SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathbb {S} _ {K _ {\mathbb {S}}} ^ {d _ {\mathbb {S}}} = \left\{\mathbf {x} _ {p} \in \mathbb {R} ^ {d _ {\mathbb {S}} + 1}: \langle \mathbf {x} _ {p}, \mathbf {x} _ {p} \rangle_ {2} = 1 / K _ {\mathbb {S}} \right\}, \tag {11}
$$
where  $K_{\mathbb{S}} > 0$  and  $\langle \cdot ,\cdot \rangle_{2}$  is the standard Euclidean inner product
$$
\langle \mathbf {x}, \mathbf {y} \rangle_ {2} = \sum_ {j = 1} ^ {d _ {\mathrm {S}} + 1} x _ {j} y _ {j}, \forall \mathbf {x}, \mathbf {y} \in \mathbb {R} ^ {d _ {\mathrm {S}} + 1}, \tag {12}
$$
Table 2 summarizes the key operators in Euclidean, hyperbolic, and spherical spaces with arbitrary curvatures. It is worth noting that the three model spaces discussed above can cover any curvature value within the range of  $(-\infty, \infty)$ . However, there is a potential issue with the hyperboloid and hypersphere becoming divergent as their respective curvatures approach zero. This results in the manifolds becoming flat, and their distance and metric tensors do not become Euclidean at zero-curvature points, which could hinder curvature learning. Therefore, stereographic projections are considered to be suitable alternatives to these spaces as they maintain a non-Euclidean structure and inherit many of the properties of the hyperboloid and hypersphere.
Table 2: Relevant operators (exponential maps and distances between two points) in Euclidean, hyperbolic, and spherical spaces with arbitrary constant curvatures.
Space Model	expx_p(x)	d(x,y)
E, Euclidean	xp + x	||x - y||2
H, hyperboloid	cosh (√-K_H||x|)xp + sinh (√-K_H||x|) x/√-K_H||x|	1/√-K_H arccosh (K_H<x,y>
S, hypersphere	cos (√K_S||x|)xp + sin (√K_S||x|) x/√K_S||x|	1/√K_S arccos (K_S<x,y>
A.3 Constructing Product Manifolds
In this appendix, we provide additional details about the generation of product manifolds. In mathematics, a product manifold is a type of manifold obtained by taking the Cartesian product of two or more manifolds. Informally, a product manifold can be thought of as a space formed by taking multiple smaller spaces and merging them in a way that maintains their individual structures. For instance, the surface of a sphere can be seen as a product manifold consisting of two real lines that intersect at each point on the sphere. Product manifolds play a crucial role in various fields of mathematics, including physics, topology, and geometry.
A product manifold can be defined as the Cartesian product:
$$
\mathcal {P} = \bigotimes_ {i = 1} ^ {n _ {\mathcal {P}}} \mathcal {M} _ {K i} ^ {d _ {i}} \tag {13}
$$
Here,  $K_{i}$  and  $d_{i}$  represent the curvature and dimensionality of the space, respectively. Points  $\mathbf{x}_p \in \mathcal{P}$  are expressed using their coordinates:
$$
\mathbf {x} _ {p} = \operatorname {c o n c a t} \left(\mathbf {x} _ {p} ^ {(1)}, \mathbf {x} _ {p} ^ {(2)}, \dots , \mathbf {x} _ {p} ^ {(n _ {\mathcal {P}})}\right): \mathbf {x} _ {p} ^ {(i)} \in \mathcal {M} _ {K _ {i}} ^ {d _ {i}}. \tag {14}
$$
Also, the metric of the product manifold decomposes into the sum of the constituent metrics
$$
g _ {\mathcal {P}} = \sum_ {i = 1} ^ {n _ {\mathcal {P}}} g _ {i}, \tag {15}
$$
hence,  $(\mathcal{P},g_{\mathcal{P}})$  is also a Riemannian manifold. It should be noted that the signature of the product space is parametrized with several degrees of freedom. These include the number of components used in the product space, the type of model spaces employed, their dimensionality, and curvature. If we restrict  $\mathcal{P}$  to be composed of the Euclidean plane  $\mathbb{E}_{K_{\mathbb{E}}}^{d_{\mathbb{E}}}$ , hyperboloids  $\mathbb{H}_{K_j^{\mathbb{H}}}^{d_j^{\mathbb{H}}}$ , and hyperspheres  $\mathbb{S}_{K_k^{\mathbb{S}}}^{d_k^{\mathbb{S}}}$  of constant curvature, we can rewrite Equation 13 as
$$
\mathcal {P} = \mathbb {E} _ {K _ {\mathbb {E}}} ^ {d _ {\mathbb {E}}} \times \left(\bigotimes_ {j = 1} ^ {n _ {\mathbb {H}}} \mathbb {H} _ {K _ {j} ^ {\mathbb {H}}} ^ {d _ {j} ^ {\mathbb {H}}}\right) \times \left(\bigotimes_ {k = 1} ^ {n _ {\mathbb {S}}} \mathbb {S} _ {K _ {k} ^ {\mathbb {S}}} ^ {d _ {k} ^ {\mathbb {S}}}\right), \tag {16}
$$
where  $K_{\mathbb{E}} = 0$ ,  $K_{j}^{\mathbb{H}} < 0$ , and  $K_{k}^{\mathbb{S}} > 0$ . Hence,  $\mathcal{P}$  would have a total of  $1 + n_{\mathbb{H}} + n_{\mathbb{S}}$  component spaces, and total dimension  $d_{\mathbb{E}} + \Sigma_{j=1}^{n_{\mathbb{H}}} d_{j}^{\mathbb{H}} + \Sigma_{k=1}^{n_{\mathbb{S}}} d_{k}^{\mathbb{S}}$ . Distances between two points in the product manifold can be computed aggregating the distance contributions  $\mathfrak{d}_i$  from each manifold composing the product manifold:
$$
\mathfrak {d} _ {\mathcal {P}} \left(\overline {{\mathbf {x}}} _ {p _ {1}}, \overline {{\mathbf {x}}} _ {p _ {2}}\right) = \sqrt {\sum_ {i = 1} ^ {n _ {\mathcal {P}}} \mathfrak {d} _ {i} \left(\overline {{\mathbf {x}}} _ {p _ {1}} ^ {(i)} , \overline {{\mathbf {x}}} _ {p _ {2}} ^ {(i)}\right) ^ {2}}. \tag {17}
$$
The overline denotes that the points  $\mathbf{x}_{p_1}$  and  $\mathbf{x}_{p_2}$  have been adequately projected on to the product manifold using the exponential map before computing the distances.
A.4 The Rationale Behind using Product Manifolds of Model Spaces
Most arbitrary Riemannian manifolds do not have closed-form solutions for their exponential maps and geodesic distances between points, as well as for other relevant mathematical notions. The exponential map takes a point on the manifold and exponentiates a tangent vector at that point to produce another point on the manifold. The geodesic distance between two points on a manifold is the shortest path along the manifold connecting those points.
For simple, well-studied manifolds like Euclidean space, hyperbolic space, and spherical space, closed-form solutions for exponential maps and geodesic distances are available. This is because these manifolds have constant curvature, which allows for more straightforward calculations. However, for most other manifolds, the exponential map and geodesic distances must typically be computed numerically or approximated using specialized algorithms. These computations often involve solving differential equations, and depending on the manifold's curvature and geometry, these solutions can be quite complex and may not have closed-form expressions. For certain types of manifolds, like product manifolds where each factor is a well-understood manifold, the computation of exponential maps and geodesic distances can sometimes be simplified due to the separability of the metric. However, scaling these methods to more general, complex manifolds can be challenging and computationally intensive.
As mentioned, closed-form solutions for exponential maps and geodesic distances are not common for arbitrary Riemannian manifolds, and computational methods often rely on numerical techniques and algorithms due to the complexity of these calculations. These properties have made product manifolds the most attractive tool to endow the latent space with richer structure while retaining computational traceability, see [Gu et al., 2018, Skopek et al., 2019, Sáez de Ocáriz Borde et al., 2023b,a, Zhang et al., 2020, Fumero et al., 2021, Pfau et al., 2020]. This motivates us to use product manifolds in our setup, which keeps our work in line with the state-of-the-art approaches in the literature. We highlight that one of the central contributions of our paper is to shed light on the problem of NLGS, and provide an initial solution to the problem in the context of the current state-of-the-art in geometric
latent space modelling. Future research avenues could explore generalizations of this which account for new techniques that emerge in the literature, potentially comparing a more general set of latent manifolds.
A.5 Geometry of Negative Curvature: Further Discussion
B Further Details on the Gromov-Hausdorff Algorithm for Neural Latent Geometry Search
As discussed in Section 3, there are limitations to the Hausdorff distance as a metric for comparing point sets that represent discretized versions of continuous manifolds. The original algorithm by Taha and Hanbury [2015] for computing the Hausdorff distance assumes that the point sets reside in the same space with the same dimensionality, which is a limiting requirement. However, the Gromov-Hausdorff distance proposed in this work allows us to compare metric spaces that are not embedded in a common ambient space, but it is not exactly computable. We will focus on providing an upper bound for  $\mathrm{d_{GH}}(\mathbb{E}^n,\mathbb{S}^n)$  and describing an algorithm for obtaining upper bounds for  $\mathrm{d_{GH}}(\mathbb{E}^n,\mathbb{H}^n)$  and  $\mathrm{d_{GH}}(\mathbb{S}^n,\mathbb{H}^n)$  too. As previously mentioned,  $\mathbb{E}^n$  and  $\mathbb{S}^n$  can be isometrically embedded into  $\mathbb{E}^{6n - 6}$  in many ways, which provides a common underlying space for computing Hausdorff distances between  $\mathbb{E}^n$ ,  $\mathbb{S}^n$ , and  $\mathbb{H}^n$ . The embeddings of different spaces require making choices: in particular, there is no way to exactly compute the infimum in the definition of Gromov-Hausdorff distance given in Section 2:
$$
\mathrm {d} _ {\mathrm {G H}} (A, B) = \inf  _ {(X, f, g) \in \operatorname {E S} (A, B)} \mathrm {d} _ {\mathrm {H}} ^ {X, f, g} (f (A), g (B)). \tag {18}
$$
so we resort to numerical approximations which are later described in this appendix.
To estimate the mutual Gromov-Hausdorff distance between the infinite smooth spaces  $\mathbb{E}^2$  and  $\mathbb{H}^2$ , the first step is to approximate them using finite discrete versions. This is done by considering well-distributed collections of points in  $\mathbb{E}^2$  and  $\mathbb{H}^2$ , obtained by applying the exponential map to a collection of points in  $\mathbb{R}^2$ . Multiple isometric embeddings of  $\mathbb{E}^2$  into  $\mathbb{R}^6$  are also considered. The estimation is obtained by computing the minimum of the Hausdorff distance between the point sets obtained from the collections in  $\mathbb{R}^6$  and the isometric embedding of  $\mathbb{H}^2$  into  $\mathbb{R}^6$ . The same applies for spherical space.
B.1 Generating Points in the Corresponding Balls of Radius One
All our computations will be done in dimension two, so we will simply precise how to generate points in  $B_{\mathbb{E}^2}$ ,  $B_{\mathbb{S}^2}$  and  $B_{\mathbb{H}^2}$ . For  $B_{\mathbb{E}^2}$  and  $B_{\mathbb{S}^2}$ , we will use very elementary trigonometry. For  $B_{\mathbb{H}^2}$ , we will need an explicit description of the exponential map of  $\mathbb{H}^2$ . Using the descriptions
$$
B _ {\mathbb {E} ^ {2}} = \left\{\left(r \cos (t), r \sin (t)\right): r \in [ 0, 1 ], t \in [ 0, 2 \pi) \right\}, \tag {19}
$$
$$
B _ {\mathbb {S} ^ {2}} = \left\{\left(\sin (\beta) \cos (\alpha), \sin (\beta) \sin (\alpha), \cos (\beta)\right): \alpha \in [ 0, 2 \pi), \beta \in [ 0, 1 ] \right\}, \tag {20}
$$
it will be easy to generate collections of points in  $B_{\mathbb{E}^2}$  and  $B_{\mathbb{S}^2}$ . As we anticipated above in the outline of our strategy to estimate  $\mathrm{d}_{\mathrm{GH}}(B_{\mathbb{E}^2}, B_{\mathbb{H}^2})$ , in order to give explicit well-distributed collection of points in  $B_{\mathbb{H}^2}$ , it is enough to give a well-distributed collection of points in  $B_{\mathbb{E}^2}$  and consider its image under the exponential map  $\exp_0: B_{\mathbb{E}^2} \to B_{\mathbb{H}^2}$ , where we have identified  $B_{\mathbb{E}^2}$  with the ball of radius one of  $\mathbb{R}^2 \cong T_0\mathbb{H}^2$ , i.e. the tangent space of  $\mathbb{H}^2$  at the point 0. However, we should remark that our description of  $\mathbb{H}^n$  will not be any of the three most standard ones: namely the hyperboloid model, the Poincaré ball model, or the upper half-plane model. We describe  $\mathbb{H}^n$  as the differentiable manifold  $\mathbb{R}^n$  (of Cartesian coordinates  $x, y_1, \ldots, y_{n-1}$ ) equipped with the Riemannian metric  $g_{-1} = \mathrm{d}x^2 + e^{2x}(\mathrm{d}y_1^2 + \dots + \mathrm{d}y_{n-1}^2)$ . This metric is complete and has constant sectional curvature of -1, which implies that  $(\mathbb{R}^n, g_{-1})$  is isometric to  $\mathbb{H}^n$ . In the hyperboloid model of  $\mathbb{H}^2$ , we view this space as a submanifold of  $\mathbb{R}^3$  (although with a distorted metric, not the one induced from the ambient  $\mathbb{R}^3$ ). In this model of  $\mathbb{H}^2$ , one can explicitly describe by the following assignment  $\exp_0: \mathbb{R}^2 \to \mathbb{H}^2$  by
$$
(x, y) \mapsto \left(\frac {x \sinh \left(\sqrt {x ^ {2} + y ^ {2}}\right)}{\sqrt {x ^ {2} + y ^ {2}}}, \frac {y \sinh \left(\sqrt {x ^ {2} + y ^ {2}}\right)}{\sqrt {x ^ {2} + y ^ {2}}}, \cosh \left(\sqrt {x ^ {2} + y ^ {2}}\right)\right). \tag {21}
$$
We use this explicit formula of the exponential map with coordinates in the hyperboloid model of  $\mathbb{H}^2$ , to give a explicit formula for the exponential map with coordinates in the model of  $\mathbb{H}^2$  that we introduced above, denoted by  $(\mathbb{R}^2,g_{-1})$ . In order to change coordinates from the hyperboloid model to the Poincaré ball model, we use the following isometry (which is well-known and can be thought of as a hyperbolic version of the stereographic projection from  $\mathbb{S}^2$  to  $\mathbb{R}^2$ ):
$$
(x, y, z) \mapsto \left(\frac {x}{1 + z}, \frac {y}{1 + z}\right). \tag {22}
$$
To change coordinates from the Poincaré ball model to the upper half-plane model, one can use the following isometry:
$$
(x, y) \mapsto \left(\frac {- 2 y}{(x - 1) ^ {2} + y ^ {2}}, \frac {1 - x ^ {2} - y ^ {2}}{(x - 1) ^ {2} + y ^ {2}}\right). \tag {23}
$$
The previous assignment comes from the standard Möbius transformation  $z \mapsto \frac{z + i}{iz + 1}$  that identifies the Euclidean ball of radius one of  $\mathbb{C}$  and the upper half plane of  $\mathbb{C}$  as Riemann surfaces (which we give explicitly in this case, although it is known to exist and to be unique by the classical Riemann mapping theorem). Finally, to go from coordinates in the upper half-plane model of  $\mathbb{H}^2$  to our model  $(\mathbb{R}^2, g_{-1})$ , we use the isometry
$$
(x, y) \rightarrow (- \log y, x). \tag {24}
$$
B.2 Embedding of  $\mathbb{H}^n$  into  $\mathbb{E}^{6n - 6}$
We want to define an isometric embedding of  $B_{\mathbb{H}^n}^1$  into  $\mathbb{E}^{6n - 6}$  ( $F$  from Section 3.2). This higher dimensional space is our candidate to fit in the three geometries  $\mathbb{E}^n$ ,  $\mathbb{H}^n$  and  $\mathbb{S}^n$  to compute their Hausdorff dimensions as subspaces of  $\mathbb{E}^{6n - 6}$  and hence estimate their Gromov-Hausdorff distances. Before describing such embedding, we introduce several preliminary auxiliary functions.
Let  $\chi(t) = \sin(\pi t) \cdot e^{-\sin^{-2}(\pi t)}$  for non-integer values of  $t$ . A priori, the inverse of  $\sin(0) = 0$  does not make sense but since  $\lim_{t \to 0^+} \chi(t) = \lim_{t \to 0^-} \chi(t) = 0$ , we can set  $\chi(0) = 0$  so it is still continuous. In fact, it is smooth and, in particular, integrable. We can say the same at all points when  $t$  is an integer, so we set  $\chi(t) = 0$  for all integers  $t$  and we obtain an smooth function  $\chi$  defined on  $\mathbb{R}$ .
$$
A = \int_ {0} ^ {1} \chi (t) d t. \tag {25}
$$
$$
\psi_ {1} (x) = \sqrt {\frac {1}{A} \cdot \int_ {0} ^ {1 + x} \chi (t) \mathrm {d} t}, \tag {26}
$$
$$
\psi_ {2} (x) = \sqrt {\frac {1}{A} \cdot \int_ {0} ^ {x} \chi (t) \mathrm {d} t}. \tag {27}
$$
We set  $c$  to be the constant
$$
c = 2 \max  \left\{G _ {1}, G _ {2} \right\}, \tag {28}
$$
defined in terms of the following
$$
G _ {1} = \left\| \frac {\mathrm {d}}{\mathrm {d} x} (\sinh (x) \cdot \psi_ {1} (x)) \right\| _ {L ^ {\infty} [ - 2, 2 ]}, \tag {29}
$$
$$
G _ {2} = \left\| \frac {\mathrm {d}}{\mathrm {d} x} (\sinh (x) \cdot \psi_ {2} (x)) \right\| _ {L ^ {\infty} [ - 2, 2 ]}, \tag {30}
$$
$$
h (x, y) = \frac {\sinh (x)}{c} \left(\psi_ {1} (x) \cos (c \cdot y), \psi_ {1} (x) \sin (c \cdot y), \psi_ {2} (x) \cos (c \cdot y), \psi_ {2} (x) \sin (c \cdot y)\right), \tag {31}
$$
$$
\psi (x, y) = \left(\sinh^ {- 1} \left(y e ^ {x}\right), \log \left(\sqrt {e ^ {- 2 x} + y ^ {2}}\right)\right). \tag {32}
$$
We also define  $f_0(x,y) = \left(\int_0^{\sinh^{-1}(ye^x)}\sqrt{1 - \varepsilon(t)^2}\mathrm{d}t,\log (\sqrt{e^{-2x} + y^2}),h(\psi (x,y))\right),$
with  $\varepsilon$  being
$$
\varepsilon = \frac {G _ {1} ^ {2} + G _ {2} ^ {2}}{c ^ {2}}. \tag {33}
$$
This way, we can set
$$
f (x, y _ {1}, \dots , y _ {n - 1}) = \frac {1}{\sqrt {n - 1}} \big (f _ {0} (x, \sqrt {n - 1} y _ {1}), \dots , f _ {0} (x, \sqrt {n - 1} y _ {n - 1}) \cdot
$$
Recall that in our case, we use the function  $F(\cdot) = f(n = 2,\cdot)$  to map the set of points  $Q$  in  $\mathbb{H}^2$  to  $\mathbb{R}^6$ , and for computing
$$
\mathrm {d} _ {\mathrm {G H}} \left(B _ {\mathbb {E} ^ {2}}, B _ {\mathbb {H} ^ {2}}\right) \approx \min  _ {i, j, k} \mathrm {d} _ {\mathrm {H}} ^ {\mathbb {R} ^ {6}, f _ {k}, F} \left(P _ {i}, Q _ {j}\right) = \min  _ {i, j, k} \mathrm {d} _ {\mathrm {H}} ^ {\mathbb {R} ^ {6}} \left(f _ {k} \left(P _ {i}\right), F \left(Q _ {j}\right)\right). \tag {34}
$$
B.3 Estimation of the Gromov-Hausdorff distance between the Euclidean and Spherical Spaces
In this section we give the analytical derivation of the Gromov-Hausdorff distance between the Euclidean space  $\mathbb{E}^n$  and the Spherical space  $\mathbb{S}^n$ . We first explain the case  $n = 1$ , where we can visually understand the situation in a better way because all distances will be measured in  $\mathbb{E}^2$ . Afterwards, we replicate the same argument for arbitrary  $n$ . Recall that the Spherical model  $\mathbb{S}^n$  can be described as the metric subspace of  $\mathbb{E}^{n + 1}$  corresponding to the Euclidean sphere of radius one. Hence, as a subspace of  $\mathbb{R}^{n + 1}$ , it corresponds to  $\{(x_0,\ldots ,x_n):\sum_{i = 0}^n x_i^2 = 1\}$ .
Figure 8: The one-dimensional model of spherical geometry  $\mathbb{S}^1$  isometrically embedded in  $\mathbb{R}^2$ .
The ball of radius one of  $\mathbb{S}^1$ , denoted by  $B_{\mathbb{S}^1}$ , is highlighted in red. Our estimation of the Gromov-Hausdorff distance between  $\mathbb{E}^1$  and  $\mathbb{S}^1$  is motivated by the following observation. In the  $y$ -axis, the red arc ranges from  $\frac{1 + \cos(1)}{2}$  to 1. If we consider the following blue segment, representing the ball of
Figure 9: Comparison between  $B_{\mathbb{E}^1}$  (in red) and  $B_{\mathbb{S}^1}$  (in blue) inside  $\mathbb{E}^2$
radius one of  $\mathbb{E}^1$  (denoted by  $B_{\mathbb{E}^1}$ ), we get
$$
\sup  _ {x \in B _ {\mathbb {S} ^ {1}}} d _ {\mathbb {E} ^ {2}} (x, B _ {\mathbb {E} ^ {1}}) = \frac {1 - \cos (1)}{2} \approx 0. 2 3. \tag {35}
$$
However, it can be seen that  $\sup_{x\in B_{\mathbb{E}^1}}d_{\mathbb{E}^2}(x,B_{\mathbb{S}^1}) = 0.279$ . More generally, if the blue line is chosen to be at the height  $1 - x$ , for some  $x$  lying in the closed interval  $[0,1 - \cos (1)]$ , it is not hard to see that, for such embedding  $f:B_{\mathbb{E}^1}\to \mathbb{E}^2$ ,
$$
d h ^ {\mathbb {E} ^ {2}} \left(f \left(B _ {\mathbb {E} ^ {1}}\right), B _ {\mathbb {S} ^ {1}}\right) = \max  \left\{x, 1 - x, \sqrt {(1 - \sin (1)) ^ {2} + (1 - \cos (1) - x) ^ {2}} \right\}, \tag {36}
$$
whose minimum is attained exactly when  $x = \sqrt{(1 - \sin(1))^2 + (1 - x - \cos(1))^2}$ , i.e. when
$$
x = \frac {(1 - \sin (1)) ^ {2} + (1 - \cos (1)) ^ {2}}{2 - 2 \cos (1)} \approx 0. 2 5 7. \tag {37}
$$
This can be seen in the following picture. The two pink lines represent the biggest lengths between
Figure 10: Comparison between  $B_{\mathbb{E}^1}$  (in red) and  $B_{\mathbb{S}^1}$  (in blue) inside  $\mathbb{E}^2$  and a schematic of the biggest lengths (in pink) between the two point sets.
points in  $B_{\mathbb{E}^1}$  and  $B_{\mathbb{S}^1}$ , whose length is approximately equal to 0.257. Since we cannot compute exactly the Gromov-Hausdorff distance  $\mathrm{d}_{\mathrm{GH}}(\mathbb{S}^1, \mathbb{E}^1)$ , choices have to be made. We have discussed why we would expect it to be somewhere in between 0.23 and 0.257. Since we are considering a very simple embedding for these estimations, it is reasonable to expect  $\mathrm{d}_{\mathrm{GH}}(\mathbb{S}^1, \mathbb{E}^1)$  to get closer to the lowest number when considering embeddings of these spaces in more complicated higher dimensional spaces, as the definition of the Gromov-Hausdorff distance allows.
For an arbitrary  $n$ , we reduce the estimation of  $\mathrm{d}_{\mathrm{GH}}(B_{\mathbb{E}^n}, B_{\mathbb{S}^n})$  to the one-dimensional case as follows. Analogously as we did for  $n = 1$ , given any value of  $x$ , we can consider  $B_{\mathbb{E}^n}$  isometrically embedded in  $\mathbb{E}^{n + 1}$  by the map  $f_x: B_{\mathbb{E}^n} \to \mathbb{R}^{n + 1}$  defined by  $f((x_0, x_1, \ldots, x_n)) = (x_0, x_1, \ldots, x_n, x)$ . We view  $B_{\mathbb{S}^n}$  isometrically embedded into  $\mathbb{R}^{n + 1}$  as the ball of radius one of  $\mathbb{S}^n \subset \mathbb{R}^{n + 1}$  with centre in the north pole of the sphere, i.e. the point with coordinates  $(0, 0, \ldots, 0, 1)$ . Given any unit vector  $u$  in  $\mathbb{R}^{n + 1}$  orthogonal to  $\vec{n} = (0, 0, \ldots, 0, 1)$ , we define  $\pi_u$  to be the two-dimensional plane linearly spanned by  $u$  and  $\vec{n}$ . It is clear that, as  $u$  ranges over the orthogonal space of  $(0, 0, \ldots, 0, 1)$ , the intersections  $B_{\mathbb{S}^n} \cap \pi_u$  cover the whole  $B_{\mathbb{S}^n}$  and the intersections  $f_x(B_{\mathbb{E}^n}) \cap \pi_u$  cover the whole  $f_x(B_{\mathbb{E}^n})$ . Hence, in order to compute  $\mathrm{d}_{\mathrm{H}}^{\mathbb{E}^{n + 1}}(f_x(B_{\mathbb{E}^n}), B_{\mathbb{S}^n})$ , it suffices to compute  $\mathrm{d}_{\mathrm{H}}^{\mathbb{E}^{n + 1}}(f_x(B_{\mathbb{E}^n}) \cap \pi_u, B_{\mathbb{S}^n} \cap \pi_u)$  for all unit vectors  $u$  orthogonal to  $\vec{n}$ . Moreover, for two such unit vectors  $u$  and  $v$ , there is a rigid motion (a rotation) of the whole ambient space  $\mathbb{R}^{n + 1}$ , that fixes  $\vec{n}$  and that maps isometrically  $f_x(B_{\mathbb{E}^n}) \cap \pi_u$  to  $f_x(B_{\mathbb{E}^n}) \cap \pi_v$  and  $B_{\mathbb{S}^n} \cap \pi_u$  to  $B_{\mathbb{S}^n} \cap \pi_v$ . In particular,
$$
\mathrm {d} _ {\mathrm {H}} ^ {\mathbb {E} ^ {n + 1}} \left(f _ {x} \left(B _ {\mathbb {E} ^ {n}}\right) \cap \pi_ {u}, B _ {\mathbb {S} ^ {n}} \cap \pi_ {u}\right) = \mathrm {d} _ {\mathrm {H}} ^ {\mathbb {E} ^ {n + 1}} \left(f _ {x} \left(B _ {\mathbb {E} ^ {n}}\right) \cap \pi_ {v}, B _ {\mathbb {S} ^ {n}} \cap \pi_ {v}\right).
$$
So we can do the previous computation for the specific value of  $u = (0,0,\dots ,1,0)$  (for clarity, where we mean the unit vector where the  $n$ -th coordinate is equal to 1 and the rest are zero). Crucially, the projection of  $\mathbb{R}^{n + 1}$  unto  $\mathbb{R}^2$  (by projecting onto the last two coordinates) restrict to isometries on  $B_{\mathbb{S}^n}\cap \pi_u$  and  $B_{\mathbb{E}^n}\cap \pi_u$ , from where it is deduced, analogously as in 36, that, as long as  $x\in [0,1 - \cos (1)]$ , we have the following:
$$
\mathrm {d} _ {\mathrm {H}} ^ {\mathbb {E} ^ {n + 1}} (f _ {x} (B _ {\mathbb {E} ^ {n}}) \cap \pi_ {u}, B _ {\mathbb {S} ^ {n}} \cap \pi_ {u}) = \max \left\{x, 1 - x, \sqrt {(1 - \sin (1)) ^ {2} + (1 - \cos (1) - x) ^ {2}} \right\}.
$$
B.4 Computational Implementation of the Gromov-Hausdorff Distance between the Remaining Model Spaces of Constant Curvature
In this appendix, we provide further details regarding how the Gromov-Hausdorff distances between the remaining manifolds (between  $B_{\mathbb{E}^2}$  and  $\bar{B}_{\mathbb{H}^2}$ , and  $B_{\mathbb{S}^2}$  and  $B_{\mathbb{H}^2}$ ) were approximated computationally. Note that as discussed in Section 3.3, these distances will be leveraged to compare not only model spaces but product manifolds as well.
B.4.1 Discretizing the Original Continuous Manifolds
This section discusses how the points in the corresponding balls of radius one described in Appendix B.1 are generated from a practical perspective. The Euclidean, hyperbolic, and spherical