surfaces on the sphere, i.e., the
level sets of $\langle\phi(x),\theta\rangle$ for a fixed
$\theta\in\mathbb{R}^{d}$ (b), and the generalized stereographic Radon
transform integration surfaces on the sphere, i.e. the level sets of $\langle
h\circ\phi(x),\theta\rangle$ for a fixed $\theta\in\mathbb{R}^{d^{\prime}}$.

Let $\mathbb{S}^{d}$ denote the $d$-dimensional sphere in $\mathbb{R}^{d+1}$
defined as $\mathbb{S}^{d}:=\\{s\in\mathbb{R}^{d+1}:\|s\|_{2}=1\\}$. The
stereographic projection
$\phi:\mathbb{S}^{d}\setminus\\{s_{n}\\}\to\mathbb{R}^{d}$ maps a point
$s\in\mathbb{S}^{d}$ (excluding the “North Pole," $s_{n}=[0,\ldots,0,1]$) to a
point $x\in\mathbb{R}^{d}$ by the formula:

$[x]_{i}=\frac{2[s]_{i}}{1-[s]_{d+1}},\quad\text{for }i=1,2,\ldots,d.$ (1)

This projection is a bijective and smooth mapping between
$\mathbb{S}^{d}\setminus\\{s_{n}\\}$ and $\mathbb{R}^{d}$, i.e., the
hyperplane tangent to the sphere at the south pole, $s_{d+1}=-1$, which is
commonly used as a way to visualize spherical geometries in an Euclidean
space.

###  2.2 Radon Transform

The Radon transform is a fundamental tool in integral geometry, widely used
for image reconstruction, particularly in computed tomography (CT). It
transforms a function defined in the $d$-dimensional Euclidean space,
$\mathbb{R}^{d}$, into an infinite set of its one-dimensional slices, i.e.,
its integrals over hyperplanes parametrized with unit vectors
$\theta\in\mathbb{S}^{d-1}$. Radon transform is instrumental in medical
imaging and has recently attracted ample attention from the machine learning
community for its favorable characteristics for measuring distances between
probability measures [6, 44, 43] and, more generally, positive measures [3,
72], and more recently on theoretical analysis of deep neural networks [59,
73].

More formally, the classic Radon transform, denoted by $\mathcal{R}$, maps a
function $f$ in the Lebesgue space $L^{1}(\mathbb{R}^{d})$ — the space of
integrable functions on $\mathbb{R}^{d}$ — to its integrals over the
hyperplanes of $\mathbb{R}^{d}$. Formally, it is defined as:

$\displaystyle\mathcal{R}f(t,\theta):=\int_{\mathbb{R}^{d}}f(x)\delta(t-\langle
x,\theta\rangle)dx,$ (2)

for $(t,\theta)\in\mathbb{R}\times\mathbb{S}^{d-1}$, where $\delta(\cdot)$ is
the one-dimensional Dirac delta function, and $\langle\cdot,\cdot\rangle$
denotes the Euclidean inner product. Here, $\mathcal{R}$ maps from
$L^{1}(\mathbb{R}^{d})$ to $L^{1}(\mathbb{R}\times\mathbb{S}^{d-1})$. Each
hyperplane, $H(t,\theta)=\\{x\in\mathbb{R}^{d}\ |\ \langle
x,\theta\rangle=t\\}$, corresponds to a level set of a function
$g:\mathbb{R}^{d}\times\mathbb{S}^{d-1}\rightarrow\mathbb{R}$, defined as
$g(x,\theta)=\langle x,\theta\rangle$. For a fixed $\theta$, the set of all
integrals over hyperplanes orthogonal to $\theta$ yields the continuous
function $\mathcal{R}f(\cdot,\theta):\mathbb{R}\rightarrow\mathbb{R}$,
representing a projection or slice of $f$.

Notably, the Radon transform $\mathcal{R}:L^{1}(\mathbb{R}^{d})\rightarrow
L^{1}(\mathbb{R}\times\mathbb{S}^{d-1})$ is a linear bijection with a closed-
form inversion formula. See the Appendix for details.

###  2.3 Generalized Radon Transform

The Generalized Radon Transform (GRT) extends the foundational concept of the
classic Radon transform, as introduced by Radon in [68], from integration over
hyperplanes in $\mathbb{R}^{d}$ to integration over more complex structures,
namely hypersurfaces or $(d-1)$-dimensional manifolds. This broader scope of
the GRT has been developed and explored in various works [4, 19, 23, 27, 48,
35], in many applications from impedance and acoustic tomography [48] to
machine learning [43].

The GRT of a function $f\in L^{1}(\mathbb{R}^{d})$ involves the integration of
$f$ over hypersurfaces in $\mathbb{R}^{d}$. These hypersurfaces are defined as
the level sets of a ‘defining function’ [43],
$g:\mathbb{R}^{d}\times(\mathbb{R}^{d^{\prime}}\setminus\\{0\\})\to\mathbb{R}$,
characterized by $H_{t,\theta}=\\{x\in\mathbb{R}^{d}|\,g(x,\theta)=t\\}$. The
GRT of $f$, denoted as $\mathcal{G}f$, is formally given by:

$\mathcal{G}f(t,\theta):=\int_{\mathbb{R}^{d}}f(x)\delta(t-g(x,\theta))\,dx,$
(3)

where $\delta$ represents the Dirac delta function, enabling the integration
over the specific level sets defined by $g$. Note that $g(x,\theta)=\langle
x,\theta\rangle$ recovers the classic Radon transform.

The injectivity of the GRT is essential for defining distances between
measures through their generalized slices. The choice of $g$ identifies
whether GRT is injective or not. [43] enumerate a set of necessary conditions
on $g$ to construct a bijective GRT. However, these conditions are not
sufficient and are limited in practical utility, as they do not provide
guidance on crafting specific defining functions that would ensure
injectivity.

To achieve a practical injective GRT, [12] recently introduced a variation of
the GRT. They posited that by setting $g(x,\theta)=\langle h(x),\theta\rangle$
for an injective function
$h:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d^{\prime}}$, one effectively applies
the classic Radon transform, which is bijective, to the image of $h$. This
approach leads to an injective GRT, thus resolving the issue of invertibility.
For consistency of notation with [12], we denote this variation of the GRT as:

$\displaystyle\mathcal{H}f(t,\theta):=\int_{\mathbb{R}^{d}}f(x)\delta(t-\langle
h(x),\theta\rangle)\,dx,$ (4)

where $\mathcal{H}:L^{1}(\mathbb{R}^{d})\rightarrow
L^{1}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$. Importantly, a neural
network can effectively parametrize the injective function $h$. For instance,
$h$ could be a normalizing flow [42]. Alternatively, to avoid the complexities
associated with normalizing flows, one can define $h(x)$ as
$[x^{T},\rho(x)^{T}]^{T}$, where $x^{T}$ and $\rho(x)^{T}$ are transposed
vectors that are concatenated. Here,
$\rho:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d^{\prime}-1}$ is any nonlinear
function parametrized as a neural network of choice [12]. In our experiments,
we will use $\mathcal{H}$ as the default GRT due to its injectivity.

###  2.4 GRT of Radon Measures

Skipping much of the theoretical details (refer to the appendix), for a Radon
measure $\mu\in\mathcal{M}(\mathbb{R}^{d})$, its Generalized Radon Transform
$\mathcal{G}(\mu)=\nu$ with respect to the defining function $g$, is defined
as the measure $\nu\in\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$
such that for each $\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$,

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\nu(t,\theta)=\int_{\mathbb{R}^{d}}(\mathcal{G}^{*}(\psi))(x)\,d\mu(x),$
(5)

where $\mathcal{G^{*}}$ is the dual operator (aka adjoint operator), which for
any $\psi\in L^{\infty}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$, is
defined as

$\displaystyle\mathcal{G}^{*}(\psi)(x)=\int_{\mathbb{S}^{d^{\prime}-1}}\psi(g(x,\theta),\theta)\,d\sigma_{d^{\prime}}(\theta)\quad\forall
x\in\mathbb{R}^{d},$ (6)

where $\sigma_{d^{\prime}}$ is the uniform probability measure defined in
$\mathbb{S}^{d^{\prime}-1}$. Importantly, the dual GRT operator satisfies,

$\mathcal{G}(\mu)(\psi)=\mu(\mathcal{G}^{*}(\psi)).$

Lastly, with a slight abuse of notation, we denote the corresponding slice for
$\theta\in\mathbb{S}^{d^{\prime}-1}$ as
$\mathcal{G}(\mu)_{\theta}=g(\cdot,\theta)_{\\#}\mu\in\mathcal{M}(\mathbb{R})$.
Note that $f_{\\#}\mu$ denotes the pushforward of measure $\mu$ with respect
to $f$, defined as $f_{\\#}\mu(A)=\mu(f^{-1}(A))$. When $\mu$ is a positive or
a probability measure in $\mathbb{R}^{d}$, then $\mathcal{G}(\mu)_{\theta}$ is
a positive/probability measure in $\mathbb{R}$.

###  2.5 Wasserstein and Sliced Wasserstein Distances

Let $M$ denote a Riemannian manifold endowed with the distance
$d(\cdot,\cdot):M\times M\rightarrow\mathbb{R}_{+}$. For $1\leq p<\infty$, let
$\mu,\nu\in\mathcal{P}_{p}(M):=\\{\mu\in\mathcal{P}(M)|\,\int_{M}d^{p}(x,x_{0})\,d\mu(x)<\infty~{}\text{for
some}~{}x_{0}\in M\\}$ be two probability measures defined on manifold $M$
with a finite $p$-th moment. Then, the optimal transport (OT) problem [74]
seeks to transport the mass in $\mu$ into $\nu$ such that the expected
traversed distance is minimized. This leads to the $p$-Wasserstein distance:

$\displaystyle
W_{p}^{p}(\mu,\nu):=\inf_{\gamma\in\Gamma(\mu,\nu)}\int_{M\times
M}d^{p}(x,y)d\gamma(x,y),$ (7)

where $\Gamma(\mu,\nu)$ denotes the joint probability measures
$\gamma\in\mathcal{P}(M\times M)$ with marginals $\mu$ and $\nu$.
Unfortunately, for discrete probability measures with $N$ particles, solving
(7) generally has a $\mathcal{O}(N^{3}\log N)$ complexity. However, for
$\mu,\nu\in\mathcal{P}_{p}(\mathbb{R})$ the problem can be solved in
$\mathcal{O}(N\log N)$:

$\displaystyle
W_{p}^{p}(\mu,\nu)=\int_{0}^{1}\|F_{\mu}^{-1}(t)-F_{\nu}^{-1}(t)\|^{p}dt$ (8)

where $F^{-1}_{\mu}$ and $F^{-1}_{\nu}$ are the quantile functions of $\mu$
and $\nu$. Notably, similar efficient solvers are developed for when
$\mu,\nu\in\mathcal{P}_{p}(\mathbb{S}^{1})$ [18, 36, 5]. For an injective
generalized Radon transform $\mathcal{G}$, such that for
$\mu\in\mathcal{P}_{p}(M)$ we have
$\mathcal{G}(\mu)_{\theta}\in\mathcal{P}_{p}(\mathbb{R})$, the generalized
Sliced-Wasserstein distance [67, 43] can be defined as:

$\displaystyle
SW_{\mathcal{G},p}^{p}(\mu,\nu):=\int_{\mathbb{S}^{d^{\prime}-1}}W_{p}^{p}(\mathcal{G}(\mu)_{\theta},\mathcal{G}(\nu)_{\theta})d\sigma_{d^{\prime}}(\theta),$
(9)

where $\sigma_{d^{\prime}}\in\mathcal{P}(\mathbb{S}^{d^{\prime}-1})$ is a
probability measure possessing a non-zero density on the sphere
$\mathbb{S}^{d^{\prime}-1}$, often simply the uniform measure. Recently, [5]
introduced the concept of Spherical Sliced Wasserstein distance, which
involves projecting spherical measures onto great circles, resulting in
$\mathcal{G}(\mu)_{\theta}\in\mathcal{P}_{p}(\mathbb{S}^{1})$, and utilizing
circular OT to measure distances between these slices. Notably, circular OT
still necessitates solving an optimization problem to register cumulative
distribution functions on a circle (i.e., finding an optimal cut). This
results in a slower solver compared to OT on $\mathbb{R}$.

##  3 Method

Based on the extended definition of GRT for probability measures in 2.4, here
we formally introduce the “Stereographic Spherical Radon Transform” and the
corresponding sliced Wasserstein distance, “Stereographic Spherical Sliced
Wasserstein Distance,” for spherical probability measures.

###  3.1 Stereographic Spherical Radon Transform

Let $\mu\in\mathcal{M}(\mathbb{S}^{d})$ denote a Radon measure defined on
$\mathbb{S}^{d}$ that does not assign mass to the North Pole, i.e.,
$\mu(\\{s_{n}\\})=0$. We denote the stereographic projection as
$\phi:\mathbb{S}^{d}\setminus\\{s_{n}\\}\to\mathbb{R}^{d}$, which is a
bijection, and we have that $\phi_{\\#}\mu$ is a Radon measure defined in
$\mathbb{R}^{d}$. The Stereographic Spherical Radon transform, of $\mu$ is
defined as

$\displaystyle\mathcal{S}_{\mathcal{R}}(\mu):=\mathcal{R}(\phi_{\\#}\mu)\in\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d-1}),$
(10)

and we also define its generalized version as

$\displaystyle\mathcal{S}_{\mathcal{G}}(\mu):=\mathcal{G}(\phi_{\\#}\mu)\in\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}).$
(11)

Similarly we define
$\mathcal{S}_{\mathcal{H}}(\mu):=\mathcal{H}(\phi_{\\#}\mu)$, which provides
an invertible transformation from $\mathcal{M}(\mathbb{S}^{d})$ to
$\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.

######  Proposition 1.

For $\mu\in\mathcal{M}(\mathbb{S}^{d})$ that does not give mass to the North
Pole $\\{s_{n}\\}$ the Stereographic Spherical Radon transforms
$\mathcal{S}_{\mathcal{G}}$ and $\mathcal{S}_{\mathcal{H}}$ satisfy the
following properties:

  1. (1)

$\mathcal{S}_{\mathcal{G}}(\mu),\mathcal{S}_{\mathcal{H}}(\mu)\in\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.
In addition $\mathcal{S}_{\mathcal{G}}$ and $\mathcal{S}_{\mathcal{H}}$
preserves mass, and if $\mu$ is a positive measure, then
$\mathcal{S}_{\mathcal{G}}(\mu),\mathcal{S}_{\mathcal{H}}(\mu)$ are positive
measures too. Finally, if
$\mu\in\mathcal{P}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$, then
$\mathcal{S}_{\mathcal{G}}(\mu),\mathcal{S}_{\mathcal{H}}(\mu)$ are
probability measures defined on $\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}$.

  2. (2)

The disintegration theorem gives a unique
$\mathcal{S}_{\mathcal{G}}(\mu)-$a.s. set of measures
$(\mathcal{S}_{\mathcal{G}}(\mu)_{\theta})_{\theta\in\mathbb{S}^{d^{\prime}-1}}\subset\mathcal{M}(\mathbb{R})$,
and a unique $\mathcal{S}_{\mathcal{H}}(\mu)-$a.s. set of measures
$(\mathcal{S}_{\mathcal{H}}(\mu)_{\theta})_{\theta\in\mathbb{S}^{d^{\prime}-1}}\subset\mathcal{M}(\mathbb{R})$
such that for any $\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$,

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{S}_{\mathcal{G}}(\mu)(t,\theta)=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\,d\mathcal{S}_{\mathcal{G}}(\mu)_{\theta}(t)\,d\sigma_{d^{\prime}}(\theta),\qquad$
(12)

and similarly for $\mathcal{S}_{\mathcal{H}}$. Then, it holds that

$\mathcal{S}_{\mathcal{G}}(\mu)_{\theta}=(g(\cdot,\theta)\circ\phi)_{\\#}\mu,$
(13)

and the same holds for $\mathcal{S}_{\mathcal{H}}$.

  3. (3)

$\mathcal{S}_{\mathcal{H}}$ is invertible.

The proof of this proposition is included in the appendix Section D. Being
equipped with the proposed stereographic spherical Radon transform, we are now
ready to define our proposed distance.

###  3.2 Stereographic Spherical Sliced Wasserstein

Let $\mu,\nu\in\mathcal{P}_{p}(\mathbb{S}^{d})$ denote two probability
measures on the unit sphere in $\mathbb{R}^{d+1}$. We introduce the formal
definition of the Stereographic Spherical Sliced Wasserstein ($S3W$) distances
as follows:

$\displaystyle S3W_{\mathcal{G},p}^{p}(\mu,\nu)$
$\displaystyle:=\int_{\mathbb{S}^{d^{\prime}-1}}W_{p}^{p}(\mathcal{S}_{\mathcal{G}}(\mu)_{\theta},\mathcal{S}_{\mathcal{G}}(\nu)_{\theta})\,d\sigma_{d^{\prime}}(\theta).$

In this context,
$\sigma_{d^{\prime}}\in\mathcal{P}(\mathbb{S}^{d^{\prime}-1})$ typically
represents a probability measure possessing a non-zero density on the sphere
$\mathbb{S}^{d^{\prime}-1}\subset\mathbb{R}^{d^{\prime}}$. However, for the
sake of simplicity and in line with common practice, we opt to consider the
uniform measure on $\mathbb{S}^{d^{\prime}-1}$. Similarly, we can define
$S3W_{\mathcal{H},p}^{p}$ by integrating over
$W_{p}^{p}(\mathcal{S}_{\mathcal{H}}(\mu)_{\theta},\mathcal{S}_{\mathcal{H}}(\nu)_{\theta})$.

######  Theorem 2.

The proposed $S3W_{\mathcal{G},p}(\cdot,\cdot)$ and
$S3W_{\mathcal{H},p}(\cdot,\cdot)$ are well-defined. Furthermore,
$S3W_{\mathcal{G},p}(\cdot,\cdot)$ is generally a pseudo-metric in
$\mathcal{P}_{p}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$, i.e., it is non-
negative, symmetric and satisfies triangular inequality. In addition,
$S3W_{\mathcal{H},p}(\cdot,\cdot)$ defines a metric in
$\mathcal{P}_{p}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$.

The proof is in the appendix Section D.

###  3.3 Distance Distortion

The Stereographic Projection, while being conformal (i.e., preserving angles),
severely distorts distances. To demonstrate the extent of this distortion,
consider points $s=(\epsilon,0,\ldots,0,\sqrt{1-\epsilon^{2}})$ and
$s^{\prime}=(-\epsilon,0,\ldots,0,\sqrt{1-\epsilon^{2}})$. Then, as
$\epsilon\rightarrow 0$, we have $\arccos(\langle s,s^{\prime}\rangle)\to 0$,
while $\|\phi(s)-\phi(s^{\prime})\|\to\infty$! This distortion implies that
the transportation cost after the stereographic projection would be
significantly different from the transportation cost on the sphere, making the
naive application of this projection with optimal transport unsuitable.

In this section, we aim to construct an injective function $h$ that closely
approximates the arclength on the sphere $\mathbb{S}^{d}$ with the Euclidean
distance in the embedded space. Specifically, we seek to satisfy:

$\|h(\phi(s))-h(\phi(s^{\prime}))\|\approx\arccos(\langle
s,s^{\prime}\rangle),~{}\forall s,s^{\prime}\in\mathbb{S}^{d}.$

We introduce two variants of $h$: an analytical function $h_{1}(\cdot)$ and a
neural network-based learnable function $h_{NN}$, both mapping to a nearly-
isometric embedding from $\phi(s)$.

We start by defining the analytical function:

$\displaystyle
h_{1}(x):=\arccos\left(\frac{\|x\|^{2}-1}{\|x\|^{2}+1}\right)\frac{x}{\|x\|},\quad\forall
x\in\mathbb{R}^{d}$ (14)

and provide the following proposition.

######  Proposition 3.

For $s_{0},s,s^{\prime}\in\mathbb{S}^{d}$, where $s_{0}$ denotes the South
Pole, the stereographic projection $\phi$, and $h_{1}$ as defined in Eq. (14)
we have:

  * •

$h_{1}(\phi(s))=\angle(s,s_{0})\frac{s[1:d]}{\|s[1:d]\|}$. Thus, we have

$\|h_{1}(\phi(s))-h_{1}(\phi(s^{\prime}))\|\leq 2\pi$

  * •

The following inequality holds,

$\displaystyle\arccos(\langle s,$ $\displaystyle
s^{\prime}\rangle)\leq\|h_{1}(\phi(s))-h_{1}(\phi(s^{\prime}))\|+\epsilon(s,s^{\prime})$

where the equality holds when $s$, $s^{\prime}$, and $s_{0}$ are in the same
great circle, and $\epsilon(s,s^{\prime})\to 0$ as
$d_{\mathbb{S}^{d}}(s,s^{\prime})\to 0$.

The proof is in the appendix Section G.

Next, we consider training a neural network to obtain a nearly-isometric
Euclidean embedding. To do so, we define
$h(x):=[h^{T}_{1}(x)/C,\rho^{T}(x)]^{T}$ where
$\rho:\mathbb{R}^{d}\to\mathbb{R}^{d^{\prime}-d}$ is a neural network, and
$C\geq 2\pi$ a constant. We first note that such
$h:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d^{\prime}}$ is injective, then train
$\rho$ by minimizing:

$\displaystyle\mathcal{L}(\rho)=\mathbb{E}_{s,s^{\prime}}\left[(\arccos(\langle
s,s^{\prime}\rangle)-\|h(\phi(s))-h(\phi(s^{\prime}))\|)^{2}\right],$ (15)

where $s$ and $s^{\prime}$ are sampled according to the uniform distribution
in the sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, i.e.,
$(s,s^{\prime})\sim\sigma_{d+1}\times\sigma_{d+1}$. Figure 2 illustrates the
arclength versus the distance in the embedding for random pairs of samples
$s,s^{\prime}\in\mathbb{S}^{d}$ with respect to the various scenarios proposed
in this section. It is evident that an injective function $h$ parameterized
with a neural network can yield a nearly-isometric embedding.

To avoid any potential confusion, we emphasize that irrespective of the
specific choice of $h$, as long as it maintains injectivity,
$S3W_{\mathcal{H},p}$ remains a valid metric in
$\mathcal{P}_{p}(\mathbb{S}^{d})$. The discussion in this section is, however,
significant, particularly when we aim to ensure that the transportation cost
in the embedding closely resembles the spherical distance.

Figure 2:  Spherical distance (i.e., the arclength) versus the distance after
stereographic projection, where CC denotes Pearson’s correlation coefficient.
From left to right, when the injective function $h=id$, and the distance is
$\|\phi(s)-\phi(s^{\prime})\|$ (a), when $h(x)=h_{1}(x)$ (See Eq. (14)) and
the distance is $\|h(\phi(s))-h(\phi(s^{\prime}))\|$ (b), when $h(x)=h_{1}(x)$
and the distance is
$\min(\|h(\phi(s))-h(\phi(s^{\prime}))\|,2\pi-\|h(\phi(s))-h(\phi(s^{\prime}))\|)$
(c), and finally when $h(x)=[h_{1}(x)/C,\rho(x)]$ where $\rho(x)$ is a trained
neural network minimizing Eq. 15 and $C\geq 2\pi$ (d).

###  3.4 Rotationally Invariant Extension of $S3W$

Rotational symmetry is a significant property when dealing with probability
metrics on the sphere. This symmetry implies that the probability distribution
remains unchanged under rotations, making it a key consideration in spherical
statistics and related applications. It simplifies calculations and enhances
our understanding of the underlying processes on a spherical surface. Notably,
the spherical OT leads to a rotationally symmetric metric on
$\mathcal{P}_{p}(\mathbb{S}^{d})$. However, our proposed Stereographic
Spherical Sliced Wasserstein ($S3W$) metric is not rotationally invariant.
Here, we propose a rotationally invariant variation of $S3W$, which leads to a
robust and easy-to-implement rotationally invariant metric.

Let $\mathrm{SO}(d+1)$ denote the special orthogonal group in
$\mathbb{R}^{d+1}$, and let $R\in\mathrm{SO}(d+1)$ denote a rotation matrix.
For $\mu\in\mathcal{P}_{p}(\mathbb{S}^{d})$ we denote the rotated measure as
$R_{\\#}\mu$. Given probability measures
$\mu,\nu\in\mathcal{P}_{p}(\mathbb{S}^{d})$, we define the rotationally
invariant extension of $S3W$ as:

$\displaystyle
RI\text{-}S3W_{\mathcal{G},p}(\mu,\nu):=\mathbb{E}_{R\sim\omega}[S3W_{\mathcal{G},p}(R_{\\#}\mu,R_{\\#}\nu)],$
(16)

where $\omega$ denotes the normalized Haar measure on $\mathrm{SO}(d+1)$
(i.e., the uniform probability measure which is left-rotation-invariant111The
Haar measure on a locally compact group $G$ is the unique, up to positive
constants, left-translation-invariant regular Borel measure on $G$. If
$G=\mathrm{SO}(d+1)$, the group translations are rotations. Here, we normalize
it to obtain a probability measure.). We similarly define
$\text{RI-S3W}_{\mathcal{H},p}$ using $\text{S3W}_{\mathcal{H},p}$.

######  Theorem 4.

$RI\text{-}S3W_{\mathcal{G},p}(\cdot,\cdot)$ and
$RI\text{-}S3W_{\mathcal{H},p}(\cdot,\cdot)$ are well-defined. Furthermore,
$RI\text{-}S3W_{\mathcal{G},p}(\cdot,\cdot)$ is generally a pseudo-metric in
$\mathcal{P}_{p}(\mathbb{S}^{d})$, i.e., it is non-negative, symmetric and
satisfies triangular inequality. In addition,
$RI\text{-}S3W_{\mathcal{H},p}(\cdot,\cdot)$ defines a metric in
$\mathcal{P}_{p}(\mathbb{S}^{d})$.

The proof is included in the appendix Section E.

##  4 Numerical Implementation Details

Stereographic Projection (SP). A key issue in implementing the SP for $S3W$
concerns the numerical handling of points near the North Pole. To ensure
numerical stability, we introduce an $\epsilon$-cap around the North Pole,
which serves to effectively establish an upper bound for the norm of the
projected points. Specifically, any point where $x_{d+1}>1-\epsilon$ is
initially mapped to the circle $x_{d+1}=1-\epsilon$, and then projected using
SP. We refer to this modified SP as $\phi_{\epsilon}$.

The $S3W$ distances. Let $\phi_{\epsilon}$ denote the Stereographic Projection
operator that excludes the $\epsilon$-cap around $s_{n}\in\mathbb{S}^{d}$, and
$h:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d^{\prime}}$ an injective defining
function, $L$ the number of slices, and
$\theta_{l}\in\mathbb{S}^{d^{\prime}-1}$ a slicing direction. For simplicity,
we initially consider two data distributions with an equal number of samples
and uniform mass distribution (the general case involving different numbers of
samples and non-uniform mass distribution is discussed in the appendix). Let
$\hat{\mu}=\frac{1}{M}\Sigma_{m=1}^{M}\delta_{x_{m}}$ and
$\hat{\nu}=\frac{1}{M}\Sigma_{m=1}^{M}\delta_{y_{m}}$ denote the empirical
distributions, where $\delta_{x_{m}}$ denotes a Dirac measure centered at
$x_{m}$. Then, $S3W_{\mathcal{H},p}^{p}(\hat{\mu},\hat{\nu})$ can be
approximated via the following Monte Carlo estimator:

$\displaystyle\frac{1}{L}\sum_{l=1}^{L}\sum_{m=1}^{M}\left|\langle
h(\phi_{\epsilon}(x_{\pi_{l}[m]})),\theta_{l}\rangle-\langle
h(\phi_{\epsilon}(y_{\pi^{\prime}_{l}[m]})),\theta_{l}\rangle\right|^{p}$

where $\pi_{l}[m]$ and $\pi^{\prime}_{l}[m]$ are the sorted indices of the
projected samples on $\theta_{l}$ (see Algorithm 1 for the procedure).

Algorithm 1 $S3W$

Input: $\\{x_{i}\\}_{i=1}^{M}\sim\mu$, $\\{y_{j}\\}_{j=1}^{M}\sim\nu$, $L$
projections, $p$-th order, $\epsilon$ for excluding the $\epsilon$-cap around
$s_{n}$.

Initialize: $h$ (injective map), $\\{\theta_{l}\\}_{l=1}^{L}$ (projections)

Compute $\\{u_{i}=h(\phi_{\epsilon}(x_{i}))\\}$ and
$\\{v_{j}=h(\phi_{\epsilon}(y_{j}))\\}$

Initialize distance $d=0$

for $l=1$ to $L$ do

Compute ${u^{l}_{i}}={\langle u_{i},\theta_{l}\rangle}$, ${v^{l}_{j}}={\langle
v_{j},\theta_{l}\rangle}$

Sort $\\{u^{l}_{i}\\}$, $\\{v_{j}^{l}\\}$, s.t $u^{l}_{\pi_{l}[i]}\leq
u^{l}_{\pi_{l}[i+1]}$, $v^{l}_{\pi_{l}^{\prime}[j]}\leq
v^{l}_{\pi_{l}^{\prime}[j+1]}$

$d=d+\frac{1}{L}\sum_{i=1}^{M}|u^{l}_{\pi_{l}[i]}-v^{l}_{\pi_{l}^{\prime}[i]}|^{p}$

end for

Return $d^{\frac{1}{p}}$

The $RI\text{-}S3W$ distances. The Monte Carlo approximation of
$RI\text{-}S3W$ distances can be written as:

$\displaystyle
RI\text{-}S3W_{p}(\hat{\mu},\hat{\nu})\approx\frac{1}{N_{R}}\sum_{n=1}^{N_{R}}S3W_{p}((R_{n})_{\\#}\hat{\mu},(R_{n})_{\\#}\hat{\nu}),$

where $\\{R_{n}\\}_{n=1}^{N_{R}}\subset\mathrm{SO}(d+1)$ are random rotation
matrices. To generate $R_{n}$, we adopt the GeoTorch library [51], which
provides a direct method to sample from $\mathrm{SO}(d+1)$. Our approach is
efficient with vectorization and parallel processing on GPU. We note that
generating $R_{n}$ is generally $O(N_{R}\cdot d^{3})$, which could become
expensive for high-dimensional data or a large number of rotations. Instead,
we could amortize this cost by presampling a rotation pool which could then be
subsampled for every distance calculation. We denote this implementation as
$ARI\text{-}S3W$, which involves a trade-off in memory and potential increased
bias. In practice, we observe highly favorable performance in a variety of
settings. We discuss this further in the appendix Section H.

Computational Complexity. Stereographic projection requires $O(Nd)$ where
$N=n+m$ is the total number of data points from the source and target, and
$(d+1)$ is the data dimensionality. If $n\gg m$, then we let $N=n$, and the
same analysis holds. Applying $h(\cdot)=h_{1}(\cdot)$ to the projected data
also requires $O(Nd)$. Thereafter, slicing the data is done in $O(LNd)$, where
$L$ is the number of projections; sorting is done in $O(LN\log N)$; and
finally, the distance calculation requires $O(LN)$. The overall time
complexity for $S3W$ is therefore $O(LN(d+\log N))$. For $RI\text{-}S3W$, the
cost of calculating $S3W$ for all rotations is $O(N_{R}LN(d+\log N))$.
Generating the random rotations takes an overhead of $O(N_{R}d^{3})$ (which
can be amortized), and applying these rotations to the data takes
$O(N_{R}Nd^{2})$. The total complexity is $O(N_{R}(d^{3}+Nd^{2}+LN(d+\log
N))$. If we amortize generating the rotation matrices, then the per-operation
cost becomes $O(N_{R}N(d^{2}+Ld+L\log N)).$

Runtime Comparison We compare the runtime to compute different distances
between the uniform distribution and a von Mises-Fisher distribution on
$\mathbb{S}^{100}$. The results in Figure 3 are averaged over $50$ iterations
for varying sample sizes of each distribution. We use $L=200$ projections for
all sliced methods, $N_{R}=10$ rotations for $ARI\text{-}S3W_{2}$ and
$RI\text{-}S3W_{2}$, and a pool size of $100$ for $ARI\text{-}S3W_{2}$. We do
not include the time required to generate the rotation pool in our
$ARI\text{-}S3W_{2}$ measurements. Results for Wasserstein and Sinkhorn are
based on the Python OT library [26].

Figure 3: Runtime comparison for $ARI\text{-}S3W_{2}$, $RI\text{-}S3W_{2}$,
$SW_{2}$, $SSW_{1}$ (using level median formula), $SSW_{2}$ with binary search
(BS) and antipodal closed form (for uniform distribution), Wasserstein, and
Sinkhorn with geodesic distance as cost function.

##  5 Experiments

Here, we present key results from our numerical experiments. We defer much of
the visualization, discussions, and further results to the appendix Section H.
All our experiments were executed on a Linux server with an AMD EPYC 7713
64-Core Processor, 8 $\times$ 32GB DIMM DDR4, 3200 MHz, and a NVIDIA RTX A6000
GPU.

###  5.1 Gradient Flows On The Sphere

Similar to the work of [5], we apply our proposed distances as a loss function
for the gradient flow problem. We consider a challenging target probability
measure $\nu$ with 12 von Mises-Fisher distributions (vMFs), and aim to solve
$\text{argmin}_{\mu}S3W(\mu,\nu)$. Suppose we have access to the target
measure only via i.i.d. samples $\\{y_{j}\\}_{j=1}^{M}$, i.e.,
$\hat{\nu}=\frac{1}{M}\sum_{j=1}^{M}\delta_{y_{j}}$ where $M=2400$. We
initialize $2400$ particles uniformly sampled on $\mathbb{S}^{2}$, and
directly optimize these particles with full-batch projected gradient descent
on the surface of the sphere. Figure 4 shows the converged loss curves after
$500$ iterations and reports the runtime, negative log-likelihood (NLL), and
the logarithm of the 2-Wasserstein distance between the distributions for
$SSW$, $S3W$, $RI\text{-}S3W$ with $N_{R}\in\\{1,5\\}$, and $ARI\text{-}S3W$
with $N_{R}=30$ and pool size of $1000$. We observe that the proposed
distances provide on-par or better performance while being significantly
faster than $SSW$ (up to $20$X). Additionally, we provide mini-batch results
in the appendix Section H.2. When the target is only known up to a constant,
we use sliced-Wasserstein variational inference [80] (see the appendix,
Section H.7).

(a)

Method |  Runtime(s) |  NLL $\downarrow$ |  $\log W_{2}$ $\downarrow$
---|---|---|---
$SSW$ (LR=$0.01$) |  94.97 $\pm$ 0.96 |  -4974.50 $\pm$ 2.27 |  -3.27 $\pm$ 0.17
$SSW$ (LR=$0.05$) |  94.98 $\pm$ 0.27 |  -4974.46 $\pm$ 3.50 |  -3.32 $\pm$ 0.13
$S3W$ |  3.32 $\pm$ 0.62 |  -4753.81 $\pm$ 83.70 |  -2.57 $\pm$ 0.19
$RI\text{-}S3W$ (1) |  4.00 $\pm$ 0.83 |  -4957.22 $\pm$ 34.70 |  -3.15 $\pm$ 0.26
$RI\text{-}S3W$ (5) |  10.58 $\pm$ 0.27 |  -4983.56 $\pm$ 6.62 |  -3.50 $\pm$ 0.17
$ARI\text{-}S3W$ (30) |  42.88 $\pm$ 0.05 |  -5025.37 $\pm$ 5.57 |  -4.37 $\pm$ 0.21

Figure 4: Learning a mixture of $12$ vMFs. $ARI\text{-}S3W$ (30) has $30$
rotations, pool size of $1000$. $S3W$ variants use $\text{LR}=0.01$. $SSW$ has
an additional $\text{LR}=0.05$ for better comparison. The plots show
convergence of different distances w.r.t. iterations and runtime. The table
summarizes numerical results for $10$ independent runs. We provide more
details of the plots in the appendix Section H.3.1.

###  5.2 Self-Supervised Learning (SSL)

We now show that our method can be an effective loss for contrastive SSL on
the sphere. We adopt the contrastive objective proposed in [76], composed of
alignment and uniformity loss terms, and replace the Gaussian kernel
uniformity loss with an $S3W$-based loss:

$\mathcal{L}=\frac{1}{n}\sum_{i=1}^{n}\lVert
z^{A}_{i}-z^{B}_{i}\rVert^{2}_{2}+\frac{\lambda}{2}\left(\text{S3W}_{2}(z^{A},\nu)+\text{S3W}_{2}(z^{B},\nu)\right),$

where $z^{A},z^{B}\in\mathbb{R}^{n\times(d+1)}$ are two encoded views of the
same images, $\nu=\text{Unif}(\mathbb{S}^{d})$ is the uniform distribution on
$\mathbb{S}^{d}$ and $\lambda>0$ is the regularization coefficient. We run our
experiments on CIFAR-10 using a ResNet18 encoder. Figure 5 visualizes the
learned embeddings when $d=2$, and Table 1 assesses the quality of the learned
embeddings for $d=9$ using the standard linear classifier evaluation. The
details of these experiments are included in the appendix Section H.6. We can
see that the proposed metrics consistently lead to embeddings that are
competitive both in terms of performance and runtime.

Figure 5: Projected features on $\mathbb{S}^{2}$ for CIFAR-10. Method |  Acc.(%) E/P $\uparrow$ |  Time(s/ep.)
---|---|---
Supervised |  92.38 / 91.77 |  —–
Hypersphere [76] |  79.76 / 74.57 |  24.28
SimCLR [11] |  79.69 / 72.78 |  20.94
$SSW$ [5] |  70.46 / 64.52 |  33.14
$SW$ |  74.45 / 68.35 |  21.09
$S3W$ |  78.54 / 73.84 |  21.36
$RI\text{-}S3W$ $(5)$ |  79.97 / 74.27 |  21.59
$ARI\text{-}S3W$ $(5)$ |  79.92 / 75.07 |  21.51
Table 1: Standard linear evaluation on CIFAR-10. Accuracy is for encoded (E)
features and projected (P) features on $\mathbb{S}^{9}$ (encoder feature
dimension is 10). Time reported is per epoch of pretraining.

###  5.3 Sliced-Wasserstein Autoencoder (SWAE)

We now adopt the SWAE framework proposed by [45] and demonstrate the
application of $S3W$ distances in generative modeling. Let
$\varphi:\mathcal{X}\rightarrow\mathbb{S}^{d}$ denote an encoder network and
$\psi:\mathbb{S}^{d}\rightarrow\mathcal{X}$ denote a decoder network. The SWAE
framework aims to enforce that the encoded data follow a prior distribution in
the latent space. In our experiments, we use a mixture of vMF distributions
with $10$ components on $\mathbb{S}^{2}$ as our prior distribution, which we
denote as $q$. Then, the training objective for the modified SWAE is:

$\min_{\varphi,\psi}\mathbb{E}_{x\sim p}[c(x,\psi(\varphi(x))]+\lambda\cdot
S3W(\varphi_{\\#}p,q)$

where $\lambda$ is the regularization coefficient, $c(\cdot,\cdot)$ the
reconstruction loss, for which we use the standard Binary Cross Entropy (BCE)
loss, and $p$ denotes the data distribution. Details about the network
architectures and results on the MNIST benchmark can be found in the appendix
Section H.8.

Method |  $\log W_{2}$ $\downarrow$ |  NLL $\downarrow$ |  BCE $\downarrow$ |  Time (s/ep.)
---|---|---|---|---
Supervised |  -0.5132 |  0.0060 |  0.6319 |  5.3243
$SSW$ |  -2.1949 |  0.0052 |  0.6323 |  15.4651
$SW$ |  -3.3229 |  -0.0007 |  0.6348 |  5.4661
$S3W$ |  -3.3381 |  0.0025 |  0.6318 |  5.7511
$RI$-$S3W$ $(5)$ |  -3.1424 |  -0.0043 |  0.6376 |  7.5443
$ARI$-$S3W$ $(5)$ |  -3.3853 |  0.0028 |  0.6332 |  5.8316
Table 2: CIFAR-10 results for SWAE. We evaluate the latent regularization loss
($\log(W_{2})$ and NLL), along with the BCE reconstruction loss on the test
data.

###  5.4 Earth Density Estimation

We extend our $S3W$ distances to the task of density estimation with
normalizing flows on $\mathbb{S}^{2}$. Our focus is on the three datasets
introduced by [54], representing the Earth’s surface as a perfect spherical
manifold: Earthquake [57], Flood [7], and Fire [25]. Similar to [5], we use an
exponential map normalizing flow model [69] (see H.5) optimizing
$\min_{T}S3W(T_{\\#}\nu,q)$, where $T$ is the transformation induced by the
model, $\nu$ is the data distribution known via samples
$\\{y_{j}\\}_{j=1}^{M}$, and $q$ is the prior distribution on
$\mathbb{S}^{2}$. The learned density at $y\in\mathbb{S}^{2}$ can then be
approximated as $f_{\nu}(y)=q(T(y))|\operatorname{det}J_{T}(y)|$ where
$J_{T}(y)$ denotes the Jacobian of $T$ at $y$.

Method |  Quake $\downarrow$ |  Flood $\downarrow$ |  Fire $\downarrow$
---|---|---|---
Stereo |  2.04 ± 0.19 |  1.85 ± 0.03 |  1.34 ± 0.11
$SW$ |  1.12 ± 0.07 |  1.58 ± 0.02 |  0.55 ± 0.18
$SSW$ |  0.84 ± 0.05 |  1.26 ± 0.03 |  0.24 ± 0.18
$S3W$ |  0.88 ± 0.09 |  1.33 ± 0.05 |  0.36 ± 0.04
$RI\text{-}S3W(1)$ |  0.79 ± 0.07 |  1.25 ± 0.02 |  0.15 ± 0.06
$ARI\text{-}S3W(50)$ |  0.78 ± 0.06 |  1.24 ± 0.04 |  0.10 ± 0.04
Table 3: Earth datasets results. We use a pool size of $100$ for
$ARI\text{-}S3W$. Stereo denotes the approach introduced by [28] which
involves stereographically projecting the sphere $\mathbb{S}^{d}$ onto
$\mathbb{R}^{d}$ and then performing RealNVP. The results are compared with
methods cited from [20].

##  6 Conclusion

We introduced a new class of sliced Wasserstein ($SW$) distances for spherical
data using the stereographic projection (SP). We rigorously addressed the
distortion issue caused by SP and presented several high-speed, high-
performing variants of our approach. $S3W$ maps data to a generalized
hypersurface with minimal distortion (via composing SP with a novel injective
function) and efficiently computes $SW$ over an order of magnitude faster than
existing baselines in many settings. $RI\text{-}S3W$ encodes rotation
invariance into $S3W$, further boosting its performance. Although
$RI\text{-}S3W$ is highly parallelizable, we achieved additional efficiency
with our implementation of amortization (denoted as $ARI\text{-}S3W$). Our
approach opens up interesting directions to bridging the spherical manifold
and classical $SW$ literature via pushforwards to appropriate hypersurfaces.

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##  Appendix A Notation

  * •

$(\mathbb{R}^{d},\|\cdot\|)$: d-dimensional Euclidean space, where $\|\cdot\|$
is the Euclidean norm (or $2-$norm): given
$x=[x_{1},\dots,x_{d}]\in\mathbb{R}^{d}$,
$\|x\|=\sqrt{x_{1}^{2}+\dots+x_{d}^{2}}$. Sometimes we will write
$\|\cdot\|_{2}$ to stress that we are considering this “$2-$norm”.

  * •

$\langle\cdot,\cdot\rangle$: canonical inner product in Euclidean spaces.

  * •

$L^{p}(\Omega)$, where $p\geq 1$ and $\Omega\subseteq\mathbb{R}^{d}$:
functional space defined by

$L^{p}(\Omega):=\\{f:\Omega\to\mathbb{R}|\,\int_{\Omega}|f|^{p}<\infty\\}$

endowed with the norm

$\|f\|_{p}=\left(\int_{\Omega}|f|^{p}\right)^{\frac{1}{p}}.$

In the extreme case $p=\infty$, we have

$\displaystyle
L^{\infty}(\Omega):=\\{f:\Omega\to\mathbb{R}|\,\sup|f|<\infty\\}.$ (17)

In our case, $\Omega$ will be $\mathbb{R}^{d}$,
$\mathbb{R}\times\mathbb{S}^{d-1}$,
$\mathbb{R}\times\mathbb{R}^{d^{\prime}}\setminus\\{0\\}$ or $\mathbb{S}^{d}$.

  * •

$C(X)$: space of real-valued continuous functions defined on the space $X$.

  * •

$C_{0}(\mathbb{R}^{d})$: set of continuous functions “that vanish at
infinity”.

  * •

$C^{\infty}$: class of infinite differentiable functions.

  * •

$\mathbb{S}^{d-1}$: the unit sphere in $\mathbb{R}^{d}$, defined as
$\mathbb{S}^{d-1}:=\\{x\in\mathbb{R}^{d}:\|x\|^{2}=1\\}.$

  * •

$\sigma_{d}$: the uniform probability measure defined in the sphere
$\mathbb{S}^{d-1}\subset\mathbb{R}^{d}$.

  * •

$s_{n}=[0,\dots,0,1]\in\mathbb{R}^{d+1}$ is the North Pole and
$s_{0}=[0,\dots,0,-1]\in\mathbb{R}^{d+1}$ is the South Pole in
$\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$.

  * •

$\phi$: stereographic projection (SP).

  * •

$\phi_{\epsilon}$: stereographic projection operator whose domain excludes the
$\epsilon$-cap around $s_{n}$ in the sphere $\mathbb{S}^{d}$.

  * •

$v^{T}$: transpose vector.

  * •

$H(t,\theta)=\\{x\in\mathbb{R}^{d}\ |\ \langle x,\theta\rangle=t\\}$
hyperplane.

  * •

$H_{t,\theta}=\\{x\in\mathbb{R}^{d}|\,g(x,\theta)=t\\}$ “level set” of
$g:\mathbb{R}^{d}\times\mathbb{S}^{d-1}\to\mathbb{R}$ at level $t$ with fixed
spherical variable $\theta$.

  * •

$\mathcal{R}(\cdot)$: Radon transform with
$\mathcal{R}:L^{1}(\mathbb{R}^{d})\to
L^{1}(\mathbb{R}\times\mathbb{S}^{d-1})$.

  * •

$\mathcal{G}(\cdot)$: Generalized Radon Transform (GRT) with

$\mathcal{G}(f)(t,\theta)=\int_{\mathbb{R}^{d}}f(x)\delta(t-g(x,\theta))\,dx.$

In this formulation, $d^{\prime}\geq d$ in general, and
$g:\mathbb{R}^{d}\times(\mathbb{R}^{d^{\prime}}\setminus\\{0\\})\to\mathbb{R}$
is a function which satisfies the following:

    1. (H.1)

$g(x,\theta)$ is $C^{\infty}$ function on
$\mathbb{R}^{d}\times(\mathbb{R}^{d^{\prime}}\setminus\\{0\\})$.

    2. (H.2)

$g(x,\theta)$ is homogeneous of degree one in $\theta$, i.e.,
$g(x,\lambda\theta)=\lambda g(x,\theta)$ for all $\lambda\in\mathbb{R}$.

    3. (H.3)

$g(x,\theta)$ is non-degenerate with respect to $x$ in the sense that
$\forall(x,\theta)\in\mathbb{R}^{d}\times(\mathbb{R}^{d^{\prime}}\setminus\\{0\\}),\,\frac{\partial
g}{\partial x}(x,\theta)\neq 0$.

    4. (H.4)

The mixed Hessian of $g$ is strictly positive, i.e.,
$\textrm{det}\left(\frac{\partial^{2}g}{\partial
x^{i}\partial\theta^{j}}\right)>0$.

By property (H.2), $\mathcal{G}:L^{1}(\mathbb{R}^{d})\to
L(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$. We refer [4, 19, 27, 23, 48,
35, 43] for more details.

  * •

$\mathcal{H}(\cdot)$: a variant and simplified version of the Generalized
Radon Transform defined by $\mathcal{H}:L^{1}(\mathbb{R}^{d})\to
L^{1}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$,

$\mathcal{H}(f)(t,\theta)=\int_{\mathbb{R}^{d}}f(x)\delta(t-\langle
h(x),\theta\rangle)dx,$

where $h:\mathbb{R}^{d}\to\mathbb{R}^{d^{\prime}}$ is injective.

  * •

$h_{1}$: function defined by (14).

  * •

$(t,\theta)$: the first and second inputs for (generalized) Radon transform,
where $t\in\mathbb{R}$ and variable $\theta$ lies in a pre-defined sphere:
$\theta\in\mathbb{S}^{d-1}$ for the classic Radon transform ($\mathcal{R}$),
and $\theta\in\mathbb{S}^{d^{\prime}-1}$ for the generalized cases
$\mathcal{G}$ and $\mathcal{H}$.

  * •

$\mathcal{R}^{*},\mathcal{G}^{*},\mathcal{H}^{*}$: the Hermitian adjoint
operators for $\mathcal{R},\mathcal{G},\mathcal{H}$. In particular,
$\mathcal{R}^{*}:L^{\infty}(\mathbb{R}\times\mathbb{S}^{d-1})\to
L^{\infty}(\mathbb{R}^{d})$, and
$\mathcal{G}^{*},\mathcal{H}^{*}:L^{\infty}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})\to
L^{\infty}(\mathbb{R}^{d})$ with

$\displaystyle\mathcal{R}^{*}(\psi)(x)=\int_{\mathbb{S}^{d-1}}\psi(\langle
x,\theta\rangle,\theta)d\sigma_{d}(\theta)$
$\displaystyle\mathcal{G}^{*}(\psi)(x)=\int_{\mathbb{S}^{d^{\prime}-1}}\psi(g(x,\theta),\theta)d\sigma_{d^{\prime}}(\theta)$
$\displaystyle\mathcal{H}^{*}(\psi)(x)=\int_{\mathbb{S}^{d^{\prime}-1}}\psi(\langle
h(x),\theta\rangle,\theta)d\sigma_{d^{\prime}}(\theta)$

  * •

$\mathcal{M}(\Omega)$: Set of all real Radon measures (finite regular Borel
measures not necessarily positive, that is, it includes signed measures)
defined on $\Omega$. In this article $\Omega$ can be $\mathbb{R}^{d}$,
$\mathbb{R}\times\mathbb{S}^{d-1}$, or
$\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}$.

$\mathcal{M}(\Omega)$ is endowed with the total variation norm

$\|\mu\|_{TV}=\mu^{+}(\Omega)+\mu^{-}(\Omega)$

where $\mu^{\pm}$ are the positive and negative parts of $\mu$.

  * •

$\mathcal{M}_{+}(\Omega)$: Set of all positive Radon measures defined on
$\Omega$. It is endowed with the total variation norm
$\|\mu\|_{TV}=\mu(\Omega)$.

  * •

$\mathcal{P}(\Omega)$: Set of all probability measures defined on $\Omega$
(i.e., positive measures with $\|\mu\|_{TV}=\mu(\Omega)=1$).

  * •

$\mathcal{P}_{p}(\Omega)$: Set of all probability measures defined on $\Omega$
with $p-$th finite moment, that is,

$\mathcal{P}_{p}(\Omega)=\\{\mu\in\mathcal{P}(\Omega):\,\int_{\Omega}\|x\|^{p}d\mu<\infty\\}$

We have the following relation:

$\mathcal{P}_{p}(\Omega)\subseteq\mathcal{P}(\Omega)\subseteq\mathcal{M}_{+}(\Omega)\subseteq\mathcal{M}(\Omega).$

  * •

$T_{\\#}\mu$: push-forward of the measure $\mu\in\mathcal{M}(X)$ by the
measurable map $T:X\to Y$, which defines a measure on $\mathcal{M}(Y)$ such
that $T_{\\#}\mu(B)=\mu\left(\\{x\in X:\,T(x)\in B\\}\right)$ for every
measurable set $B\subseteq Y$.

  * •

$W_{p}$: $p$-Wasserstein distance.

  * •

$\hat{\mu}=\frac{1}{M}\Sigma_{m=1}^{M}\delta_{x_{m}}$: empirical distribution.

  * •

$\text{Unif}(X)$: Uniform distribution on a measure space $X$.

  * •

$F_{\mu}$, $F_{\mu}^{-1}$: cumulative distribution function (CDF) and quantile
function of the measure $\mu$, respectively.

  * •

$\mathcal{S}_{\mathcal{R}}$, $\mathcal{S}_{\mathcal{G}}$, and
$\mathcal{S}_{\mathcal{H}}$: Stereographic Spherical Radon Transforms (SSRT),

$\mathcal{S}_{\mathcal{R}}(\mu)=\mathcal{R}(\phi_{\\#}\mu),\qquad\mathcal{S}_{\mathcal{G}}(\mu)=\mathcal{G}(\phi_{\\#}\mu),\qquad\mathcal{S}_{\mathcal{H}}(\mu)=\mathcal{H}(\phi_{\\#}\mu).$

  * •

$S3W_{\mathcal{R},p}$, $S3W_{\mathcal{G},p}$, and $S3W_{\mathcal{H},p}$:
Stereographic Spherical Sliced Wasserstein (S3W) distances

$S3W_{\mathcal{R},p}^{p}(\mu,\nu)=\int_{\mathbb{S}^{d-1}}W_{p}^{p}(\mathcal{S}_{\mathcal{R}}(\mu)_{\theta},\mathcal{S}_{\mathcal{R}}(\nu)_{\theta})\,d\sigma_{d}(\theta),$
$S3W_{\mathcal{G},p}^{p}(\mu,\nu)=\int_{\mathbb{S}^{d^{\prime}-1}}W_{p}^{p}(\mathcal{S}_{\mathcal{G}}(\mu)_{\theta},\mathcal{S}_{\mathcal{G}}(\nu)_{\theta})\,d\sigma_{d^{\prime}}(\theta),$
$S3W_{\mathcal{H},p}^{p}(\mu,\nu)=\int_{\mathbb{S}^{d^{\prime}-1}}W_{p}^{p}(\mathcal{S}_{\mathcal{H}}(\mu)_{\theta},\mathcal{S}_{\mathcal{H}}(\nu)_{\theta})\,d\sigma_{d^{\prime}}(\theta).$

  * •

$\mathbb{E}$: expected value.

  * •

$\mathrm{O}(n)$, $\mathrm{SO}(n)$: orthogonal group (rotations and
reflections) and special orthogonal group (rotations) of $n\times n$ real
matrices, respectively.

  * •

$\omega$: Haar probability measure on the compact group $\mathrm{SO}(d+1)$.

  * •

$\approx$: approximately.

  * •

$x\sim\mu$: element $x$ sampled from the distribution $\mu$.

  * •

$\nabla_{x}$: gradient with respect to the variable $x$.

  * •

$RI-S3W_{\mathcal{G},p}$: Rotationally Invariant Extension of S3W defined by
(16).

  * •

$ARI-S3W$: Amortized Rotationally Invariant Extension of S3W used in our
experiments.

  * •

$d_{\mathbb{S}^{d}}(\cdot,\cdot)$: The great circle distance on the sphere
$\mathbb{S}^{d}$ defined by $d_{\mathbb{S}^{d}}(s_{1},s_{2})=\arccos(\langle
s_{1},s_{2}\rangle)$ for all $s_{1},s_{2}\in\mathbb{S}^{d}$.

  * •

$\angle(s_{1},s_{2})$: the angle between the points $s_{1}$ and $s_{2}$
defined by $\angle(s_{1},s_{2})=\arccos(\langle
s_{1},s_{2}\rangle){\in[0,\pi]}$.

##  Appendix B Radon Transform for Radon Measures

This section will review the classic Radon transform in the general Radon
measure setting. In the next section, we introduce the Generalized Radon
Transform (GRT) for general Radon measures.

As we discussed in the main section, given a function $f\in
L^{1}(\mathbb{R}^{d})$, the Radon transform, $\mathcal{R}(f)$, is a function
in $L^{1}(\mathbb{R}\times\mathbb{S}^{d-1})$. In particular, for each
$(t,\theta)\in\mathbb{R}\times\mathbb{S}^{d-1}$, we have:

$\displaystyle\mathcal{R}(f)(t,\theta)$
$\displaystyle=\int_{\mathbb{R}^{d}}f(x)\delta(t-\langle x,\theta\rangle)\,dx$
$\displaystyle=\int_{\mathbb{R}^{d-1}}f(t\theta+U_{\theta}\xi)\,d\xi$

where $U_{\theta}\in\mathbb{R}^{d\times(d-1)}$ is any matrix such that its
columns are formed by an orthonormal basis of $\theta^{\perp}$ (the subspace
which is perpendicular to the vector $\theta$), see equation (22) in [6]. It
is straightforward to verify the above definition is well-defined (i.e.
$\mathcal{R}(f)$ is independent of the choice of $U_{\theta}$).

Notably, the Radon transform $\mathcal{R}:L^{1}(\mathbb{R}^{d})\rightarrow
L^{1}(\mathbb{R}\times\mathbb{S}^{d-1})$ is a linear bijection with a closed-
form inversion formula. For a function $f\in L^{1}(\mathbb{R}^{d})$, the Radon
transform $\mathcal{R}f$ can be inverted using the inverse Radon transform,
denoted by $\mathcal{R}^{-1}$. The inversion formula is given by:

$f(x)=\mathcal{R}^{-1}(\mathcal{R}f)(x)=\int_{\mathbb{S}^{d-1}}\big{(}\mathcal{R}f(\cdot,\theta)*\eta(\cdot)\big{)}(\langle
x,\theta\rangle)\,d\sigma_{d}(\theta),$ (18)

where $\eta(\cdot)$ is a one-dimensional high-pass filter with corresponding
Fourier transform $\mathcal{F}\eta(\omega)=c|\omega|^{d-1}$, which appears due
to the Fourier slice theorem [34], and ‘$*$’ is the convolution operator. This
integral formula effectively reconstructs $f$ from its Radon transform and, in
the medical imaging community, is referred to as ‘filtered back projection.’

Besides, the fact that the Radon transform $\mathcal{R}$ is a bounded linear
operator from $L^{1}(\mathbb{R}^{d})$ to
$L^{1}(\mathbb{R}\times\mathbb{S}^{d-1})$, yields to the definition of the
dual operator 222Some references call the back projection as adjoint (aka
Hermitian conjugate) operator e.g. [6, 73]., denoted as $\mathcal{R}^{*}$,
which is generally called “back projection” [34, 70]. By definition of the
dual operator and Riesz representation theorem that identifies the dual space
of $L^{1}$ with $L^{\infty}$, for each $\psi\in
L^{\infty}(\mathbb{R}\times\mathbb{S}^{d-1})$, we have

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d-1}}\psi(t,\theta)\,\mathcal{R}(f)(t,\theta)\,dt\,d\sigma_{d}(\theta)=\int_{\mathbb{R}^{d}}f(x)\,\mathcal{R}^{*}(\psi)(x)\,dx.$
(19)

Moreover, $\mathcal{R}^{*}:L^{\infty}(\mathbb{R}\times\mathbb{S}^{d-1})\to
L^{\infty}(\mathbb{R}^{d})$ has closed form: For each $x\in\mathbb{R}^{d}$ and
$\psi\in L^{\infty}(\mathbb{R}\times\mathbb{S}^{d-1})$,

$\displaystyle\mathcal{R}^{*}(\psi)(x)=\int_{\mathbb{S}^{d-1}}\psi(\langle
x,\theta\rangle,\theta)\,d\sigma_{d}(\theta),$ (20)

where $\sigma_{d}$ is the uniform probability measure defined in the sphere
$\mathbb{S}^{d-1}\subset\mathbb{R}^{d}$ .

As discussed in the main section, the classic Radon transform can describe the
distribution in projected 1D space. Indeed, suppose $f$ is the density of a
probability measure $\mu$, and $\theta\in\mathbb{S}^{d-1}$, then function
$\mathcal{R}(f)(\cdot,\theta):\mathbb{R}\to\mathbb{R}_{\geq 0}$ is exactly the
density of the projected probability measure of $\mu$ into the space spanned
by $\theta$. We denote this projection $\theta_{\\#}\mu$ by using pushforward
notation where we identify $\theta\in\mathbb{S}^{d-1}$ with the function

$\displaystyle\theta:\mathbb{R}^{d}\to\mathbb{R}^{d}$ $\displaystyle\qquad
x\mapsto\langle x,\theta\rangle\,\theta,$ (21)

which range lies of the line generated by the direction $\theta$. Thus, we
write

$\mathcal{R}(f)(\cdot,\theta)=\theta_{\\#}\mu\in\mathcal{M}(\mathbb{R}).$ (22)

However, when $\mu$ is not a continuous measure, continuous (e.g. empirical
distribution), then $\theta_{\\#}\mu$ does not admit a density function. To
address this limitation and extend $\mathcal{R}$ to any king of finite
measures, we follow a measure-theoretic approach of the Radon transform as in
[6] by using equation (19). We refer the reader also to [34][Ch.1, Sec.5]).

For this extension, we first recall the classical Riesz-Markov representation
theorem and the lemma for the identity between Radon measures.

######  Theorem 5 (Riesz-Markov Representation theorem).

Suppose $\Omega$ is a separable and locally compact space, then the dual space
of the Banach space $(C_{0}(\Omega),\|\cdot\|_{\infty})$, denoted as
$(C_{0}(\Omega)^{*},\|\cdot\|_{*})$, is isomorphic to
$(\mathcal{M}(\Omega),\|\cdot\|_{TV})$. In particular, for each bounded linear
functional $\xi\in C_{0}(\Omega)^{*}$, there exists a unique
$\nu\in\mathcal{M}(\Omega)$ such that

$\xi(\psi)=\int_{\Omega}\psi(x)\,d\nu(x),\qquad\text{ and
}\qquad\|\xi\|_{*}=\|\nu\|_{TV}.$

######  Lemma 6 (Identity of Radon measures).

Given $\mu_{1},\mu_{2}\in\mathcal{M}(\Omega)$ where
$\Omega\subset\mathbb{R}^{d}$, the following are equivalent:

  1. (1)

$\nu_{1}=\nu_{2}$

  2. (2)

$\int_{\Omega}\psi(x)\,d\nu_{1}(x)=\int_{\Omega}\psi(x)\,d\nu_{2}(x),\qquad\forall\psi\in
C_{0}(\Omega)$

  3. (3)

$\int_{\Omega}\psi(x)\ d\nu_{1}(x)=\int_{\Omega}\psi(x)\
d\nu_{2}(x),\qquad\forall\psi\in L^{\infty}(\Omega)$

We will refer to the spaces $C_{0}(\Omega)$ and $L^{\infty}(\Omega)$ as the
spaces of test functions for $\mathcal{M}(\Omega)$.

###### Proof.

If $\nu_{1}=\nu_{2}$, we directly have $(2),(3)$. Since $C_{0}(\Omega)\subset
L^{\infty}(\Omega)$, then $(3)$ implies (2). All we need to prove is
$(2)\Rightarrow(1)$. By the Riesz-Markov Representation theorem 5,
$\mathcal{M}(\Omega)$ is isometric to $(C_{0}(\Omega)^{*},\|\cdot\|_{*})$. By
the identity in the dual space, we have $\nu_{1}=\nu_{2}$ given (2). ∎

Based on the above theorem, for $\mu\in\mathcal{M}(\mathbb{R}^{d})$, its Radon
transform $\mathcal{R}(\mu)=\nu$ is defined as the measure
$\nu\in\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d-1})$ such that for each
$\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d-1})$,

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d-1}}\psi(t,\theta)\,d\nu(t,\theta)=\int_{\mathbb{R}^{d}}(\mathcal{R}^{*}(\psi))(x)\,d\mu(x),$
(23)

where the dual operator $\mathcal{R^{*}}$ is defined in (20). Synthetically,
one usually writes

$\mathcal{R}(\mu)(\psi)=\mu(\mathcal{R}^{*}(\psi)).$

Note that, by the above lemma, we can choose the
$L^{\infty}(\mathbb{R}\times\mathbb{S}^{d-1})$ space of test functions. In
particular, by setting $\psi=1_{A}$ for any Borel set
$A\subset\mathbb{R}\times\mathbb{S}^{d-1}$, we can verify that $\nu$ is Radon
measure if $\mu$ is Radon measure. Similarly, $\nu$ is a positive Radon
measure if $\mu$ is positive. In addition, one can obtain
$\|\nu\|_{TV}=\|\mu\|_{TV}$. Thus, if $\mu$ is a probability measure on
$\mathbb{R}^{d}$, then $\nu=\mathcal{R}(\mu)$ is a probability measure defined
on $\mathbb{R}\times\mathbb{S}^{d-1}$.

By the disintegration theorem in classic measure theory, there exists a
$\nu-$a.s. unique set of measures
$(\nu_{\theta})_{\theta\in\mathbb{S}^{d-1}}\subset\mathcal{M}(\mathbb{R})$
such that for any $\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d-1})$, we have

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d-1}}\psi(t,\theta)\,d\nu(t,\theta)=\int_{\mathbb{S}^{d-1}}\int_{\mathbb{R}}\psi(t,\theta)\,d\nu_{\theta}(t)\,d\sigma_{d}(\theta).$
(24)

Similar to (22) for the classic Radon measure, given a fixed
$\theta\in\mathbb{S}^{d-1}$, $\mathcal{R}(\mu)_{\theta}$ describes the
projected measure of $\mu$ into the 1D space spanned by $\theta$
[6][Proposition 6], that is:

$\mathcal{R}(\mu)_{\theta}=\nu_{\theta}=\theta_{\\#}\mu\in\mathcal{M}(\mathbb{R}).$

##  Appendix C Generalized Radon Transform for Probability Measures

Extending the Radon transform as in (3) and (4) is natural. In this section,
we review those definitions.

Given
$g:\mathbb{R}^{d}\times(\mathbb{R}^{d^{\prime}}\setminus\\{0\\})\to\mathbb{R}$,
such that $g$ satisfies the technical assumptions [H.1–H.4] listed in the
Notation section and given in [4] and [43], the Generalized Radon Transform
(GRT) is defined as

$\mathcal{G}(f)(t,\theta)=\int_{\mathbb{R}^{d}}f(x)\delta(t-g(x,\theta))\,dx,\qquad\forall(t,\theta)\in\mathbb{R}\times(\mathbb{R}^{d^{\prime}}\setminus\\{0\\}).$

Note, for each
$(t,\theta)\in\mathbb{R}\times(\mathbb{R}^{d^{\prime}}\setminus\\{0\\})$, we
set the corresponded hyper-surface as
$H_{t,\theta}=\\{x\in\mathbb{R}^{d}\mid\,g(x,\theta)=t\\}$. By the homogeneity
property (H.2), $H_{\lambda
t,\lambda\theta}=H_{t,\theta},\,\forall\lambda\in\mathbb{R}$. Then,

$\mathcal{G}f(\lambda
t,\lambda\theta)=\mathcal{G}f(t,\theta),\qquad\forall(t,\theta,\lambda)\in\mathbb{R}\times(\mathbb{R}^{d^{\prime}}\setminus\\{0\\})\times\mathbb{R}.$

Thus, the hypersurface can be parameterized by
$(t,\theta)\in\mathbb{R}\times\mathbb{S}^{d-1}$, and we can redefine
$\mathcal{G}$ as a mapping from space $L^{1}(\mathbb{R}^{d})$ to
$L^{1}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.

Furthermore, as $\mathcal{G}$ is an linear operator from
$L^{1}(\mathbb{R}^{d})$ to $L^{1}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$,
we can define the dual operator $\mathcal{G}^{*}$ of $\mathcal{G}$ as in [35]
(also called the adjoint operator). In analogy with the Radon transform, this
is done by the duality between $L^{1}$ and $L^{\infty}$: For each $\psi\in
L^{\infty}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$,

$\displaystyle\mathcal{G}^{*}(\psi)(x)=\int_{\mathbb{S}^{d^{\prime}-1}}\psi(g(x,\theta),\theta)\,d\sigma_{d^{\prime}}(\theta),\qquad\forall
x\in\mathbb{R}^{d},$ (25)

where $\sigma_{d^{\prime}}$ is the uniform probability measure defined in
$\mathbb{S}^{d^{\prime}-1}$.

In the specific case, $g(x,\theta)=\langle h(x),\theta\rangle$, where
$h:\mathbb{R}^{d}\to\mathbb{R}^{d^{\prime}}$ is a injective function, the
induced GRT becomes

$\mathcal{H}(f)(t,\theta)=\int_{\mathbb{S}^{d^{\prime}-1}}f(x)\delta(t-\langle
h(x),\theta\rangle)\,dx,\qquad\forall(t,\theta)\in\mathbb{R}\times\mathbb{S}^{d^{\prime}-1},$

with corresponding dual operator

$\displaystyle\mathcal{H}^{*}(\psi)(x)=\int_{\mathbb{S}^{d^{\prime}-1}}\psi(\langle
h(x),\theta\rangle,\theta)\,d\sigma_{d^{\prime}}(\theta),\qquad\forall\psi\in
L^{\infty}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}),\,x\in\mathbb{R}^{d}.$
(26)

In this case, that is $g(x,\theta)=\langle h(x),\theta\rangle$ for
$h=(h_{1},\dots,h_{d^{\prime}}):\mathbb{R}^{d}\to\mathbb{R}^{d^{\prime}}$,
properties [H.1-H.4] read as follows: First, notice that the real inner
product in $\mathbb{R}^{d}$ defines, in each variable, a $C^{\infty}$ function
which is homogeneous of degree one. In particular, (H.2) is trivially
satisfied. Then, $h$ must satisfy:

  1. (a)

In place of (H.1): $h\in C^{\infty}$.

  2. (b)

In place of (H.3): For all
$(x,\theta)\in\mathbb{R}^{d}\times\mathbb{S}^{d^{\prime}-1}$,
$\left(\langle\frac{\partial h}{\partial
x^{1}}(x),\theta\rangle,\dots,\langle\frac{\partial h}{\partial
x^{d}}(x),\theta\rangle\right)\not=0$.

  3. (c)

In place of (H.4): $\mathrm{det}\left(\left[\frac{\partial h_{i}}{\partial
x^{j}}\right]_{1\leq i\leq 1,1\leq j\leq d^{\prime}}\right)>0$.

In particular, if $d=d^{\prime}$, let $\mathbf{J}_{h}$ denote the Jacobian of
$h=(h_{1},\dots,h_{d}):\mathbb{R}^{d}\to\mathbb{R}^{d}$, that is, the
$d^{\prime}\times d$ matrix $\mathbf{J}_{h}=\left[\frac{\partial
h_{i}}{\partial x^{j}}\right]_{ij}$. Then, condition (b) can be written as the
matrix-vector multiplication $\mathbf{J}_{h}(x)\cdot\theta\not=0$, and
condition (c) is equivalent $\mathrm{det}(\mathbf{J}_{h}(x))>0$ for all
$x\in\mathbb{R}^{d}$, and so in this case (b) is redundant. Therefore, when
$d=d^{\prime}$, $h:\mathbb{R}^{d}\to\mathbb{R}^{d}$ must satisfy:

$h\in C^{\infty}\qquad\text{ and
}\qquad\mathrm{det}\left(\mathbf{J}_{h}\right)>0.$

Inspired by the framework of classic Radon transform for Radon measure, we can
extend the idea into the GRT setting. In particular, given
$\mu\in\mathcal{M}(\mathbb{R}^{d})$, we define the measures $\mathcal{G}(\mu)$
and $\mathcal{H}(\mu)$ as follows: For each test function $\psi\in
C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$,

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{G}(\mu)(t,\theta)=\int_{\mathbb{R}^{d}}\mathcal{G}^{*}(\psi)(x)\,d\mu(x)$
(27)
$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{H}(\mu)(t,\theta)=\int_{\mathbb{R}^{d}}\mathcal{H}^{*}(\psi)(x)\,d\mu(x)$
(28)

######  Proposition 7.

Given $\mu\in\mathcal{M}(\mathbb{R}^{d})$, the measures
$\mathcal{G}(\mu),\mathcal{H}(\mu)$ is defined in (27) and (28) have the
following properties:

  1. (1)

$\mathcal{G}(\mu)$ is well-defined, i.e.,
$\mathcal{G}(\mu)\in\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.

  2. (2)

If $\mu\in\mathcal{M}_{+}(\mathbb{R}^{d})$, then
$\mathcal{G}(\mu)\in\mathcal{M}_{+}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.
Similarly, if $\mu\in\mathcal{P}(\mathbb{R}^{d})$, then
$\mathcal{G}(\mu)\in\mathcal{P}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.

  3. (3)

$\mathcal{G}$ preserves the the total variation norm, i.e.,
$\|\mu\|_{TV}=\|\mathcal{G}(\mu)\|_{TV}$. In particular, $\mathcal{G}$
preserves mass.

  4. (4)

If $\mu\in\mathcal{P}(\mathbb{R}^{d})$, then
$\mathcal{G}(\mu)\in\mathcal{P}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.

  5. (5)

Given $\mu\in\mathcal{M}(\mathbb{R}^{d})$, the disintegration theorem gives a
unique $\mathcal{G}(\mu)-$a.s. set of measures
$(\mathcal{G}(\mu)_{\theta})_{\theta\in\mathbb{S}^{d^{\prime}-1}}\subset\mathcal{M}(\mathbb{R})$
such that for any $\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$,

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{G}(\mu)(t,\theta)=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\,d\mathcal{G}(\mu)_{\theta}(t)\,d\sigma_{d^{\prime}}(\theta)$

Then, it holds that

$\mathcal{G}(\mu)_{\theta}=g(\cdot,\theta)_{\\#}\mu.$

  6. (6)

Properties $(1)-(5)$ hold true for $\mathcal{H}$ in place of $\mathcal{G}$.

  7. (7)

Given $\mu\in\mathcal{M}(\mathbb{R}^{d})$, the disintegration theorem gives a
unique $\mathcal{H}(\mu)-$a.s. set of measures
$(\mathcal{H}(\mu)_{\theta})_{\theta\in\mathbb{S}^{d^{\prime}-1}}\subset\mathcal{M}(\mathbb{R})$
such that for any $\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$,

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{H}(\mu)(t,\theta)=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\,d\mathcal{H}(\mu)_{\theta}(t)\,d\sigma_{d^{\prime}}(\theta)$

Then, it holds that

$\mathcal{H}(\mu)_{\theta}=\theta_{\\#}h_{\\#}\mu,$

or equivalently, $\mathcal{H}(\mu)_{\theta}=(\theta\circ h)_{\\#}\mu$, where
the function $\theta$ is defined as in (21).

  8. (8)

$\mathcal{H}(\mu)=\mathcal{R}(h_{\\#}\mu)$ for every
$\mu\in\mathcal{M}(\mathbb{R}^{d})$.

  9. (9)

When $h$ is injective, $\mathcal{H}$ is invertible.

###### Proof.

  1. (1)

Let $\mu\in\mathcal{M}(\mathbb{R}^{d})$. By [4, 35], we have that (6) is the
dual operator of $\mathcal{G}$. In fact, by functional analysis theory,
$L^{\infty}$ is the dual space of Banach space $L^{1}$, and thus,
$\mathcal{G^{*}}:L^{\infty}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})\to
L^{\infty}(\mathbb{R}^{d})$ is well-defined. In particular,
$\int_{\mathbb{R}^{d}}\mathcal{G}^{*}(\psi)(x)\,d\mu(x)$ is finite for all
$\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})\subset
L^{\infty}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$. Then, expression (27)
is finite for all $\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$,
and moreover defines $\mathcal{G}(\mu)$ as an element in the dual space of
$C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$. By Theorem 5, we have
$\mathcal{G}(\mu)\in\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.

  2. (2)

Let $\mu\in\mathcal{M}_{+}(\mathbb{R}^{d})$ and $\psi\in
C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$ with $\psi(t,\theta)\geq
0,\forall t,\theta$. Thus, by definition (6), $\mathcal{G}^{*}\psi(x)\geq
0,\forall x\in\mathbb{R}^{d}$. Therefore,

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{G}\mu(t,\theta)=\int_{\mathbb{R}^{d}}\mathcal{G}^{*}(\psi)(x)\,d\mu(x)\geq
0$

which follows from the facts $\mathcal{G}^{*}(\psi)\geq 0$ and
$\mu\in\mathcal{M}_{+}(\mathbb{R}^{d})$. Since the above property holds for
all non-negative test function $\psi\in
C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$, we obtain that
$\mathcal{G}(\mu)\in\mathcal{M}_{+}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.

  3. (3)

Set $\psi=1_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\in
L^{\infty}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$. By definition of
$\mathcal{G}^{*}$, we have

$\mathcal{G}^{*}(\psi)(x)=\int_{\mathbb{S}^{d^{\prime}-1}}1\,d\sigma_{d^{\prime}}(\theta)=1,\qquad\forall
x\in\mathbb{R}^{d}.$

That is,
$\mathcal{G}^{*}(1_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}})=1_{\mathbb{R}^{d}}$.
Now, let $\mu\in\mathcal{M}(R^{d})$. We have that

$\displaystyle\mu(\mathbb{R}^{d})$
$\displaystyle=\int_{\mathbb{R}^{d}}1_{\mathbb{R}^{d}}\,d\mu=\int_{\mathbb{R}^{d}}\mathcal{G}^{*}(1_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}})\,d\mu=\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}1_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\,d\mathcal{G}(\mu)=\mathcal{G}(\mu)(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}).$
(29)

Let $\mu^{+}$ and $\mu^{-}$ are the positive and negative parts of $\mu$, that
is, $\mu=\mu^{+}-\mu^{-}$, where
$\mu^{\pm}\in\mathcal{M}_{+}(\mathbb{R}^{d})$. By property (2) we have that
$\mathcal{G}(\mu^{\pm})\in\mathcal{M}_{+}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.
By linearity of $\mathcal{G}$, we have

$\mathcal{G}(\mu)=\mathcal{G}(\mu^{+}-\mu^{-})=\underbrace{\mathcal{G}(\mu^{+})}_{\mathcal{M}_{+}}-\underbrace{\mathcal{G}(\mu^{-})}_{\mathcal{M}_{+}},$

which by the uniqueness of the Hahn-Jordan decomposition theorem implies that
the positive and negative parts of the measure $\mathcal{G}(\mu)$ are
$\mathcal{G}(\mu^{+})$ and $\mathcal{G}(\mu^{-})$, respectively. That is,

$\mathcal{G}(\mu)^{\pm}=\mathcal{G}(\mu^{\pm}).$

Then,

$\displaystyle\|\mathcal{G}(\mu)\|_{TV}$
$\displaystyle=\mathcal{G}(\mu)^{+}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})+\mathcal{G}(\mu)^{-}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$
$\displaystyle=\mathcal{G}(\mu^{+})(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})+\mathcal{G}(\mu^{-})(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$
$\displaystyle=\mu^{+}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})+\mu^{-}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})=\|\mu\|_{TV},$

that is, $\|\mathcal{G}(\mu)\|_{TV}=\|\mu\|_{TV}$.

  4. (4)

If $\mu\in\mathcal{P}(\mathbb{R}^{d})$, combining the property
$\mathcal{G}(\mu)\in\mathcal{M}_{+}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$
with the property $\|\mathcal{G}(\mu)\|_{TV}=\|\mu\|_{TV}=1$, we obtain that
$\mathcal{G}(\mu)\in\mathcal{P}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.

  5. (5)

We will follow the ideas in [6][Proposition 6]. For any test function $\psi\in
C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$ we have

$\displaystyle\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\
d\mathcal{G}(\mu)_{\theta}(t)\,d\sigma_{d^{\prime}}(\theta)$
$\displaystyle=\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\
d\mathcal{G}(\mu)(t,\theta)\qquad(\textit{disintegration theorem})$
$\displaystyle=\int_{\mathbb{R}^{d}}\mathcal{G}^{*}(\psi)(x)\,d\mu(x)\qquad(\textit{definition
of }\mathcal{G}(\mu))$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\mathbb{S}^{d^{\prime}-1}}\psi(g(x,\theta),\theta)\,d\sigma_{d^{\prime}}(\theta)\,d\mu(x)\qquad(\textit{definition
of }\mathcal{G}^{*})$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}^{d}}\psi(g(x,\theta),\theta)\,d\mu(x)\,d\sigma_{d^{\prime}}(\theta)$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\,d(g(\cdot,\theta)_{\\#}\mu)(t)\,d\sigma_{d^{\prime}}(\theta),$

where the fourth equality follows from Fubini theorem and the fact $\psi$ is
bounded and $\mu,\sigma_{d^{\prime}}$ are finite measures; in the fifth
equality, $t=g(x,\theta)$ and we use the definition of pushforward. By
uniqueness of the disintegration of measures we have
$\mathcal{G}(\mu)_{\theta}=g(\cdot,\theta)_{\\#}\mu$. This also can be shown
by choosing an arbitrary $\psi_{0}\in C_{0}(\mathbb{R})$, defining
$\psi(t,\theta):=\psi_{0}(t),\,\forall(t,\theta)\in\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}$,
so $\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$, and therefore
we have

$\displaystyle\int_{\mathbb{R}}\psi_{0}(t)\,d\mathcal{G}(\mu)_{\theta}(t)$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\,d\mathcal{G}(\mu)_{\theta}(t)\,d\sigma_{d^{\prime}}(\theta)$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\,d(g(\cdot,\theta)_{\\#}\mu)(t)\
d\sigma_{d^{\prime}}(\theta)$
$\displaystyle=\int_{\mathbb{R}}\psi_{0}(t)\,d(g(\cdot,\theta)_{\\#}\mu)(t).$

  6. (6)

For $\mathcal{H}$, properties $(1)-(5)$ can be derived similarly.

  7. (7)

As in $(5)$, we will follow the ideas in [6][Proposition 6]. For any test
function $\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$ we have

$\displaystyle\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\
d\mathcal{H}(\mu)_{\theta}(t)\,d\sigma_{d^{\prime}}(\theta)$
$\displaystyle=\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\
d\mathcal{H}(\mu)(t,\theta)\qquad(\textit{disintegration theorem})$
$\displaystyle=\int_{\mathbb{R}^{d}}\mathcal{H}^{*}(\psi)(x)\,d\mu(x)\qquad(\textit{definition
of }\mathcal{H}(\mu))$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\mathbb{S}^{d^{\prime}-1}}\psi(g(x,\theta),\theta)\,d\sigma_{d^{\prime}}(\theta)\,d\mu(x)\qquad(\textit{definition
of }\mathcal{H}^{*})$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}^{d}}\psi(\langle
h(x),\theta\rangle,\theta)\,d\mu(x)\,d\sigma_{d^{\prime}}(\theta)\qquad(\textit{Fubini's
Theorem})$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(\langle
y,\theta\rangle,\theta)\,d(h_{\\#}\mu)(y)\,d\sigma_{d^{\prime}}(\theta)\qquad(\textit{definition
of pushforward})$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\,d(\theta_{\\#}h_{\\#}\mu)(t)\,d\sigma_{d^{\prime}}(\theta)\qquad(\textit{definition
of pushforward}).$

Thus, by the uniqueness of the disintegration of measures, we have
$\mathcal{H}(\mu)_{\theta}=\theta_{\\#}h_{\\#}\mu=(\theta\circ h)_{\\#}\mu$.

  8. (8)

Let $\mu\in\mathcal{M}(\mathbb{R}^{d})$, and let $\psi\in
C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$ be any test function.
Following the ideas from $(7)$, we have

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{H}(\mu)(t,\theta)$
$\displaystyle=\int_{\mathbb{R}^{d}}\mathcal{H}^{*}(\psi)(x)\,d\mu(x)$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\mathbb{S}^{d^{\prime}-1}}\psi(\langle
h(x),\theta\rangle,\theta)\,d\sigma_{d^{\prime}}(\theta)\,d\mu(x)$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\mathbb{S}^{d^{\prime}-1}}\psi(\langle
y,\theta\rangle,\theta)\,d\sigma_{d^{\prime}}(\theta)\,dh_{\\#}\mu(y)$
$\displaystyle=\int_{\mathbb{R}^{d}}(\mathcal{R}^{*}(\psi))(y)\,dh_{\\#}\mu(y)$
$\displaystyle=\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{R}(h_{\\#}\mu)(t,\theta),$

where in the third equality $y=h(x)$ and we use the definition of pushforward;
the first and fifth equation follows from the equations (27) and (5),
respectively. Therefore, $\mathcal{H}(\mu)=\mathcal{R}(h_{\\#}\mu)$.

  9. (9)

By Proposition 7 in [6], we have the mapping

$\displaystyle h_{\\#}(\mu)$
$\displaystyle\mapsto\mathcal{R}(h_{\\#}(\mu))=\mathcal{H}(\mu)$ (30)

is invertible. If $h$ is injective, we have $\mu\mapsto h_{\\#}(\mu)$ is
invertible. Thus, $\mu\mapsto\mathcal{H}(\mu)$ is invertible.

∎

##  Appendix D Stereographic Spherical Radon Transform and Stereographic
Spherical Sliced Wasserstein Distance for Probability Measures

Based on the extended definition of GRT for probability measures, in this
section, we formally introduce the “Stereographic Spherical Radon Transform”
and the corresponding sliced Wasserstein distance, “Stereographic Spherical
Sliced Wasserstein distance” for general probability measures.

Given $\mu\in\mathcal{M}(\mathbb{S}^{d})$ and the stereographic projection
$\phi:\mathbb{S}^{d}\setminus\\{s_{n}\\}\to\mathbb{R}^{d}$, which is a
bijection, we have that $\phi_{\\#}\mu$ is a measure defined in
$\mathbb{R}^{d}$. The generalized Stereopgraphic Spherical Radon transform, of
$\mu$ has two versions $\mathcal{S}_{\mathcal{G}}(\mu)$ and
$\mathcal{S}_{\mathcal{H}}(\mu)$ defined as follows:

$\mathcal{S}_{\mathcal{G}}(\mu):=\mathcal{G}(\phi_{\\#}\mu)\qquad\text{ and
}\qquad\mathcal{S}_{\mathcal{H}}(\mu):=\mathcal{H}(\phi_{\\#}\mu).$ (31)

That is, from (27) and (28), we have that for each test function $\psi\in
C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{S}_{\mathcal{G}}(\mu)(t,\theta)=\int_{\mathbb{R}^{d}}\mathcal{G}^{*}(\psi)(x)\,d\phi_{\\#}\mu(x)=\int_{\mathbb{S}^{d}\setminus\\{s_{n}\\}}\mathcal{G}^{*}(\psi)(\phi(s))d\mu(s),$
(32)
$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{S}_{\mathcal{H}}(\mu)(t,\theta)=\int_{\mathbb{R}^{d}}\mathcal{H}^{*}(\psi)(x)\,d\phi_{\\#}\mu(x)=\int_{\mathbb{S}^{d}\setminus\\{s_{n}\\}}\mathcal{H}^{*}(\psi)(\phi(s))\,d\mu(s).$
(33)

######  Proposition 8.

Let $\mu\in\mathcal{M}(\mathbb{S}^{d})$ that does not give mass to the North
Pole $\\{s_{n}\\}$. The Stereographic Spherical Radon transforms defined in
(31) have the following properties:

  1. (1)

$\mathcal{S}_{\mathcal{G}}(\mu),\mathcal{S}_{\mathcal{H}}(\mu)\in\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$.
In addition $\mathcal{S}_{\mathcal{G}}$ and $\mathcal{S}_{\mathcal{H}}$
preserves mass, and if $\mu$ is a positive measure, then
$\mathcal{S}_{\mathcal{G}}(\mu),\mathcal{S}_{\mathcal{H}}(\mu)$ are positive
measures too. Finally, if
$\mu\in\mathcal{P}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$, then
$\mathcal{S}_{\mathcal{G}}(\mu),\mathcal{S}_{\mathcal{H}}(\mu)$ are
probability measures defined on $\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}$.

  2. (2)

The disintegration theorem gives a unique
$\mathcal{S}_{\mathcal{G}}(\mu)-$a.s. set of measures
$(\mathcal{S}_{\mathcal{G}}(\mu)_{\theta})_{\theta\in\mathbb{S}^{d^{\prime}-1}}\subset\mathcal{M}(\mathbb{R})$,
and a unique $\mathcal{S}_{\mathcal{H}}(\mu)-$a.s. set of measures
$(\mathcal{S}_{\mathcal{H}}(\mu)_{\theta})_{\theta\in\mathbb{S}^{d^{\prime}-1}}\subset\mathcal{M}(\mathbb{R})$
such that for any $\psi\in C_{0}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$,

$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{S}_{\mathcal{G}}(\mu)(t,\theta)$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\,d\mathcal{S}_{\mathcal{G}}(\mu)_{\theta}(t)\,d\sigma_{d^{\prime}}(\theta),\qquad\text{and}$
$\displaystyle\int_{\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}}\psi(t,\theta)\,d\mathcal{S}_{\mathcal{H}}(\mu)(t,\theta)$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}-1}}\int_{\mathbb{R}}\psi(t,\theta)\,d\mathcal{S}_{\mathcal{H}}(\mu)_{\theta}(t)\,d\sigma_{d^{\prime}}(\theta).$
(34)

Then, it holds that

$\mathcal{S}_{\mathcal{G}}(\mu)_{\theta}=(g(\cdot,\theta)\circ\phi)_{\\#}\mu,\qquad\text{
and }\qquad\mathcal{S}_{\mathcal{H}}(\mu)_{\theta}=(\theta\circ
h\circ\phi)_{\\#}\mu,$ (35)

where we recall that the function $\theta$ is defined as in (21)

  3. (3)

$\mathcal{S}_{\mathcal{H}}$ is invertible.

###### Proof.

  1. (1)

If $\mu\in\mathcal{M}(\mathbb{S}^{d})$ is such that does not give mass to
$\\{s_{n}\\}$, then $\phi_{\\#}\mu$ is a measure defined in $\mathbb{R}^{d}$
with the same total mass. Thus, since
$\mathcal{S}_{\mathcal{G}}(\mu)=\mathcal{G}(\phi_{\\#}\mu)$, the conclusion
follows from by (1)–(4) in Proposition (7). Similar to
$\mathcal{S}_{\mathcal{H}}(\mu)$.

  2. (2)

By definition of $\mathcal{S}_{\mathcal{G}}$ and (5) in Proposition (7) we
have

$\mathcal{S}_{\mathcal{G}}(\mu)_{\theta}=g(\cdot,\theta)_{\\#}(\phi_{\\#}\mu)=g(\phi(\cdot),\theta)_{\\#}\mu.$

Similarly, by definition of $\mathcal{S}_{\mathcal{H}}$ and (7) in Proposition
(7) we have

$\mathcal{S}_{\mathcal{H}}(\mu)_{\theta}=(\theta\circ
h)_{\\#}\phi_{\\#}\mu=(\theta\circ h\circ\phi)_{\\#}\mu.$

  3. (3)

Finally, since the stenographic projection
$\phi:\mathbb{S}^{d}\setminus\\{s_{n}\\}\to\mathbb{R}^{d}$ is invertible, then
following composition

$\displaystyle\mathcal{M}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$
$\displaystyle\to\mathcal{M}(\mathbb{R}^{d})\to\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$
$\displaystyle\mu$ $\displaystyle\longmapsto\phi_{\\#}\mu\ \
\longmapsto\mathcal{H}(\phi_{\\#}\mu)=\mathcal{S}_{\mathcal{H}}(\mu)$

is invertible as each mapping is invertible. Similarly,

$\displaystyle\mathcal{P}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$
$\displaystyle\to\mathcal{P}(\mathbb{R}^{d})\to\mathcal{M}(\mathbb{R}\times\mathbb{S}^{d^{\prime}-1})$
$\displaystyle\mu$ $\displaystyle\longmapsto\phi_{\\#}\mu\ \
\longmapsto\mathcal{H}(\phi_{\\#}\mu)=\mathcal{S}_{\mathcal{H}}(\mu)$

is invertible. Thus $\mu\mapsto\mathcal{S}_{\mathcal{H}}(\mu)$ is an
invertible mapping either general measures or for probability measures.

∎

######  Remark 9.

If $g(x,\theta)=\langle x,\theta\rangle$, then the generalized Radon transform
$\mathcal{G}$ coincides with the classical Radon transform $\mathcal{R}$, and
the corresponding Stereographic Spherical Radon transform coincides with the
one defined in (10):

$\mathcal{S}_{\mathcal{R}}(\mu)=\mathcal{R}(\phi_{\\#}\mu)$

for any Radon measure $\mu\in\mathcal{M}(\mathbb{S}^{d})$ that does not assign
mass to $\\{s_{n}\\}$

We introduce the notation
$\phi_{g}^{\theta},\phi_{h}^{\theta}:\mathbb{S}^{d}\backslash\\{s_{n}\\}\rightarrow\mathbb{R}^{d^{\prime}}$

$\phi_{g}^{\theta}(z):=g(\phi(z),\theta)\theta,\qquad\phi_{h}^{\theta}(z):=\langle
h(\phi(z),\theta\rangle\theta,$

having range in the line generated by $\theta\in\mathbb{S}^{d^{\prime}-1}$.
The expressions (13) in (2) in the above Proposition can be written as

$\mathcal{S}_{\mathcal{G}}(\mu)_{\theta}={\phi_{g}^{\theta}}_{\\#}\mu,\qquad\mathcal{S}_{\mathcal{H}}(\mu)_{\theta}={\phi_{h}^{\theta}}_{\\#}\mu$

and we call them slices of the SSRT of $\mu$.

Given two measures $\mu_{1},\mu_{2}\in\mathcal{P}(\mathbb{S}^{d})$, we
introduce the formal definition of Stereographic Spherical Sliced Wasserstein
(S3W) distance as follows:

$\displaystyle S3W_{\mathcal{G},p}^{p}(\mu_{1},\mu_{2})$
$\displaystyle:=\int_{\mathbb{S}^{d^{\prime}-1}}W_{p}^{p}(\mathcal{S}_{\mathcal{G}}(\mu_{1})_{\theta},\mathcal{S}_{\mathcal{G}}(\mu_{2})_{\theta})\,d\sigma_{d^{\prime}}(\theta)$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}-1}}W_{p}^{p}({\phi_{g}^{\theta}}_{\\#}\mu_{1},{\phi_{g}^{\theta}}_{\\#}\mu_{2})\,d\sigma_{d^{\prime}}(\theta)$
(36) $\displaystyle S3W_{\mathcal{H},p}^{p}(\mu_{1},\mu_{2})$
$\displaystyle:=\int_{\mathbb{S}^{d^{\prime}-1}}W_{p}^{p}(\mathcal{S}_{\mathcal{H}}(\mu_{1})_{\theta},\mathcal{S}_{\mathcal{H}}(\mu_{2})_{\theta})\,d\sigma_{d^{\prime}}(\theta)$
$\displaystyle=\int_{\mathbb{S}^{d^{\prime}}}W_{p}^{p}({\phi_{h}^{\theta}}_{\\#}\mu_{1},{\phi_{h}^{\theta}}_{\\#}\mu_{2})\,d\sigma_{d^{\prime}}(\theta),$
(37)

where $\sigma_{d^{\prime}}\in\mathcal{P}(\mathbb{S}^{d^{\prime}-1})$ is the
uniform measure on $\mathbb{S}^{d^{\prime}-1}$.

######  Theorem 10.

The above definitions $S3W_{\mathcal{G},p}(\cdot,\cdot)$ and
$S3W_{\mathcal{H},p}(\cdot,\cdot)$ are well-defined. Furthermore,
$S3W_{\mathcal{G},p}(\cdot,\cdot)$ is a pseudo-metric in
$\mathcal{P}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$, i.e., it is non-negative,
symmetric and satisfies triangular inequality. In addition,
$S3W_{\mathcal{H},p}(\cdot,\cdot)$ defines a metric in
$\mathcal{P}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$.

###### Proof.

Given probability measures
$\mu_{1},\mu_{2}\in\mathcal{P}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$, by (1) in
Proposition 8 we have that
$\mathcal{S}_{\mathcal{G}}(\mu_{1}),\mathcal{S}_{\mathcal{G}}(\mu_{2})$ are
probability measures defined on $\mathbb{R}\times\mathbb{S}^{d^{\prime}-1}$.
Thus, by (2) in Proposition 8,
$\mathcal{S}_{\mathcal{G}}(\mu_{1})_{\theta},\mathcal{S}_{\mathcal{G}}(\mu_{2})_{\theta}$
are probability measure on $\mathbb{R}$. Then,
$W_{p}^{p}(\mathcal{S}_{\mathcal{G}}(\mu_{1})_{\theta},\mathcal{S}_{\mathcal{G}}(\mu_{2})_{\theta})$
is well-defined and thus $S3W_{\mathcal{G},p}^{p}(\cdot,\cdot)$ is well
defined. Analogously, $S3W_{\mathcal{H},p}^{p}(\cdot,\cdot)$ is well-defined.

Since the Wasserstein distance $W_{p}(\cdot,\cdot)$ in
$\mathcal{P}(\mathbb{R})$ is non-negative and symmetric and
$\sigma_{d^{\prime}}$ is a positive measure we have that
$S3W_{\mathcal{G},p}(\cdot,\cdot)$ and $S3W_{\mathcal{H},p}(\cdot,\cdot)$ are
non-negative and symmetric.

For triangular inequality, let
$\mu_{1},\mu_{2},\mu_{3}\in\mathcal{P}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$,

$\displaystyle S3W_{\mathcal{G},p}(\mu_{1},\mu_{3})$
$\displaystyle=\left(\int_{\mathbb{S}^{d^{\prime}-1}}W_{p}^{p}(\mathcal{S}_{\mathcal{G}}(\mu_{1})_{\theta},\mathcal{S}_{\mathcal{G}}(\mu_{3})_{\theta})d\sigma_{d^{\prime}}(\theta)\right)^{1/p}$
$\displaystyle\leq\left(\int_{\mathbb{S}^{d^{\prime}-1}}\left(W_{p}(\mathcal{S}_{\mathcal{G}}(\mu_{1})_{\theta},\mathcal{S}_{\mathcal{G}}(\mu_{2})_{\theta})+W_{p}(\mathcal{S}_{\mathcal{G}}(\mu_{2})_{\theta},\mathcal{S}_{\mathcal{G}}(\mu_{3})_{\theta})\right)^{p}d\sigma_{d^{\prime}}(\theta)\right)^{1/p}$
$\displaystyle\leq\left(\int_{\mathbb{S}^{d^{\prime}-1}}W_{p}^{p}(\mathcal{S}_{\mathcal{G}}(\mu_{1})_{\theta},\mathcal{S}_{\mathcal{G}}(\mu_{2})_{\theta})\right)^{1/p}+\left(\int_{\mathbb{S}^{d^{\prime}-1}}W_{p}^{p}(\mathcal{S}_{\mathcal{G}}(\mu_{2})_{\theta},\mathcal{S}_{\mathcal{G}}(\mu_{3})_{\theta})\right)^{1/p}$
$\displaystyle=S3W_{\mathcal{G},p}(\mu_{1},\mu_{2})+S3W_{\mathcal{G},p}(\mu_{2},\mu_{3})$

where the first inequality holds from the fact $W_{p}(\cdot,\cdot)$ in
$\mathcal{P}(\mathbb{R})$ is a metric; the second inequality follows from
Minkowski inequality in $L^{p}(\mathbb{S}^{d^{\prime}-1})$. Then,
$S3W_{\mathcal{G},p}(\cdot,\cdot)$ is a Pseudo-metric. Analogously for
$S3W_{\mathcal{H},p}(\cdot,\cdot)$.

In addition, suppose $S3W_{\mathcal{H},p}(\mu_{1},\mu_{2})=0$, we have
$\mathcal{S}_{\mathcal{H}}(\mu_{1})_{\theta}=\mathcal{S}_{\mathcal{H}}(\mu_{2})_{\theta}$
where the equality holds $\sigma_{d^{\prime}}(\theta)-$a.s. Then, from (34) we
can deduce that
$\mathcal{S}_{\mathcal{H}}(\mu_{1})=\mathcal{S}_{\mathcal{H}}(\mu_{2})$. Since
$\mathcal{S}_{\mathcal{H}}$ is invertible by Proposition (8), we have
$\mu_{1}=\mu_{2}$. Thus, $S3W_{\mathcal{H},p}(\cdot,\cdot)$ is a metric.

∎

Lastly, and similar to the max-sliced Wasserstein distance, here we define the
max-S3W distance as:

$\displaystyle\text{Max-}S3W_{\mathcal{G},p}^{p}(\mu_{1},\mu_{2}):=\sup_{\theta\in\mathbb{S}^{d^{\prime}-1}}~{}~{}W_{p}^{p}(\mathcal{S}_{\mathcal{G}}(\mu_{1})_{\theta},\mathcal{S}_{\mathcal{G}}(\mu_{2})_{\theta}),$
(38)

and similarly for $\mathcal{H}$ we define
$\text{Max-}S3W_{\mathcal{H},p}^{p}$.

##  Appendix E Rotationally Invariant Stereographic Spherical Sliced
Wasserstein Distances (RI-S3W)

In this section, we discuss the Rotationally Invariant $S3W$ distance (16):

$RI\text{-}S3W_{\mathcal{G},p}(\mu,\nu):=\mathbb{E}_{R\sim\omega}[S3W_{\mathcal{G},p}(R_{\\#}\mu,R_{\\#}\nu)].$

First, we describe the uniform (Haar) distribution in $\mathrm{O}(d+1)$
denoted as $\omega$. Precisely, we will describe how a $(d+1)\times(d+1)$
orthonormal matrix $R$ with rows $v_{1},\dots,v_{d+1}$ can be randomly
generated as a sample of $\omega$. The procedure relies on the Gram-Schmidt
algorithm:

First, we randomly select $v_{1}\in\mathbb{S}^{d}$, in particular, we denote
it as $v_{1}\sim\sigma_{d+1}$, that is, we sample $v_{1}$ according to
$\sigma_{d+1}$ which is the uniform distribution in $\mathbb{S}^{d}$.

Next, $v_{2}$ should selected from
$\text{span}(v_{1})^{\perp}=H(v_{1},0)=\\{v:\langle v_{1},v\rangle=0\\}$ and
satisfy $\|v_{2}\|=1$, where $\text{span}(v_{1})$ is the 1D sub-space in
$\mathbb{R}^{d}$ spanned by $v_{1}$. Denote
$\mathbb{S}^{d-1}(v_{1}):=\mathbb{S}^{d}\cap H(v_{1},0)$. Note, it is
essentially a $d-1$ sphere. We randomly select
$v_{2}\in\mathbb{S}^{d-1}(v_{1})$, denoted as $v_{2}\sim\sigma_{d}$ given
$v_{1}$, where $\sigma_{d}$ denotes the uniform distribution in sphere
$\mathbb{S}^{d-1}(v_{1})$.

In the third step, similarly, let
$\mathbb{S}^{d-2}(v_{1},v_{2}):=\mathbb{S}^{d}\cap H(v_{1},0)\cap H(v_{2},0)$,
and sample $v_{3}\sim\sigma_{d-1}$ given $v_{1},v_{2}$, where $\sigma_{d-1}$
is the uniform distribution in the $d-2$ sphere
$\mathbb{S}^{d-2}(v_{1},v_{2})$.

Continue this process recursively until step $d+1$. Note that at the last
iteration, we have an orthonormal set of vectors $\\{v_{1},\ldots,v_{d}\\}$,
and $\mathbb{S}^{0}(v_{1},\ldots v_{d})$ is a $0$-dimensional sphere, that is,
it is a set of two points $\\{u_{1},u_{2}\\}$. We randomly select $v_{d+1}$ as
one of those two points in $\mathbb{S}^{0}(v_{1},\ldots v_{d})$, denoting
$v_{d+1}\sim\sigma_{1}$, where $\sigma_{1}$ is the uniform distribution in
$\mathbb{S}^{0}(v_{1},\ldots,v_{d})$ (that is,
$\sigma_{1}=0.5\delta_{u_{1}}+0.5\delta_{u_{2}}$).

Thus, the uniform probability measure $\omega$ in $\mathrm{O}(d+1)$ can be
described in the following way: Consider an arbitrary Borel set
$A\subset\mathrm{O}(d+1)$, and let $A_{i}:=\\{v\in\mathbb{S}^{d}:\,v\text{ is
the $i-$th row of }R,\,R\in A\\}$ for $i=1,2,\ldots d+1$, then

$\displaystyle\omega(A):=\int_{v_{1}\in A_{1},\ldots,v_{d+1}\in
A_{d+1}}d\sigma_{1}(v_{d+1})\ d\sigma_{2}(v_{d})\ldots
d\sigma_{d}(v_{2})\,d\sigma_{d+1}(v_{1}).$ (39)

where we are using the above notation, that is, $\sigma_{d+1}$ is the uniform
distribution on $\mathbb{S}^{d}$, and $\sigma_{d+1-k}$ is the uniform
distribution on $\mathbb{S}^{d}(v_{1},\dots,v_{k})$ for $|\leq k\leq d$ (i.e.,
$\sigma_{d}$ is the uniform distribution on $\mathbb{S}^{d}(v_{1})$, $\ldots$
, $\sigma_{2}$ is the uniform distribution on
$\mathbb{S}^{d}(v_{1},\dots,v_{d-1})$, $\sigma_{1}$ is the uniform
distribution on $\mathbb{S}^{d}(v_{1},\dots,v_{d})$). In particular, $\omega$
is constant, in the sense that its density function is constant: Indeed, let
$f_{\sigma_{d+1}},\dots,f_{\sigma_{2}}$ denote the density functions of
$\sigma_{d+1},\dots,\sigma_{2}$ (which are all constant equal to the
reciprocal of the “surface area” of the corresponding sphere). Then, let
$f_{\omega}$ denote the density function of probability measure $\omega$, and
so, for each $R=[v_{1}^{T},v_{2}^{T},\ldots,v_{d+1}^{T}]\in\mathrm{O}(d+1)$

$\displaystyle f_{\omega}(R)=f_{\sigma_{d+1}}(v_{1})\cdot\dots\cdot
f_{\sigma_{2}}(v_{d})\cdot{\sigma_{1}}(v_{d+1})=\frac{1}{2}\prod_{i=1}^{d}\frac{1}{\|\mathbb{S}^{i}\|_{i}}$
(40)

Thus we call $\omega$ uniform distribution. We mention that it is also called
a Haar measure for the special orthogonal group $\mathrm{SO}(d+1)$. Finally,
we can normalize it so that we have a probability distribution.

Next, we discuss the proof of Theorem 4. We first introduce the following
lemma.

######  Lemma 11.

Consider $s_{n}=[0,\dots,0,1]\in\mathbb{R}^{d+1}$ the North Pole of
$\mathbb{S}^{d}$. Let $s_{1}\in\mathbb{S}^{d}$ and
$A=\\{R\in\mathrm{O}(d+1):\,Rs_{1}=s_{n}\\}$. Then
$A=\\{R\in\mathrm{O}(d+1):\text{ the $(d+1)-$th row of $R$ is }s_{1}\\}$.
Furthermore, $\omega(A)=0$.

###### Proof.

Let $B=\\{R\in\mathrm{O}(d+1):\,\text{the $(d+1)-$th row of $R$ is }s_{1}\\}$.
It is straightforward to verify $B\subset A$. For the other direction, for any
$R=[v_{1}^{T},\ldots,v_{d+1}^{T}]\in A$ we have

$\displaystyle v_{i}^{T}s_{1}=0$ $\displaystyle\text{ if }i\neq d+1$
$\displaystyle v_{d+1}^{T}s_{1}=1$ $\displaystyle\text{ if }i=d+1$

Thus, $Rs_{1}=s_{n}$, and we have $A\subset B$.

Next, we will show $\omega(A)=0$. Let
$A^{\prime}=\\{R\in\mathrm{O}(d+1):\text{ 1st row of $R$ is }s_{1}\\}$. We
have $\omega(A^{\prime})=\omega(A)$. Then $A^{\prime}_{1}=\\{s_{1}\\}$. and
(39), we have

$\displaystyle\omega(A^{\prime})$
$\displaystyle\leq\int_{v_{1}=s_{1}}\left(\int_{v_{2},\ldots
v_{d}\in\mathbb{S}^{d}}d\sigma_{1}(v_{d+1})\ldots
d\sigma_{d}(v_{2})\right)d\sigma_{d+1}(v_{1})=0,$

which completes the proof. ∎

###### Proof of Theorem 4.

.

We first claim that (16) is well-defined.

Let $S(\mu,\nu)=\\{s\in\mathbb{S}^{d}:\min(\mu(\\{s\\}),\nu(\\{s\\}))>0\\}$.
Since $\mu,\nu$ are probability measures, we have that $S(\mu,\nu)$ is a
finite set. For each $s\in S$, let
$A(s)=\\{R\in\mathrm{O}(d+1):\,Rs=s_{n}\\}$. Note that
$S3W_{\mathcal{G},p}(R_{\\#}\mu,R_{\\#}\nu)$ is well-defined for each
$R\in\mathrm{O}(d+1)\setminus A(\mu,\nu)$, where $A(\mu,\nu)=\bigcup_{s\in
S(\mu,\nu)}A(s)$. By lemma 11,

$\omega(A(\mu,\nu))=\omega\left(\bigcup_{s\in
S(\mu,\nu)}A(s)\right)\leq\sum_{s\in S(\mu,\nu)}\omega(A(s))=0.$

Thus, we have that

$\displaystyle
RI-S3W_{\mathcal{G},p}(\mu,\nu):=\mathbb{E}_{R\sim\omega}[S3W_{\mathcal{G},p}(R_{\\#}\mu,R_{\\#}\nu)]=\int_{R\in\mathrm{O}(d+1)\setminus
A(\mu,\nu)}S3W_{\mathcal{G},p}(R_{\\#}\mu,R_{\\#}\nu)\,d\omega(R)$

is well-defined.

Now, we will verify its properties.

By Theorem 10, $S3W_{\mathcal{G},p}$ is non-negative, and symmetric. Thus
$RI\text{-}S3W_{\mathcal{G},p}(\cdot,\cdot)$ is also non-negative and
symmetric.

For the triangle inequality, let
$\mu_{1},\mu_{2},\mu_{3}\in\mathcal{P}(\mathbb{S}^{d})$. Similarly to before,
let $A(\mu_{1},\mu_{2},\mu_{3})=\bigcup_{s\in
S(\mu_{1},\mu_{2},\mu_{3})}{A(s)}$. Since
$\omega(A(\mu_{1},\mu_{2},\mu_{3}))=0$, we have

$\displaystyle
RI-S3W_{\mathcal{G},p}(\mu_{1},\mu_{3})=\int_{\mathrm{O}(d+1)}S3W_{\mathcal{G},p}(R_{\\#}\mu,R_{\\#}\nu)\,d\omega(R)=\int_{\mathrm{SO}(d+1)\setminus
A(\mu_{1},\mu_{2},\mu_{3})}S3W_{\mathcal{G},p}(R_{\\#}\mu,R_{\\#}\nu)\,d\omega(R)$
$\displaystyle\leq\int_{\mathrm{SO}(d+1)\setminus
A(\mu_{1},\mu_{2},\mu_{3})}\left(S3W_{\mathcal{G},p}(R_{\\#}\mu_{1},R_{\\#}\mu_{2})+S3W_{\mathcal{G},p}(R_{\\#}\mu_{2},R_{\\#}\mu_{3})\right)\,d\omega(R)\qquad\text{(Triangular
inequality)}$ $\displaystyle=\int_{\mathrm{SO}(d+1)\setminus
A(\mu_{1},\mu_{2},\mu_{3})}S3W_{\mathcal{G},p}(R_{\\#}\mu_{1},R_{\\#}\mu_{2})\,d\omega(R)+\int_{\mathrm{SO}(d+1)\setminus
A(\mu_{1},\mu_{2},\mu_{3})}S3W_{\mathcal{G},p}(R_{\\#}\mu_{2},R_{\\#}\mu_{3})\,d\omega(R)$
$\displaystyle=\int_{\mathrm{SO}(d+1)}S3W_{\mathcal{G},p}(R_{\\#}\mu_{1},R_{\\#}\mu_{2})\,d\omega(R)+\int_{\mathrm{SO}(d+1)}S3W_{\mathcal{G},p}(R_{\\#}\mu_{2},R_{\\#}\mu_{3})\,d\omega(R)$
$\displaystyle=RI-S3W_{\mathcal{G},p}(\mu_{1},\mu_{2})+RI-S3W_{\mathcal{G},p}(\mu_{2},\mu_{3})$

where the second and previous to last equalities hold from the fact
$\omega(A(\mu_{1},\mu_{2},\mu_{3}))=0$. Thus,
$RI\text{-}S3W_{\mathcal{G},p}(\cdot,\cdot)$ is a well-defined pseudo-metric.

Similarly, we can prove $RI\text{-}S3W_{\mathcal{H},p}(\cdot,\cdot)$ is a
well-defined pseudo-metric.

Finally, it remains to show that $\mu=\nu$ if
$RI\text{-}S3W_{\mathcal{H},p}(\mu,\nu)=0$. If that holds, then for all
$R\in\mathrm{SO}(d+1)\setminus A(\mu,\nu)$, we have
$S3W_{\mathcal{H},p}(R_{\\#}\mu,R_{\\#}\nu)=0$. Pick one of such rotations
$R$. From Theorem 10, and the fact
$R_{\\#}\mu,R_{\\#}\nu\in\mathcal{P}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$, we
have $R_{\\#}\mu=R_{\\#}\nu$. Since

$\mathbb{R}^{d+1}\ni v\mapsto Rv\in\mathbb{R}^{d+1}$

is a one-by-one mapping, we have $\mu=\nu$.

∎

##  Appendix F Funk Radon Transform and Stereographic Spherical Radon
Transform

The Funk-Radon Transform [66, 34, 63, 64], which is also known as Minkowski-
Funk transform, or spherical Radon transform is a classical and significant
mathematical tool used in integral geometry, with profound applications in
image reconstruction and analysis. It plays a crucial role in fields like
tomography, allowing for the reconstruction of images from projection data. In
this section, we discuss the relation between the Funk Transform and the
Stereographic Spherical Radon Transform.

First, we review some basic concepts about the Funk transform. In the sphere
$\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$ centered at the origin with radius
$1$, every $(d-1)-$dimensional sub-sphere (“circle”) is the intersection of
$\mathbb{S}^{d}$ with a $d-$dimensional hyperplane, that is,

$S(s,t)=\\{s^{\prime}\in\mathbb{S}^{d}\mid\,\langle s,s^{\prime}\rangle=t\\},$

where $s\in\mathbb{S}^{d}$ is normal to the hyperplane and $t\in[-1,1]$ is the
signed distance of the hyperplane to origin. For example, if $t=\pm 1$,
$S(s,t)$ consists of only the singleton $\\{\pm s\\}$. As an another example,
when $t=0$, for any $s\in\mathbb{S}^{d}$, the sub-sphere $S(s,0)$ is a “great
circle” (a sub-sphere of dimension $d-1$ in $\mathbb{R}^{d+1}$ centered at the
origin and with radius $1$).

The Spherical Transform or Spherical Mean Operator [63] of a function defines
on $\mathbb{S}^{d}$ is formally defined as a function on
$\mathbb{S}^{d}\times[-1,1]$ by the surface integral

$\displaystyle\mathcal{U}f(s,t):=\frac{1}{\|S(s,t)\|_{d-1}}\int_{S(s,t)}f\,dS(s,t)=\frac{1}{\|S(s,t)\|_{d-1}}\int_{\mathbb{R}^{d+1}}f(s^{\prime})\delta(t-\langle
s,s^{\prime}\rangle)\,ds^{\prime},\quad(s,t)\in\mathbb{S}^{d}\times[-1,1],$
(41)

where $\|S(s,t)\|_{d-1}$ denotes the “surface area” of the sub-sphere
$S(s,t)$, and $dS(s,t)$ is the induced volume form on the surface $S(s,t)$. In
the special case $d=2$, given $f$ defined on
$\mathbb{S}^{2}\subset\mathbb{R}^{3}$, the integral in (41) is carried out
with respect to the arclength $dS$ of the circle $S(s,t)$, that is,

$\displaystyle\mathcal{U}f(s,t)=\frac{1}{2\pi\cdot\text{radius}(S(s,t))}\int_{S(s,t)}f\,dS(s,t):=\frac{1}{2\pi\cdot\text{radius}(S(s,t))}\int_{a}^{b}f(r(u))\|r^{\prime}(u)\|\,du,$

where $\text{radius}({S(s,t)})$ denotes the radius of the circle $S(s,t)$,
$r(u)$ for $a\leq u\leq b$ is a parametrization of $S(s,t)$, and
$\|r^{\prime}(u)\|$ is the magnitude of the tangent vector $r^{\prime}(u)$ to
$S(s,t)$. Besides, in the special case $d=3$, given $f$ defined on
$\mathbb{S}^{3}\subset\mathbb{R}^{4}$, we have that $S(s,t)$ is
$2$-dimensional sphere and we get

$\displaystyle\mathcal{U}f(s,t)$
$\displaystyle=\frac{1}{\frac{4}{3}\pi\cdot\text{radius}(S(s,t))^{3}}\int_{S(s,t)}f\,dS(s,t)$
$\displaystyle:=\frac{1}{\frac{4}{3}\pi\cdot\text{radius}(S(s,t))^{3}}\int\int
f(r(u,v))\|r_{u}\times r_{v}\|\,du\,dv,$

where $\text{radius}({S(s,t)})$ denotes the radius of the sphere $S(s,t)$,
$r(u,v)$ is a parametrization of $S(s,t)$, and $\|r_{u}\times r_{v}\|$ is the
magnitude of the cross-product between the partial derivatives $r_{u}$ and
$r_{v}$, which is a normal vector to $S(s,t)$.

The inversion of this Spherical Mean Operator is an over-determined problem
(for instance $\mathcal{U}f(s,t)=f(s)$ for all $s\in\mathbb{S}^{d}$). In
application, one has access restrictions of $\mathcal{U}f(s,t)$ to specific
sub-spheres $S(s,t)$, and this yields different transforms.

One important example arises when we restrict $\mathcal{U}f(s,t)$ to “great
circles” $S(s,0)$ getting the classical Funk-Radon transform
$\mathcal{F}_{1}$, that is,

$\displaystyle\mathcal{F}_{1}f(s):=\mathcal{U}f(s,0),\qquad
s\in\mathbb{S}^{d}.$ (42)

Another example, studied in [1, 30, 34] arises when considering the
restriction to the family of sub-spheres containing the North Pole
$s_{n}=[0,\dots,0,1]\in\mathbb{R}^{d+1}$.

Similarly as in [63, 64], we will interested in a generalized Funck-Radon
transform given by the following restriction of $\mathcal{U}$. Let
$s=[s_{1},\dots,s_{d+1}]\in\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, and
$\xi\in[-1,1]$. Consider the point $[0,\ldots,0,\xi
s_{d+1}]\in\mathbb{R}^{d+1}$ located in the same positive axis as the North
Pole $s_{n}=[0,\dots,0,1]$ and inside the sphere $\mathbb{S}^{d}$333In [63,
64], $\xi\in[0,1)$ and so $[0,\ldots,0,\xi s_{d+1}]$ lies strictly inside the
sphere.. Then, we restrict $\mathcal{U}f$ to the integrals over the
intersections of the sphere $\mathbb{S}^{d}$ with hyperplanes containing the
point $[0,\ldots,0,\xi s_{d+1}]$, that is,

$\zeta_{\xi}^{s}:=S(s,\xi s_{d+1})=\\{s^{\prime}\in\mathbb{S}^{d}|\,\langle
s,s^{\prime}\rangle=\xi s_{d+1}\\},\qquad\xi\in[-1,1],\,s\in\mathbb{S}^{d}.$

Then, the generalized Funk-Radon transform is defined by

$\displaystyle\mathcal{U}_{\xi}f(s):=\mathcal{U}f(s,\xi
s_{d+1})=\frac{1}{\|\zeta_{\xi}^{s}\|_{d-1}}\int_{\zeta_{\xi}^{s}}f\,d\zeta_{\xi}^{s}=\frac{1}{\|\zeta_{\xi}^{s}\|_{d-1}}\int_{\mathbb{R}^{d+1}}f(s^{\prime})\delta(\xi
s_{d+1}-\langle s,s^{\prime}\rangle)\,ds^{\prime},$ (43)

where $\|\zeta_{\xi}^{s}\|_{d-1}$ is the volume (“surface area”) of sub-sphere
(circle, if $d=2$) $\zeta_{\xi}^{s}$ and $d\zeta_{\xi}^{s}$ is the integration
over the surface $\zeta_{\xi}^{s}$ (path, if $d=2$).

As said before, the goal of this section is to discuss the relationship
between a generalized Funk-Radon transform (43) and the Stereographic Radon
transform. For doing that let us first consider the stereographic projection
$\phi:\mathbb{S}^{d}\setminus\\{s_{n}\\}\to\mathbb{R}^{d}$ given by the
formula

$\phi(s)=\frac{s[1:d]}{1-s_{d+1}},\qquad\forall
s=[s_{1},\dots,s_{d},s_{d+1}]\in\mathbb{S}^{d},$ (44)

where $s[1:d]:=[s_{1},\dots,s_{d}]\in\mathbb{R}^{d}$, which corresponds to the
intersection with the equator plane $x_{d+1}=0$, rather than (1) which
corresponds to project onto the plane $x_{d+1}=-1$ (which has a factor of $2$
in (44)).

It is straightforward to verify that the inverse of $\phi$, denoted as
$\phi^{-1}$, is as follows:

$\phi^{-1}(x)=[(1-s_{d+1})x,s_{d+1}],\quad\text{where
}s_{d+1}=\frac{\|x\|^{2}-1}{\|x\|^{2}+1},\qquad\forall x\in\mathbb{R}^{d}.$
(45)

See Figures 6, 7 and 8.

Figure 6: Visualization for $d=1$ of the stereographic projection $\phi$
defined by the formula (1). The ambient space $\mathbb{R}^{d+1}$ is depicted
in green, while the unit sphere (circle) $\mathbb{S}^{d}$ is in blue. The
North and South Poles are labeled as $s_{n}$ and $s_{0}$, respectively. The
projected space $\mathbb{R}^{d}=\phi(\mathbb{S}^{d}\setminus{s_{n}})$ is
highlighted in red, corresponding to the plane (line) in $\mathbb{R}^{d+1}$
defined by all points with the last coordinate equal to $-1$. Two points,
$\theta_{1}$ and $\theta_{2}$ (blue dots), from $\mathbb{S}^{d}$ are projected
through $\phi$ obtaining the points $\phi(\theta_{1})$ and $\phi(\theta_{2})$
(red dots). Figure 7: Visualization for $d=1$ of the stereographic projection
$\phi$ defined by the formula (44). The ambient space $\mathbb{R}^{d+1}$ is
depicted in green, while the unit sphere (circle) $\mathbb{S}^{d}$ is in blue.
The North and South Poles are labeled as $s_{n}$ and $s_{0}$, respectively.
The projected space $\mathbb{R}^{d}=\phi(\mathbb{S}^{d}\setminus{s_{n}})$
coincides with the hyperplane $x_{d+1}=0$, which in the figure is nothing but
the horizontal axis. Two points, $\theta_{1}$ and $\theta_{2}$ (blue dots),
from $\mathbb{S}^{d}$ are projected through $\phi$ obtaining the points
$\phi(\theta_{1})$ and $\phi(\theta_{2})$ (red dots). The South Pole is
projected to the origin in $\mathbb{R}^{d}$ and the points in the Equator are
fixed points for this stereographic projection. Figure 8: Visualization for
$d=2$ of the stereographic projection $\phi$ defined by the formula (44). The
ambient space is the 3D space $\mathbb{R}^{3}$. The sphere $\mathbb{S}^{2}$ is
depicted in blue. The projected space
$\mathbb{R}^{2}=\phi(\mathbb{S}^{2}\setminus{s_{n}})$ is depicted in grey. The
points in the Equator are fixed points for this stereographic projection, and
circles parallel to the Equator are mapped to circles in the plane as shown in
the plot: If the height of the circle is positive, its projection is a circle
with radius greater than 1 as the case depicted in the figure; if the height
is negative, its projection is a circle with radius smaller than 1.

When we set $\xi=1$ we have the following relations between the sub-spheres
$\zeta_{1}^{s}$ and the hyperplanes $H(t,\theta)$ through the stereographic
projection $\phi$.

######  Proposition 12.

Consider the sub-sphere (circle)
$\zeta_{1}^{s}=\\{s^{\prime}\in\mathbb{S}^{d}|\,\langle
s^{\prime},s\rangle=s_{d+1}\\}$, where
$s=[s_{1},\dots,s_{d},s_{d+1}]\in\mathbb{S}^{d}$ is such that $s\neq
s_{n}=[0,\ldots,0,1]$. Then, $\phi(\zeta_{1}^{s})$ is a hyperplane in
$\mathbb{R}^{d}$, precisely,

$\phi(\zeta_{1}^{s})=\left\\{x\in\mathbb{R}^{d}\mid\,\langle
x,\theta\rangle=t\right\\},$

where $\theta=\frac{s[1:d]}{\|s[1:d]\|}$ and
$t=\frac{s_{d+1}}{\sqrt{1-s_{d+1}^{2}}}$.

Conversely, given hyperplane $H(t,\theta)=\\{x\in\mathbb{R}^{d}\mid\,\langle
x,\theta\rangle=t\\}$ for $\theta\in\mathbb{S}^{d-1}$ and $t\in\mathbb{R}$, we
have

$\phi^{-1}(H(t,\theta))=\zeta_{1}^{s},$

where $s=[\sqrt{1-s_{d+1}^{2}}\theta,s_{d+1}]$ with
$s_{d+1}=\text{sign}(t)\frac{t^{2}}{t^{2}+1}$.

###### Proof.

Let $s\in\mathbb{S}^{d}$, and consider $s^{\prime}\in\zeta_{1}^{s}$. Let
$x^{\prime}=\phi(s^{\prime})=\frac{s^{\prime}[1:d]}{1-s^{\prime}_{d+1}}$.
Then, we can write
$s^{\prime}=[(1-s_{d+1}^{\prime})x^{\prime},s^{\prime}_{d+1}]$. Besides, since

$s_{1}^{2}+\dots+s_{d}^{2}+s_{d+1}^{2}=1,$

by defining $\theta=\frac{s[1:d]}{\|s[1:d]\|}$, we can write
$s=\left[\sqrt{1-s_{d+1}^{2}}\theta,s_{d+1}\right]$. Then, we have that

$\displaystyle s_{d+1}$ $\displaystyle=\langle s^{\prime},s\rangle$
$\displaystyle=s^{\prime}_{d+1}s_{d+1}+\langle(1-s_{d+1}^{\prime})x^{\prime},\sqrt{1-s_{d+1}^{2}}\theta\rangle$
$\displaystyle=s^{\prime}_{d+1}s_{d+1}+(1-s^{\prime}_{d+1})\sqrt{1-s^{2}_{d+1}}\langle
x^{\prime},\theta\rangle.$ (46)

That is,

$\langle x^{\prime},\theta\rangle=\frac{s_{d+1}}{\sqrt{1-s_{d+1}^{2}}}.$

Thus,

$\phi(\zeta_{1}^{s})=\left\\{x\in\mathbb{R}^{d}\mid\,\langle
x,\theta\rangle=\frac{s_{d+1}}{\sqrt{1-s_{d+1}^{2}}}\right\\}.$

For the converse, given $\theta\in\mathbb{S}^{d-1}$ and $t\in\mathbb{R}$,
consider $x^{\prime}\in H(t,\theta)$. By using (45), let

$s^{\prime}=\phi^{-1}(x)=[(1-s^{\prime}_{d+1})x,s^{\prime}_{d+1}],\qquad\forall
x\in\mathbb{R}^{d}$

where $s^{\prime}_{d+1}=\frac{\|x^{\prime}\|^{2}-1}{\|x^{\prime}\|^{2}+1}$.

We set $s=[\sqrt{1-s_{d+1}^{2}}\theta,s_{d+1}]$ with
$s_{d+1}=\text{sign}(t)\sqrt{\frac{t^{2}}{t^{2}+1}}$. So, we have that
$s\in\mathbb{S}^{d}$ because

$(1-s_{d+1}^{2})\underbrace{\theta^{2}}_{=1}+s_{d+1}^{2}=1,$

and also we have that $t=\frac{s_{d+1}}{\sqrt{1-s_{d+1}^{2}}}$. Thus, we have
that

$\langle x^{\prime},\theta\rangle=t=\frac{s_{d+1}}{\sqrt{1-s_{d+1}^{2}}}$

and by reversing the above proof (see (F)), we have that it implies that

$s_{d+1}=\langle s^{\prime},s\rangle,$

and so $s^{\prime}\in\zeta_{1}^{s}.$ Combining with the fact $\phi$ is
invertible, we have

$\phi^{-1}(H(t,\theta))=\zeta_{1}^{s}.$

∎

######  Lemma 13.

Let $s\in\mathbb{S}^{d}$ different from the North Pole $s_{n}=[0,\dots,0,1]$,
the South Pole $s_{0}=[0,\dots,0,-1]$, and consider $S_{0}$ the unique great
circle (meridian) that passes through the North Pole $s_{n}$, the South Pole
$s_{0}$, and the point $s$. For each $\hat{s}\in S_{0}$, we define the
functions

$\theta(\hat{s}):=\frac{\hat{s}[1:d]}{\|\hat{s}[1:d]\|},\qquad
t(\hat{s}):=\frac{s_{d+1}}{\sqrt{1-s_{d+1}^{2}}}.$

For all $\hat{s}\in S_{0}$, $\theta(\hat{s})=\theta(s)$. In addition
$t:S_{0}\setminus\\{s_{n}\\}\to\mathbb{R}$ is a bijection.

###### Proof.

Without loss of generality, we can suppose
$s=[\cos\alpha,0,\ldots,0,\sin\alpha]$ for some
$\alpha\in(-\frac{3}{2}\pi,\frac{1}{2}\pi)$. Thus, $\theta(s)=[1,0,\ldots 0]$,
and

$S_{0}=\left\\{\hat{s}_{\alpha}=[\cos\hat{\alpha},0,\ldots,0,\sin\hat{\alpha}]:\,\hat{\alpha}\in[-\frac{3}{2}\pi,\frac{1}{2}\pi]\right\\}.$

Pick $\hat{s}_{\alpha}\in S_{0}\setminus\\{s_{n}\\}$, we have
$\theta(\hat{s}_{\alpha})=[1,0,\ldots,0]$. In addition,
$t(\hat{s}_{\alpha})=\frac{\sin\alpha}{|\cos\alpha|}$ is a one by one mapping.
∎

Notice that for $\mathbb{S}^{2}\subset{\mathbb{R}^{3}}$, if we visualize
$\mathbb{S}^{2}$ as the Earth and we pick the point $s\in\mathbb{S}^{2}$ to be
the Royal Observatory in the town of Greenwich, London, England, then $S_{0}$
is the Greenwich meridian (or “prime meridian”).

Finally, we discuss the relation between the variant Funk transform defined in
(43) with $\xi=1$ (i.e., $\mathcal{U}_{1}$) and the Stereographic Spherical
Radon transform defined in (10) as

$\mathcal{S}_{\mathcal{R}}(\mu)=\mathcal{R}(\phi_{\\#}\mu)$

for any Radon measure $\mu\in\mathcal{M}(\mathbb{S}^{d})$ that does not assign
mass to $\\{s_{n}\\}$. In particular, we can define the Stereographic
Spherical Radon transform for $L^{1}$ functions
$f:\mathbb{S}^{d}\setminus\\{s_{n}\\}\to\mathbb{R}$, as they can be viewed as
densities of continuous measures.

Indeed, let $\mu$ be the corresponding measure for $f$, that is,

$\int_{\mathbb{S}^{d}\setminus\\{s_{n}\\}}\psi_{0}(s)f(s)\,d\sigma_{d+1}(s)=\int_{\mathbb{S}^{d}\setminus\\{s_{n}\\}}\psi_{0}(s)\,d\mu(s),\qquad\forall\psi_{0}\in
C(\mathbb{S}^{d}),$

where $\sigma_{d+1}$ is the uniform measure on
$\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$.

Let $\hat{f}$ denote the density of $\phi_{\\#}\mu$. By definition of push
forward measure, we have:

$\displaystyle\int_{\mathbb{S}^{d}\setminus\\{s_{n}\\}}\psi(\phi(s))f(s)d\sigma_{d+1}(s)=\int_{\mathbb{S}^{d}\setminus\\{s_{n}\\}}\psi(\phi(s))d\mu(s)=\int_{\mathbb{R}^{d}}\psi(x)d\phi_{\\#}\mu(x)=\int_{\mathbb{R}^{d}}\psi(x)\hat{f}(x)dx,$
(47)

for all test functions $\psi\in C_{0}(\mathbb{R}^{d})$ that decay to zero. In
short, we usually write $d\mu=fd\sigma_{d+1}$, and $d\phi_{\\#}\mu=\hat{f}dx$,
where $dx$ represents the Lebesgue measure in $\mathbb{R}^{d}$.

Thus, $\mathcal{S}_{\mathcal{R}}(f)$ is defined as

$\displaystyle\mathcal{S}_{\mathcal{R}}(f)(t,\theta)=\mathcal{R}(\hat{f})(t,\theta)=\int_{\mathbb{R}^{d}}\hat{f}(x)\delta(t-\langle
x,\theta\rangle)\,dx=\int_{H(t,\theta)}\hat{f}\ dH(t,\theta),$ (48)

where $dH(t,\theta)$ is the surface/line integral over the
hyperplane/plane/line $H(t,\theta)$.

######  Proposition 14.

Let $f\in L^{1}(\mathbb{S}^{d}\setminus\\{s_{n}\\})$, and let
$s\in\mathbb{S}^{d}$. Then,

$\|\zeta^{s}_{1}\|_{d-1}\,\mathcal{U}_{1}f(s)=\mathcal{S}_{\mathcal{R}}f(t,\theta),$

where

$\theta=\frac{s[1:d]}{\|s[1:d]\|},\qquad
t=\frac{s_{d+1}}{\sqrt{1-s_{d+1}^{2}}}.$

###### Proof.

For convenience, we can extend $f$ to $\mathbb{S}^{d}$ by setting
$f(s_{n})=0$.

Without loss of generality, we suppose $s=[\cos\theta,0,\ldots,0,\sin\theta]$
where $\theta\in(-\frac{3}{2}\pi,\frac{1}{2}\pi)$.

As before, let $\mu$ be the measure having density $f$ and let $\hat{f}$
denote the density of $\phi_{\\#}\mu$. Since $\phi$ is invertible, it is
straightforward to verify that $(\phi^{-1})_{\\#}(\phi_{\\#})(\mu)=\mu$, that
is,

$\displaystyle\int_{\mathbb{R}^{d}}\psi_{0}(\phi^{-1}(x))\hat{f}(x)dx=\int_{\mathbb{S}^{d}}\psi_{0}(s)f(s)d\sigma_{d+1}(s),\qquad\forall\psi_{0}\in
C(\mathbb{S}^{d})$ (49)

In short, $d\mu=fd\sigma_{d+1}$ and $d\phi_{\\#}\mu=\hat{f}dx$.

Let $S_{0}$ denote the unique great circle (meridian) that passes through the
North Pole $s_{n}=[0,\dots,0,1]$, the South Pole $s_{0}=[0,\dots,0,-1]$ and
the point $s$. It is straightforward to verify

$\mathbb{S}^{d}=\bigcup_{\hat{s}\in S_{0}}\zeta_{1}^{\hat{s}}$

and that for any distinct $\hat{s}_{1},\hat{s}_{2}\in S_{0}$,
$\zeta_{1}^{\hat{s}_{1}}\cap\zeta_{1}^{\hat{s}_{2}}=\\{s_{n}\\}$.

Similarly,

$\mathbb{R}^{d}=\bigcup_{\hat{t}\in\mathbb{R}}H({\hat{t},\theta})$

and for any distinct $\hat{t}_{1},\hat{t}_{2}\in\mathbb{R}$,
$H({\hat{t}_{1},\theta})\cap H({\hat{t}_{2},\theta})=\emptyset$.

Thus, we can “slice” and integrate in the following way:

$\displaystyle\int_{\mathbb{S}^{d}}\psi_{0}\,d\sigma_{d+1}=\int_{\hat{s}\in
S_{0}}\int_{\zeta_{1}^{\hat{s}}}\psi_{0}\,d\zeta_{1}^{\hat{s}}\,dS_{0}(\hat{s}),$
(50)

where $d\zeta_{1}^{\hat{s}}$ denotes the surface/path integral over
$\zeta_{1}^{\hat{s}}$, and $dS_{0}$ denotes the surface/path integral over
$S_{0}$.

Also, by using Proposition 12, we can “slice” and integrate in the following
way:

$\displaystyle\int_{\mathbb{R}^{d}}\psi(x)\,dx=\int_{\hat{s}\in\mathbb{R}}\int_{H(t(\hat{s}),\theta(\hat{s}))}\psi\,dH(t(\hat{s}),\theta(\hat{s}))\,d\hat{s},$
(51)

where for each $\hat{s}\in S_{0}$ we define
$\theta(\hat{s}):=\frac{\hat{s}[1:d]}{\|\hat{s}[1:d]\|},t(\hat{s}):=\frac{\hat{s}_{d+1}}{\sqrt{1-\hat{s}_{d+1}^{2}}}$,
and where $dH(t(\hat{s}),\theta(\hat{s}))$ denotes the surface/line integral