Original: Pseudo-Riemannian Graph Convolutional Networks
Revised: Pseudo-Riemannian Graph Convolutional Networks

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: Abstract
Revised: Abstract

Original: Graph Convolutional Networks (GCNs) are powerful frameworks for learning embeddings of graph-structured data.
Revised: Graph Convolutional Networks (GCNs) are powerful frameworks for learning embeddings of graph-structured data.

Original: GCNs are traditionally studied through the lensof Euclidean geometry.
Revised: GCNs are traditionally studied through the lensof Euclidean geometry.

Original: Recent works find that non-Euclidean Riemannian manifolds provide specific inductive biases for embedding hierarchical or sphericaldata.
Revised: Recent works find that non-Euclidean Riemannian manifolds provide specific inductive biases for embedding hierarchical or sphericaldata.

Original: However, they cannot align well with data of mixed graph topologies.
Revised: However, they cannot align well with data of mixed graph topologies.

Original: Weconsider a larger class of pseudo-Riemannian manifolds that generalize hyperboloid and sphere.
Revised: Weconsider a larger class of pseudo-Riemannian manifolds that generalize hyperboloid and sphere.

Original: We develop new geodesic tools that allow for extending neural network operations into geodesically disconnected pseudo-Riemannian manifolds.
Revised: We develop new geodesic tools that allow for extending neural network operations into geodesically disconnected pseudo-Riemannian manifolds.

Original: As a consequence, we derive a pseudo-Riemannian GCN that models datain pseudo-Riemannian manifolds of constant nonzero curvature in the contextof graph neural networks.
Revised: As a consequence, we derive a pseudo-Riemannian GCN that models datain pseudo-Riemannian manifolds of constant nonzero curvature in the contextof graph neural networks.

Original: Our method provides a geometric inductive bias thatis sufficiently flexible to model mixed heterogeneous topologies like hierarchical graphs with cycles.
Revised: Our method provides a geometric inductive bias thatis sufficiently flexible to model mixed heterogeneous topologies like hierarchical graphs with cycles.

Original: We demonstrate the representational capabilities of thismethod by applying it to the tasks of graph reconstruction, node classification andlink prediction on a series of standard graphs with mixed curvatures.
Revised: We demonstrate the representational capabilities of thismethod by applying it to the tasks of graph reconstruction, node classification andlink prediction on a series of standard graphs with mixed topologies.

Original: Empiricalresults demonstrate that our method outperforms Riemannian counterparts whenembedding graphs of complex topologies.
Revised: Empiricalresults demonstrate that our method outperforms Riemannian counterparts whenembedding graphs of complex topologies.

Original:  Introduction
Revised:  Introduction

Original: Learning from graph-structured data is a pivotal task in machine learning, for which Graph Convolutional Networks (GCNs) [1, 2, 3, 4] have emerged as powerful graph representation learningtechniques.
Revised: Learning from graph-structured data is a pivotal task in machine learning, for which Graph Convolutional Networks (GCNs) [1, 2, 3, 4] have emerged as powerful graph representation learningtechniques.

Original: For this purpose, graphs are usually embedded in Riemannian manifolds equipped witha positive definite metric.
Revised: For this purpose, graphs are usually embedded in Riemannian manifolds equipped witha positive definite metric.

Original: Euclidean geometry is a special case of Riemannian manifolds of constantzero curvature that can be understood intuitively and has well-defined operations.
Revised: Euclidean geometry is a special case of Riemannian manifolds of constantzero curvature that can be understood intuitively and has well-defined operations.

Original: However, the representation power of Euclidean space is limited [5], especially when embedding complex graphsexhibiting hierarchical structures [6].
Revised: However, the representation power of Euclidean space is limited [5], especially when embedding complex graphsexhibiting hierarchical structures [6].

Original: Non-Euclidean Riemannian manifolds of constant curvaturesprovide an alternative to accommodate specific graph topologies such as hierarchical or cyclicalstructures.
Revised: Non-Euclidean Riemannian manifolds of constant curvaturesprovide an alternative to accommodate specific graph topologies.

Original: An intuitive example is the a hyperbolic manifold of constant negative curvature thatis well suited to represent hierarchical structures such as tree-like graphs [7, 8].
Revised: For example, hyperbolic manifoldof constant negative curvature has exponentially growing volume and is well suited to representhierarchical structures such as tree-like graphs [7, 8].

Original: Similarly, spherical manifold of constant positive curvature can be used to model spherical data in various fields[9, 10, 11].
Revised: Similarly, spherical manifold of constant positive curvature is suitable for embedding spherical data in various fields [9, 10, 11] including graphswith cycles.

Original: Some recent works [12, 13, 14, 15, 16] have extended GCNs to such non-Euclideanmanifolds and have shown substantial improvements with specific geometries.
Revised: Some recent works [12, 13, 14, 15, 16] have extended GCNs to such non-Euclideanmanifolds and have shown substantial improvements with specific geometries.

Original: The topologies in real-world graphs [6], however, usually exhibit highly heterogeneous topologicalstructures, which are best represented by different geometrical curvatures.
Revised: The topologies in real-world graphs [6], however, usually exhibit highly heterogeneous topologicalstructures, which are best represented by different geometrical curvatures.

Original: A globally homogeneousgeometry lacks the flexibility for modeling complex graphs, resulting in distortional representations[17].
Revised: A globally homogeneousgeometry lacks the flexibility for modeling complex graphs, resulting in distortional representations[17].

Original: Instead of using a single manifold, product manifolds [17, 18] combining multiple Riemannian manifolds have shown advantages when embedding graphs of mixed topologies.
Revised: Instead of using a single manifold, product manifolds [17, 18] combining multiple Riemannian manifolds have shown advantages when embedding graphs of mixed topologies.

Original: However, thesimple combination of spaces still does not accommodate topologically heterogeneous graphs very well.
Revised: However, thesimple combination of spaces still does not accommodate topologically heterogeneous graphs very well.

Original: Furthermore, Riemannian manifolds are equipped with a positive definite metric disallowingfor the faithful representation of the negative eigen-spectrum of input similarities [19].
Revised: Furthermore, Riemannian manifolds are equipped with a positive definite metric disallowingfor the faithful representation of the negative eigen-spectrum of input similarities [19].

Original: Going beyond Riemannian manifolds, pseudo-Riemannian manifolds equipped with indefinite metrics constitute a larger class of geometries, pseudo-Riemannian manifolds of constant nonzero curvature do not only generalize the hyperbolic and spherical manifolds, but also contain their submanifolds (Cf.
Revised: Going beyond Riemannian manifolds, pseudo-Riemannian manifolds equipped with indefinite metrics constitute a larger class of geometries, pseudo-Riemannian manifolds of constant nonzero curvature do not only generalize the hyperbolic and spherical manifolds, but also contain their submanifolds (Cf.

Original: Fig.
Revised: Fig.

Original: 1), thus providing inductive biases specific to these geometries.
Revised: 1), thus providing inductive biases specific to these geometries.

Original: Pseudo-Riemanniangeometry with constant zero curvature (i.e. Lorentzian spacetime) was applied to manifold learning for preserving local information of non-metric data [5] and embedding directed acyclic graph[20].
Revised: Pseudo-Riemanniangeometry with constant zero curvature (i.e. Lorentzian spacetime) was applied to manifold learning for preserving local information of non-metric data [5] and embedding directed acyclic graph[20].

Original: To model complex graphs containing both hierarchies and cycles, pseudo-Riemannian manifolds with constant nonzero curvature have recently been applied into graph embeddings usingnon-parametric learning [21, 22], but the representation power of these works is not on par withthe Riemannian counterparts yet, mostly because of the absence of geodesic tools to extend neuralnetwork operations into pseudo-Riemannian geometry.
Revised: To model complex graphs containing both hierarchies and cycles, pseudo-Riemannian manifolds with constant nonzero curvature have recently been applied into graph embeddings usingnon-parametric learning [21, 22], but the representation power of these works is not on par withthe Riemannian counterparts yet, mostly because of the absence of geodesic tools to extend neuralnetwork operations into pseudo-Riemannian geometry.

Original: In this paper, we take the first step to extend GCNs into pseudo-Riemannian manifolds foregoing therequirement to have a positive definite metric.
Revised: In this paper, we take the first step to extend GCNs into pseudo-Riemannian manifolds foregoing therequirement to have a positive definite metric.

Original: Exploiting pseudo-Riemannian geometry for GCNs isnon-trivial because of the geodesical disconnectedness in pseudo-Riemannian geometry.
Revised: Exploiting pseudo-Riemannian geometry for GCNs isnon-trivial because of the geodesical disconnectedness in pseudo-Riemannian geometry.

Original: There existbroken points that cannot be smoothly connected by a geodesic, leaving necessary geodesic tools undefined.
Revised: There existbroken points that cannot be smoothly connected by a geodesic, leaving necessary geodesic tools undefined.

Original: To deal with it, we develop novel geodesic tools that empower manipulating representationsin geodesically disconnected pseudo-Riemannian manifolds.
Revised: To deal with it, we develop novel geodesic tools that empower manipulating representationsin geodesically disconnected pseudo-Riemannian manifolds.

Original: We make it by finding diffeomorphicmanifolds that provide alternative geodesic operations that smoothly avoid broken cases.
Revised: We make it by finding diffeomorphicmanifolds that provide alternative geodesic operations that smoothly avoid broken cases.

Original: Subsequently, we generalize GCNs to learn representations of complex graphs in pseudo-Riemanniangeometry by defining corresponding operations such as linear transformation and tangential aggregation .
Revised: Subsequently, we generalize GCNs to learn representations of complex graphs in pseudo-Riemanniangeometry by defining corresponding operations such as linear transformation and tangential aggregation .

Original: Different from previous works, the initial features of GCN could be fully defined in the
Revised: Different from previous works, the initial features of GCN could be fully defined in the

Original: Euclidean space.
Revised: Euclidean space.

Original: Thanks to the diffeomorphic operation that is bijective and differentiable, the standard gradient descent algorithm can be exploited to perform optimization.
Revised: Thanks to the diffeomorphic operation that is bijective and differentiable, the standard gradient descent algorithm can be exploited to perform optimization.

Original: To summarize, our main contributions are as follows: 1) We present neural network operations inpseudo-Riemannian manifolds with novel geodesic tools, to stimulate the applications of pseudoRiemannian geometry in geometric deep learning.
Revised: To summarize, our main contributions are as follows: 1) We present neural network operations inpseudo-Riemannian manifolds with novel geodesic tools, to stimulate the applications of pseudoRiemannian geometry in geometric deep learning.

Original: We present a principled framework, pseudoRiemannian GCN, which generalizes GCNs into pseudo-Riemannian manifolds with indefinite metrics, providing more flexible inductive biases to accommodate complex graphs with mixed topologies.
Revised: We present a principled framework, pseudoRiemannian GCN, which generalizes GCNs into pseudo-Riemannian manifolds with indefinite metrics, providing more flexible inductive biases to accommodate complex graphs with mixed topologies.

Original: 3) Extensive evaluations on three standard tasks demonstrate that our model outperforms baselines that operate in Riemannian manifolds.
Revised: 3) Extensive evaluations on three standard tasks demonstrate that our model outperforms baselines that operate in Riemannian manifolds.

Original: Source code is open available.
Revised: Source code is open available.

Original:  Preliminaries2.1 Pseudo-Riemannian manifolds
Revised:  Preliminaries2.1 Pseudo-Riemannian manifolds

Original: By closely following the notions of [21], a pseudo-Riemannian manifold [23] ( M , g ) is a smoothmanifold M equipped with a nondegenerate and indefinite metric tensor g .
Revised: By closely following the notions of [21], a pseudo-Riemannian manifold [23] ( M , g ) is a smoothmanifold M equipped with a nondegenerate and indefinite metric tensor g .

Original: Nondegeneracy meansthat for a given ξ ∈ T x M , for any ζ ∈ T x
Revised: Nondegeneracy meansthat for a given ξ ∈ T x M , for any ζ ∈ T x

Original: M we have g x ( ξ , ζ ) = 0 , then ξ = 0 .
Revised: M we have g x ( ξ , ζ ) = 0 , then ξ = 0 .

Original: The metrictensor g induces a scalar product on the tangent space T x M for each point x ∈ M such that g x :
Revised: The metrictensor g induces a scalar product on the tangent space T x M for each point x ∈ M such that g x :

Original: T x M×T x M → R , where the tangent space T x M can be seen as the first order local approximationof M around point x .
Revised: T x M×T x M → R , where the tangent space T x M can be seen as the first order local approximationof M around point x .

Original: The elements of T x M are called tangent vectors.
Revised: The elements of T x M are called tangent vectors.

Original: Indefinity means that themetric tensor could be of arbitrary signs.
Revised: Indefinity means that themetric tensor could be of arbitrary signs.

Original: A principal special case is the Riemannian geometry, wherethe metric tensor is positive definite (i.e. ∀ ξ ∈ T x M , g x ( ξ , ξ ) > 0 iff ξ ̸ = 0 ).
Revised: A principal special case is the Riemannian geometry, wherethe metric tensor is positive definite (i.e. ∀ ξ ∈ T x M , g x ( ξ , ξ ) > 0 iff ξ ̸ = 0 ).

Original: Pseudo-hyperboloid.
Revised: Pseudo-hyperboloid.

Original: By analogy with hyperboloid and sphere in Euclidean space.
Revised: By analogy with hyperboloid and sphere in Euclidean space.

Original: Pseudohyperboloids are defined as the submanifolds in the ambient pseudo-Euclidean space R s,t +1 withthe dimensionality of d
Revised: Pseudohyperboloids are defined as the submanifolds in the ambient pseudo-Euclidean space R s,t +1 withthe dimensionality of d

Original: = s + t + 1 that uses the scalar product as ∀ x , y ∈ R s,t +1 , ⟨ x , y ⟩ t =− (cid:80) ti
Revised: = s + t + 1 that uses the scalar product as ∀ x , y ∈ R s,t +1 , ⟨ x , y ⟩ t =− (cid:80) ti

Original: =
Revised: =

Original: x
Revised: x

Original: i y i
Revised: i y i

Original: + (cid:80) s
Revised: + (cid:80) s

Original: + t j = t
Revised: + t j = t

Original: +1 x j y j .
Revised: +1 x j y j .

Original: The scalar product induces a norm ∥ x ∥ 2 t = ⟨ x , x ⟩ t that can be usedto define a pseudo-hyperboloid Q s,tβ = (cid:110) x
Revised: The scalar product induces a norm ∥ x ∥ 2 t = ⟨ x , x ⟩ t that can be usedto define a pseudo-hyperboloid Q s,tβ = (cid:110) x

Original: = ( x 0 , x 1 , · · · , x s + t ) ⊤ ∈ R s,t +1 :
Revised: = ( x 0 , x 1 , · · · , x s + t ) ⊤ ∈ R s,t +1 :

Original: ∥ x ∥ 2 t = β (cid:111)
Revised: ∥ x ∥ 2 t = β (cid:111)

Original: , whereβ is a nonzero real number parameter of curvature.
Revised: , whereβ is a nonzero real number parameter of curvature.

Original: Q s,tβ is called pseudo-sphere when β > 0 andpseudo-hyperboloid when β < 0 .
Revised: Q s,tβ is called pseudo-sphere when β > 0 andpseudo-hyperboloid when β < 0 .

Original: Since Q s,tβ is interchangeable with Q t +1 ,s − 1 − β , we consider thepseudo-hyperbololid here.
Revised: Since Q s,tβ is interchangeable with Q t +1 ,s − 1 − β , we consider thepseudo-hyperbololid here.

Original: Following the terminology of special relativity, a point in Q s,tβ can beinterpreted as an event [5], where the first t + 1 dimensions are time dimensions and the last sdimensions are space dimensions.
Revised: Following the terminology of special relativity, a point in Q s,tβ can beinterpreted as an event [5], where the first t + 1 dimensions are time dimensions and the last sdimensions are space dimensions.

Original: Hyperbolic H and spherical S manifolds can be defined as thespecial cases of pseudo-hyperboloids by setting all time dimensions except one to be zero and setting all space dimensions to be zero, respectively, i.e. H β = Q s, 1 β , S − β
Revised: Hyperbolic H and spherical S manifolds can be defined as thespecial cases of pseudo-hyperboloids by setting all time dimensions except one to be zero and setting all space dimensions to be zero, respectively, i.e. H β = Q s, 1 β , S − β

Original: = Q 0 ,tβ .
Revised: = Q 0 ,tβ .

Original: 2.2 Geodesic tools of pseudo-hyperboloid
Revised: 2.2 Geodesic tools of pseudo-hyperboloid

Original: Geodesic.
Revised: Geodesic.

Original: A generalization of a straight-line in the Euclidean space to a manifold is called thegeodesic [12, 24].
Revised: A generalization of a straight-line in the Euclidean space to a manifold is called thegeodesic [12, 24].

Original: Formally, a geodesic γ is defined as a constant speed curve γ : τ (cid:55)→ γ ( τ ) ∈
Revised: Formally, a geodesic γ is defined as a constant speed curve γ : τ (cid:55)→ γ ( τ ) ∈

Original: M , τ ∈ [0 , 1] joining two points x , y ∈ M that minimizes the length, where the length of a curveis given by L ( γ ) = (cid:82) 1 0 (cid:113)(cid:13)(cid:13) ddt γ ( τ ) (cid:13)(cid:13) γ ( τ ) dt .
Revised: M , τ ∈ [0 , 1] joining two points x , y ∈ M that minimizes the length, where the length of a curveis given by L ( γ ) = (cid:82) 1 0 (cid:113)(cid:13)(cid:13) ddt γ ( τ ) (cid:13)(cid:13) γ ( τ ) dt .

Original: The geodesic holds that γ ∗ = arg min γ L ( γ ) , such thatγ (0) = x , γ (1) =
Revised: The geodesic holds that γ ∗ = arg min γ L ( γ ) , such thatγ (0) = x , γ (1) =

Original: y , and (cid:13)(cid:13) ddτ γ ( τ ) (cid:13)(cid:13) γ ( τ ) = 1 .
Revised: y , and (cid:13)(cid:13) ddτ γ ( τ ) (cid:13)(cid:13) γ ( τ ) = 1 .

Original: By the means of the geodesic, the distance betweenx , y ∈ Q s,tβ is defined as the arc length of geodesic γ ( τ ) .
Revised: By the means of the geodesic, the distance betweenx , y ∈ Q s,tβ is defined as the arc length of geodesic γ ( τ ) .

Original: Exponential and logarithmic maps.
Revised: Exponential and logarithmic maps.

Original: The connections between manifolds and tangent space areestablished by the differentiable exponential map and logarithmic map.
Revised: The connections between manifolds and tangent space areestablished by the differentiable exponential map and logarithmic map.

Original: The exponential map at x isdefined as exp x ( ξ ) = γ (1) , which gives a way to project a vector ξ ∈ T x M to a point exp x ( ξ ) ∈ Mon the manifold.
Revised: The exponential map at x isdefined as exp x ( ξ ) = γ (1) , which gives a way to project a vector ξ ∈ T x M to a point exp x ( ξ ) ∈ Mon the manifold.

Original: The logarithmic map log x : M → T x M is defined as the inverse of the exponentialmap (i.e. log x = exp − 1 x ).
Revised: The logarithmic map log x : M → T x M is defined as the inverse of the exponentialmap (i.e. log x = exp − 1 x ).

Original: Note that since Q s,tβ is a geodesically complete manifold, the domain ofthe exponential map D x is hence defined on the entire tangent space, i.e. D x = T x Q s,tβ .
Revised: Note that since Q s,tβ is a geodesically complete manifold, the domain ofthe exponential map D x is hence defined on the entire tangent space, i.e. D x = T x Q s,tβ .

Original: However,as we will explain later, the logarithmic map log x ( y ) is only defined when there exists a a lengthminimizing geodesic between x , y ∈ Q s,tβ .
Revised: However,as we will explain later, the logarithmic map log x ( y ) is only defined when there exists a a lengthminimizing geodesic between x , y ∈ Q s,tβ .

Original: More details can be found in the Appendix C.
Revised: More details can be found in the Appendix C.

Original: Geodesical connectedness.
Revised: Geodesical connectedness.

Original: A pseudo-Riemannian manifold M is connected iff any two points of
Revised: A pseudo-Riemannian manifold M is connected iff any two points of

Original: M can be joined by a piecewise (broken) geodesic with each piece being a smooth geodesic.
Revised: M can be joined by a piecewise (broken) geodesic with each piece being a smooth geodesic.

Original: Themanifold is geodesically connected (or g-connected )
Revised: Themanifold is geodesically connected (or g-connected )

Original: iff any two points can be smoothly connectedby a geodesic, where the two points are called g-connected , otherwise called g-disconnected .
Revised: iff any two points can be smoothly connectedby a geodesic, where the two points are called g-connected , otherwise called g-disconnected .

Original: Different from Riemannian manifolds in which the geodesical completeness implies the g-connectedness(Hopf–Rinow theorem [25]), pseudo-hyperboloid is a geodesically complete but not g-connectedmanifold where there exist points that cannot be smoothly connected by a geodesic [26].
Revised: Different from Riemannian manifolds in which the geodesical completeness implies the g-connectedness(Hopf–Rinow theorem [25]), pseudo-hyperboloid is a geodesically complete but not g-connectedmanifold where there exist points that cannot be smoothly connected by a geodesic [26].

Original: Formally, in the pseudo-hyperboloid, two points x , y ∈ Q s,tβ are g-connected iff ⟨ x , y ⟩ t < | β | .
Revised: Formally, in the pseudo-hyperboloid, two points x , y ∈ Q s,tβ are g-connected iff ⟨ x , y ⟩ t < | β | .

Original: The set of g-connected points of x ∈ Q s,tβ is denoted as its normal neighborhood U x = 115 (cid:110)
Revised: The set of g-connected points of x ∈ Q s,tβ is denoted as its normal neighborhood U x = 115 (cid:110)

Original: y ∈ Q s,tβ : ⟨ x , y ⟩ t <
Revised: y ∈ Q s,tβ : ⟨ x , y ⟩ t <

Original: | β | (cid:111) .
Revised: | β | (cid:111) .

Original: For g-disconnected points x , y ∈ Q s,tβ , there does not exist a tangent vector ξ such that y =
Revised: For g-disconnected points x , y ∈ Q s,tβ , there does not exist a tangent vector ξ such that y =

Original: exp β x ( ξ ) , which implies that its inverse log β x ( · ) is only defined inthe normal neighborhood of x .
Revised: exp β x ( ξ ) , which implies that its inverse log β x ( · ) is only defined inthe normal neighborhood of x .

Original: In a nutshell, the geodesic tools for the g-disconnected cases are notwell-defined, making it impossible to define corresponding vector operations.
Revised: In a nutshell, the geodesic tools for the g-disconnected cases are notwell-defined, making it impossible to define corresponding vector operations.

Original:  Pseudo-Riemannian GCNs
Revised:  Pseudo-Riemannian GCNs

Original: In this section, we first describe how to tackle the g-disconnectedness in pseudo-Riemannian manifolds.
Revised: In this section, we first describe how to tackle the g-disconnectedness in pseudo-Riemannian manifolds.

Original: Then we present the pseudo-Riemannian GCNs based on the proposed geodesic tools.
Revised: Then we present the pseudo-Riemannian GCNs based on the proposed geodesic tools.

Original: 3.1 Diffeomorphic geodesic tools
Revised: 3.1 Diffeomorphic geodesic tools

Original: One standard way to tackle the g-disconnectedness in differential geometry is to introduce diffeomorphic manifolds in which the operations are well-defined.
Revised: One standard way to tackle the g-disconnectedness in differential geometry is to introduce diffeomorphic manifolds in which the operations are well-defined.

Original: A diffeomorphic manifold can be derived from a diffeomorphism, defined as follows.
Revised: A diffeomorphic manifold can be derived from a diffeomorphism, defined as follows.

Original: Definition 1 (Diffeomorphism [23]) .
Revised: Definition 1 (Diffeomorphism [23]) .

Original: Given two manifolds M and M ′ , a smooth map ψ : M → M ′is called a diffeomorphism if ψ is bijective and its inverse ψ − 1 is smooth as well.
Revised: Given two manifolds M and M ′ , a smooth map ψ : M → M ′is called a diffeomorphism if ψ is bijective and its inverse ψ − 1 is smooth as well.

Original: If a diffeomorphismbetween M and M ′ exists, we call them diffeomorphic and write M ≃ M ′ .
Revised: If a diffeomorphismbetween M and M ′ exists, we call them diffeomorphic and write M ≃ M ′ .

Original: For pseudo-Riemannian manifolds, the following diffeomorphism [21] decomposes pseudohyperboloid into the product manifolds of an unit sphere and the Euclidean space.
Revised: For pseudo-Riemannian manifolds, the following diffeomorphism [21] decomposes pseudohyperboloid into the product manifolds of an unit sphere and the Euclidean space.

Original: Theorem 1 (Theorem 4.1 in [21]) .
Revised: Theorem 1 (Theorem 4.1 in [21]) .

Original: For any point x ∈ Q s,tβ , there exists a diffeomorphism ψ :
Revised: For any point x ∈ Q s,tβ , there exists a diffeomorphism ψ :

Original: Q s,tβ → S t 1 × R s that maps x into the product manifolds of an unit sphere and the Euclidean space.
Revised: Q s,tβ → S t 1 × R s that maps x into the product manifolds of an unit sphere and the Euclidean space.

Original: The diffeomorphism is given in the Appendix D.1.
Revised: The diffeomorphism is given in the Appendix D.1.

Original: In light of this, we introduce a new diffeomorphism that maps x to the product manifolds of sphere with curvature − 1 /β and the Euclidean space.
Revised: In light of this, we introduce a new diffeomorphism that maps x to the product manifolds of sphere with curvature − 1 /β and the Euclidean space.

Original: Theorem 2.
Revised: Theorem 2.

Original: For any point x ∈ Q s,tβ , there exists a diffeomorphism ψ : Q s,tβ → S t − β × R s that mapsx into the product manifolds of a sphere and the Euclidean space (proof in the Appendix D.2).
Revised: For any point x ∈ Q s,tβ , there exists a diffeomorphism ψ : Q s,tβ → S t − β × R s that mapsx into the product manifolds of a sphere and the Euclidean space (proof in the Appendix D.2).

Original: Compared with Theorem 1, this diffeomorphism preserves the curvatures in the diffeomorphic components, making it satisfy some geometric properties, e.g. the mapped point x t ∈ S t − β still lies on thesurface of the pseudo-hyperboloid, making moving the tangent vectors from the pseudo-hyperboloidto the diffeomorphic manifold easy as we explained later.
Revised: Compared with Theorem 1, this diffeomorphism preserves the curvatures in the diffeomorphic components, making it satisfy some geometric properties, e.g. the mapped point x t ∈ S t − β still lies on thesurface of the pseudo-hyperboloid, making moving the tangent vectors from the pseudo-hyperboloidto the diffeomorphic manifold easy as we explained later.

Original: We call ψ as the spherical projection.
Revised: We call ψ as the spherical projection.

Original: Exponential and logarithmic maps.
Revised: Exponential and logarithmic maps.

Original: Considering that pseudo-hyperboloid Q s,tβ is gdisconnected , we propose to transfer the logmap and expmap into the diffeomorphic manifoldψ : Q s,tβ → S t − β × R s , since the product manifold S t − β × R s is g-connected .
Revised: Considering that pseudo-hyperboloid Q s,tβ is gdisconnected , we propose to transfer the logmap and expmap into the diffeomorphic manifoldψ : Q s,tβ → S t − β × R s , since the product manifold S t − β × R s is g-connected .

Original: To map tangentvectors between tangent space of Q s,tβ and S t − β × R s , we exploit pushforward that induce a linearapproximation of smooth maps on tangent spaces.
Revised: To map tangentvectors between tangent space of Q s,tβ and S t − β × R s , we exploit pushforward that induce a linearapproximation of smooth maps on tangent spaces.

Original: Definition 2 (Pushforward) .
Revised: Definition 2 (Pushforward) .

Original: Suppose that ψ : M → M ′ is a smooth map, then the differential ofψ : dψ at point x is a linear map from the tangent space of M at x to the tangent space of M ′ atψ ( x ) .
Revised: Suppose that ψ : M → M ′ is a smooth map, then the differential ofψ : dψ at point x is a linear map from the tangent space of M at x to the tangent space of M ′ atψ ( x ) .

Original: Namely, dψ : T x M → T ψ ( x ) M ′
Revised: Namely, dψ : T x M → T ψ ( x ) M ′

Original: Intuitively, pushforward can be used to push tangent vectors on T x Q s,tβ forward to tangent vectorson T ψ ( x ) S t − β × R s .
Revised: Intuitively, pushforward can be used to push tangent vectors on T x Q s,tβ forward to tangent vectorson T ψ ( x ) S t − β × R s .

Original: Based on this, the new logmap and its inverse expmap can be defined by Eq.
Revised: Based on this, the new logmap and its inverse expmap can be defined by Eq.

Original: [Equation]
Revised: [Equation]

Original: where ψ ( · ) is the spherical projection and ψ − 1 ( · ) is the inverse.
Revised: where ψ ( · ) is the spherical projection and ψ − 1 ( · ) is the inverse.

Original: The mapping of tangent vectors isachieved by pushforward operations.
Revised: The mapping of tangent vectors isachieved by pushforward operations.

Original: The operations log S t − β × R s ( · ) and exp S t − β × R s ( · ) in the productmanifolds can be defined as the concatenation of corresponding operations in different components.
Revised: The operations log S t − β × R s ( · ) and exp S t − β × R s ( · ) in the productmanifolds can be defined as the concatenation of corresponding operations in different components.

Original: [Equation]
Revised: [Equation]

Original: where || denotes the concatenation, x ′ =
Revised: where || denotes the concatenation, x ′ =

Original: ψ S ( x ) consists of spherical features x ′ t ∈ S t − β
Revised: ψ S ( x ) consists of spherical features x ′ t ∈ S t − β

Original: and Euclidean features x ′ s ∈ R s .
Revised: and Euclidean features x ′ s ∈ R s .

Original: ξ is the tangent vector induced by x on Q s,tβ .
Revised: ξ is the tangent vector induced by x on Q s,tβ .

Original: Remarkably, although we transfer logmap and expmap to the diffeomorphic manifold S t − β × R s ,the diffeomorphic operations (cid:99) log x ( · ) and (cid:100) exp x ( · ) are still bijective functions from the pseudohyperboloid to the tangent space benefited from the diffeomorphism and pushforward operations.

Original: Note that we only move the neural operations to the diffeomorphic manifold but the final distance isstill defined in the pseudo-Riemannian manifold, which implies that the pseudo-Riemannain metricand the geometric expressiveness of pseudo-hyperboloid are preserved.

Original: This is typically differentfrom simply using the metric defined in the product manifolds.

Original: We choose points where space dimension s = 0 as the reference points due to the following property.
Revised: We choose points where space dimension s = 0 as the reference points due to the following property.

Original: Theorem 3.
Revised: Theorem 3.

Original: For any reference point x = (cid:18) ts (cid:19) ∈ Q s,tβ with space dimension s = 0 , the inducedtangent space of Q s,tβ is equal to the tangent space of its diffeomorphic manifold S t − β ×
Revised: For any reference point x = (cid:18) ts (cid:19) ∈ Q s,tβ with space dimension s = 0 , the inducedtangent space of Q s,t β is equal to the tangent space of its diffeomorphic manifold S t − β ×

Original: R s , namely,
Revised: R s , namely,

Original: T x ( S t − β × R s )
Revised: T x ( S t − β × R s )

Original: =
Revised: =

Original: T x Q s,tβ .
Revised: T x Q s,tβ .

Original: (proof in the Appendix D.3).
Revised: (proof in the Appendix D.3).

Original: The intuition of proof is that if space dimension s = 0 , the pushforward (differential) function justinfluences time dimension, for which the mapping is just an identity function (see Appendix D.2).
Revised: The intuition of proof is that if space dimension s = 0 , the pushforward (differential) functionjust influences time dimension, for which the mapping is just an identity function (see Appendix D.2).

Original: This property significantly simplifies thecomputation of diffeomorphic operations.
Revised: In this way, although we transfer logmap and expmap to the diffeomorphic manifold S t − β × R s , the diffeomorphic operations (cid:99) log x ( · ) and (cid:100) exp x ( · ) are still bijective functions from the pseudohyperboloid to the tangent space of the manifold itself. Hence, our final operations are actually stilldefined in the tangent space of the pseudo-hyperboloid. Note that such property only holds whenour Theorem 2 is applied and the special reference points with space dimension s = 0 are chosen.

Original: By leveraging the new logmap and expmap , we further formulate the diffeomorphic version of tangential operations as follows.
Revised: By leveraging the new logmap and expmap , we further formulate the diffeomorphic version of tangential operations as follows.

Original: Tangential operations.
Revised: Tangential operations.

Original: For function f : R d → R d ′ , the pseudo-hyperboloid version f ⊗ : Q s,tβ →
Revised: For function f : R [Equation] ′ , the pseudo-hyperboloid version f [Equation]

Original: Q s ′ ,t ′ β with s + t = d and s ′ + t ′ =
Revised: [Equation] β with s + t = d and s ′ + t ′ =

Original: d ′ can be defined by the means of (cid:99) log β x ( · ) and (cid:100)
Revised: d ′ can be defined by the means of (cid:99) log β x ( · ) and (cid:100)

Original: exp β x ( · ) as Eq.
Revised: exp β x ( · ) as Eq.

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where x is the reference point.
Revised: where x is the reference point.

Original: Note that this function is a morphism (i.e. ( f ◦ g ) ⊗ = f ⊗ ◦ g ⊗ )and direction preserving (i.e. f ⊗ ( · ) /
Revised: Note that this function is a morphism (i.e. ( f ◦ g ) ⊗ = f ⊗ ◦ g ⊗ )and direction preserving (i.e. f ⊗ ( · ) /

Original: ∥ f ⊗ ( · ) ∥
Revised: ∥ f ⊗ ( · ) ∥

Original: = f ( · ) /
Revised: = f ( · ) /

Original: ∥ f ( · ) ∥ ) [12], making it a natural way to define pseudo-hyperboloid version of vector operations such like scalar multiplication, matrix-vectormultiplication, tangential aggregation and point-wise non-linearity and so on.
Revised: ∥ f ( · ) ∥ ) [12], making it a natural way to define pseudo-hyperboloid version of vector operations such like scalar multiplication, matrix-vectormultiplication, tangential aggregation and point-wise non-linearity and so on.

Original: Parallel transport.
Revised: Parallel transport.

Original: Parallel transport is the generalization of Euclidean translation into manifolds.
Revised: Parallel transport is the generalization of Euclidean translation into manifolds.

Original: Formally, for any two points x and y connected by a geodesic, parallel transport P β x → y ( ξ ) : T x M →
Revised: Formally, for any two points x and y connected by a geodesic, parallel transport P β x → y ( ξ ) : T x M →

Original: T y M is an isomorphism between two tangent spaces by moving one tangent vector ζ ∈ T x
Revised: T y M is an isomorphism between two tangent spaces by moving one tangent vector ζ ∈ T x

Original: Mwith tangent direction ξ ∈ T x M to another tangent space T y M .
Revised: Mwith tangent direction ξ ∈ T x M to another tangent space T y M .

Original: The parallel transport in pseudohyperboloid can be defined as the combination of Riemannian parallel transport [27].
Revised: The parallel transport in pseudohyperboloid can be defined as the combination of Riemannian parallel transport [27].

Original: However, theparallel transport has not been defined when there does not exist a geodesic between x and y .
Revised: However, theparallel transport has not been defined when there does not exist a geodesic between x and y .

Original: i.e.,the tangent vector ζ induced by x can not be transported to the tangent space of points outside ofthe normal neighborhood
Revised: i.e.,the tangent vector ζ induced by x can not be transported to the tangent space of points outside ofthe normal neighborhood

Original: U x .
Revised: U x .

Original: Intuitively, the normal neighborhoods satisfy the following property.
Revised: Intuitively, the normal neighborhoods satisfy the following property.

Original: Theorem 4.
Revised: Theorem 4.

Original: For any point x ∈ Q s,tβ , the union of the normal neighborhood of x and the normalneighborhood of its antipodal point − x cover the entire manifold.
Revised: For any point x ∈ Q s,tβ , the union of the normal neighborhood of x and the normalneighborhood of its antipodal point − x cover the entire manifold.

Original: Namely, U x ∪ U − x = Q s,tβ (proofin the Appendix D.4).
Revised: Namely, U x ∪ U − x = Q s,tβ (proofin the Appendix D.4).

Original: This theorem ensures that if a point y / ∈ U x , its antipodal point − y ∈
Revised: This theorem ensures that if a point y / ∈ U x , its antipodal point − y ∈

Original: U x .
Revised: U x .

Original: Besides, T y M is parallelto T − y M .
Revised: Besides, T y M is parallelto T − y M .

Original: Hence, P β x → y can be alternatively defined as P β x →− y for broken points.
Revised: Hence, P β x → y can be alternatively defined as P β x →− y for broken points.

Original: This result iscrucial to define the pseudo-hyperbolic addition, such as bias translation, detailed in section 3.2.
Revised: This result iscrucial to define the pseudo-hyperbolic addition, such as bias translation, detailed in section 3.2.

Original: Broken geodesic distance.
Revised: Broken geodesic distance.

Original: By the means of geodesic, the induced distance between x andy in pseudo-hyperboloid is defined as the arc length of geodesic γ ( τ ) , given by d γ ( x , y )
Revised: By the means of geodesic, the induced distance between x andy in pseudo-hyperboloid is defined as the arc length of geodesic γ ( τ ) , given by d γ ( x , y )

Original: = 194 (cid:113) ∥ log x ( y ) ∥ 2 t .
Revised: = 191 (cid:113) ∥ log x ( y ) ∥ 2 t .

Original: For broken cases in which log x ( y ) is not defined, one approach is to use approximation like [21].
Revised: For broken cases in which log x ( y ) is not defined, one approach is to use approximation like [21].

Original: Different from that, we define following closed-form distance, given by Eq.
Revised: Different from that, we define following closed-form distance, given by Eq.

Original: [Equation]
Revised: [Equation]

Original:  The intuition is that when x , y ∈ Q s,tβ are g-disconnected , we consider the distance as d γ ( x , y ) =
Revised: [Equation] The intuition is that when x , y ∈ Q s,t β are g-disconnected , we consider the distance as d γ ( x , y ) =

Original: d γ ( x , − x ) + d γ ( − x , y ) or d γ ( x , y ) =
Revised: d γ ( x , − x ) + d γ ( − x , y ) or d γ ( x , y ) =

Original: d γ ( x , − y )
Revised: d γ ( x , − y )

Original: + d γ ( − y , y ) .
Revised: + d γ ( − y , y ) .

Original: Since d γ ( x , − x ) = d γ ( − y , y ) =
Revised: Since d γ ( x , − x ) = d γ ( − y , y ) =

Original: π (cid:112) | β | is a constant and d γ ( − x , y ) =
Revised: π (cid:112) | β | is a constant and d γ ( − x , y ) =

Original: d γ ( x , − y ) , the distance between broken points can be calculated as d γ ( x , y ) =
Revised: d γ ( x , − y ) , the distance between broken points can be calculated as d γ ( x , y ) =

Original: π (cid:112) | β | + d γ ( x , − y ) .
Revised: π (cid:112) | β | + d γ ( x , − y ) .

Revised: To clarify theoretical contributions, our Theorem 2 is nessasary for our Theorem 3 while Theorem3 is nessasary for transforming the GCN operations directly into the tangent space of the pseudohyperboloid.

Original: To clarify theoretical contributions, we are the first to formulate the diffeomorphic expmap , logmapand tangential operations of pseudo-hyperboloid to avoid broken cases.
Revised: Besides, we are the first to formulate the diffeomorphic expmap , logmap and tangentialoperations of pseudo-hyperboloid to avoid broken cases.

Original: The theoretical propertiesof parallel transport and geodesic distance are discussed in the literature [21, 27].
Revised: The theoretical properties of parallel transport and geodesic distance are discussed in the literature [21, 27].

Original: We re-formulateparallel transport to avoid broken issues and use the broken geodesic to define the distance, which isdifferent from the approximated distance in [21].
Revised: However, we re-formulate paralleltransport with Theorem 4 to avoid broken issues and propose a new distance measure using thebroken geodesic (Eq.4 ), which is different from the approximated distance in [21].

Original: 3.2 Model architecture
Revised: 3.2 Model architecture

Original: GCNs can be interpreted as performing neighborhood aggregation after a linear transformation onnode features of each layer.
Revised: GCNs can be interpreted as performing neighborhood aggregation after a linear transformation onnode features of each layer.

Original: We present pseudo-Riemannian GCNs ( Q -GCN) by deriving corresponding operations with the developed geodesic tools in the Q s,tβ .
Revised: We present pseudo-Riemannian GCNs ( Q -GCN) by deriving corresponding operations with the developed geodesic tools in the Q s,tβ .

Original: Feature initialization.
Revised: Feature initialization.

Original: We first map the features from Euclidean space to pseudo-hyperboloid,considering that the input features of nodes usually live in Euclidean space.
Revised: We first map the features from Euclidean space to pseudo-hyperboloid,considering that the input features of nodes usually live in Euclidean space.

Original: Following the feature transformation from Euclidean space to pseudo-hyperboloid in [21], we initialize the node featuresby performing a differentiable mapping φ : R t +1 ∗ ×
Revised: Following the featuretransformation from Euclidean space to pseudo-hyperboloid in [21], we initialize the node featuresby performing a differentiable mapping φ : R t +1 ∗ ×

Original: R s → Q s,tβ that can be implemented by adouble projection [21] based on Theorem 2, i.e. φ = ψ − 1 ◦ ψ .
Revised: R s → Q s,tβ that can be implemented by adouble projection [21] based on Theorem 2, i.e. φ = ψ − 1 ◦ ψ .

Original: The intuition is that we first mapthe Euclidean features into diffeomorphic manifolds S t − β × R s via ψ ( · ) , and then map them into thepseudo-hyperboloid Q s,tβ via ψ − 1 ( · ) , where the mapping functions are given by Eq.
Revised: The intuition is that we first mapthe Euclidean features into diffeomorphic manifolds S t − β × R s via ψ ( · ) , and then map them into thepseudo-hyperboloid Q s,tβ via ψ − 1 ( · ) , where the mapping functions are given by Eq.

Original: [Equation]
Revised: [Equation]

Original: where x = ( t , s ) ⊤ ∈ Q s,tβ with t ∈ R t ∗ and s ∈ R s .
Revised: where x = ( t , s ) ⊤ ∈ Q s,tβ with t ∈ R t ∗ and s ∈ R s .

Original: z = ( u , v ) ⊤ ∈ S t − β ×
Revised: z = ( u , v ) ⊤ ∈ S t − β ×

Original: R s with u ∈ S t − β and 217 v ∈ R s .
Revised: R s with u ∈ S t − β and 216 v ∈ R s .

Original: Tangential aggregation.
Revised: Tangential aggregation.

Original: The linear combination of neighborhood features is lifted to the tangentspace, which is an intrinsic operation in differential manifolds [28, 13].
Revised: The linear combination of neighborhood features is lifted to the tangentspace, which is an intrinsic operation in differential manifolds [28, 13].

Original: Specifically, Q -GCN aggregates neighbours’ embeddings in the tangent space of the reference point o before passing througha tangential activation function, and then projects the updated representation back to the manifold.
Revised: Specifically, Q -GCN aggregates neighbours’ embeddings in the tangent space of the reference point o before passing througha tangential activation function, and then projects the updated representation back to the manifold.

Original: Formally, at each layer ℓ , the updated features of each node i are defined as Eq.
Revised: Formally, at each layer ℓ , the updated features of each node i are defined as Eq.

Original: [Equation]
Revised: [Equation]

Original: where σ ( · ) is the activation function, β ℓ and β ℓ +1 are two layer-wise curvatures, N ( i ) denotesthe one-hop neighborhoods of node i , and the ⊗ , ⊕ denote two basic operations, i.e. tangentialtransformation and bias translation, respectively.
Revised: where σ ( · ) is the activation function, β ℓ and β ℓ +1 are two layer-wise curvatures, N ( i ) denotesthe one-hop neighborhoods of node i , and the ⊗ , ⊕ denote two basic operations, i.e. tangentialtransformation and bias translation, respectively.

Original: Tangential transformation.
Revised: Tangential transformation.

Original: We perform Euclidean transformations on the tangent space by leveraging the expmap and logmap in Eq.
Revised: We perform Euclidean transformations on the tangent space by leveraging the expmap and logmap in Eq.

Original: Specifically, we first project the hidden feature into thetangent space of south pole o =
Revised: Specifically, we first project the hidden feature into thetangent space of south pole o =

Original: [ | β | , 0 , , ..., 0] using logmap and then perform Euclidean matrixmultiplication.
Revised: [ | β | , 0 , , ..., 0] using logmap and then perform Euclidean matrixmultiplication.

Original: Afterwards, the transformed features are mapped back to the manifold using expmap .
Revised: Afterwards, the transformed features are mapped back to the manifold using expmap .

Original: Formally, at each layer ℓ , the tangential transformation is given by W ℓ ⊗ β h ℓ := (cid:100) exp 
Revised: Formally, at each layer ℓ , the tangential transformation is given by W ℓ ⊗ β h ℓ := (cid:100) exp [Equation] o ( W

Original: β o ( W ℓ (cid:99) log β o ( h ℓ )) ,where ⊗ β denotes the pseudo-hyperboloid tangential multiplication, and W ℓ ∈ R d ′ ×
Revised: [Equation] ( h [Equation] )) ,where ⊗ β denotes the pseudo-hyperboloid tangential multiplication, and W ℓ ∈ R d ′ ×

Original: d denotes thelayer-wise learnable matrix in Euclidean space.
Revised: d denotes thelayer-wise learnable matrix in Euclidean space.

Original: Bias translation.
Revised: Bias translation.

Original: It is noteworthy that simply stacking multiple layers of the tangential transformation would collapse the composition [28, 12], i.e. exp β o ...
Revised: It is noteworthy that simply stacking multiple layers of the tangential transformation would collapse the composition [28, 12], i.e. exp β o ...

Original: ( W 1 log β
Revised: ( W 1 log β

Original: o (exp β o ( W 0 log β
Revised: o (exp β o ( W 0 log β

Original: o ( x ))))
Revised: o ( x ))))

Original: =
Revised: =

Original: exp β
Revised: exp β

Original: o ( W 0 × W 1 × ...
Revised: o ( W 0 × W 1 × ...

Original: × log β
Revised: × log β

Original: o ( x )) , which means that these multiplications can simply be performed in Euclidean space except the first logmap and last expmap .
Revised: o ( x )) , which means that these multiplications can simply be performed in Euclidean space except the first logmap and last expmap .

Original: To avoid model collapsing , weperform bias translation after the tangential transformation.
Revised: To avoid model collapsing , weperform bias translation after the tangential transformation.

Original: By the means of pseudo-hyperboloidparallel transport , the bias translation can be performed by parallel transporting a tangent vectorb ℓ ∈ T
Revised: By the means of pseudo-hyperboloidparallel transport , the bias translation can be performed by parallel transporting a tangent vectorb ℓ ∈ T

Original: o Q s,tβ to the tangent space of the point of interest.
Revised: o Q s,tβ to the tangent space of the point of interest.

Original: Finally, the transported tangent vectoris mapped back to the manifold with expmap .
Revised: Finally, the transported tangent vectoris mapped back to the manifold with expmap .

Original: Considering that (cid:100) exp( · ) is only defined at pointx ∈ Q s,tβ with the space dimension x s = 0 , we perform the original exp Q s,t β ( · ) at the point ofinterest.
Revised: Considering that (cid:100) exp( · ) is only defined at pointx ∈ Q s,t β with the space dimension x s = 0 , we perform the original exp Q s,t β ( · ) at the point ofinterest.

Original: The bias translation is formally given by:
Revised: The bias translation is formally given by:

Original: [Equation]
Revised: [Equation]

Original: 7) where (cid:101) h ℓ = W ℓ ⊗ β h ℓ , ⊕ β denotes the pseudo-hyperboloid addition.
Revised: where (cid:101) h ℓ = W ℓ ⊗ β h ℓ , ⊕ β denotes the pseudo-hyperboloid addition.

Original: For the broken cases where⟨ o , (cid:101) h ℓ ⟩ t ≥ | β | , the parallel transport P β o → (cid:101) h ℓ is not defined.
Revised: For the broken cases where⟨ o , (cid:101) h ℓ ⟩ t ≥ | β | , the parallel transport P β o → (cid:101) h ℓ is not defined.

Original: In this case, we parallel transport b ℓ to thetangent space of the antipodal point − (cid:101) h ℓ , and then perform exp β − (cid:101) h ℓ to map it back to the manifold.
Revised: In this case, we parallel transport b ℓ to thetangent space of the antipodal point − (cid:101) h ℓ , and then perform exp β − (cid:101) h ℓ to map it back to the manifold.

Original: Note that the case ⟨ o , (cid:101) h ℓ ⟩ t =
Revised: Note that the case ⟨ o , (cid:101) h ℓ ⟩ t =

Original: | β | occurs if and only if h = − o , in which case P β o →− (cid:101) h ℓ (cid:0) b ℓ (cid:1) =
Revised: | β | occurs if and only if h = − o , in which case P β o →− (cid:101) h ℓ (cid:0) b ℓ (cid:1) =

Original: P β o →− o (cid:0) b ℓ (cid:1) = − b .
Revised: P β o →− o (cid:0) b ℓ (cid:1) = − b .

Original: 3.3 Model training
Revised: 3.3 Model training

Original: Having introduced all the building blocks, Q -GCN stacks multiple pseudo-Riemannian GCN layersand the final embeddings at the last layer can then be used to perform downstream tasks.
Revised: Having introduced all the building blocks, Q -GCN stacks multiple pseudo-Riemannian GCN layersand the final embeddings at the last layer can then be used to perform downstream tasks.

Original: For graph reconstruction, the objective is to map all nodes into a low-dimensional space such that the connected nodes are closer than unconnected nodes.
Revised: For graph reconstruction, the objective is to map all nodes into a low-dimensional space such that the connected nodes are closer than unconnected nodes.

Original: Following [17, 21], we minimize the loss function
Revised: Following [17, 21], we minimize the loss function

Original: L (Θ) =
Revised: L (Θ) =

Original: (cid:80) ( u,v ) ∈D log
Revised: (cid:80) ( u,v ) ∈D log

Original: e − d ( u , v ) (cid:80) v ′∈E ( u ) e − d ( u , v ′ ) under the set of connected relations D in the graph, where
Revised: e − d ( u , v ) (cid:80) v ′∈E ( u ) e − d ( u , v ′ ) under the set of connected relations D in the graph, where

Original: E ( u ) = { v | ( u, v ) / ∈ D, v ̸ = u } is the set of negative examples for node u , d ( · ) is the distancefunction defined in Eq.
Revised: E ( u ) = { v | ( u, v ) / ∈ D, v ̸ = u } is the set of negative examples for node u , d ( · ) is the distancefunction defined in Eq.

Original: For node classification, we map the output of the last layer of Q -GCNto the tangent space, and then perform Euclidean multinomial logistic regression.
Revised: For node classification, we map the output of the last layer of Q -GCNto the tangent space, and then perform Euclidean multinomial logistic regression.

Original: For link prediction, we utilize the Fermi-Dirac decoder [8] to compute probability scores for edges, formally givenby P ( e uv ∈ E| Θ)
Revised: For link prediction, we utilize the Fermi-Dirac decoder [8] to compute probability scores for edges, formally givenby P ( e uv ∈ E| Θ)

Original: = 1 e ( d ( u , v ) − r ) /t
Revised: = 1 e ( d ( u , v ) − r ) /t

Original: +1 , where r and t are hyperparameters, d ( u , v ) is the distancefunction in the embedding space.
Revised: +1 , where r and t are hyperparameters, d ( u , v ) is the distancefunction in the embedding space.

Original: We then train Q -GCN by minimizing the cross-entropy loss usingnegative sampling.
Revised: We then train Q -GCN by minimizing the cross-entropy loss usingnegative sampling.

Original: Optimization.
Revised: Optimization.

Original: Although the model is built in Q s,tβ , the trainable parameters are all defined in Euclidean space through the diffeomorphic mappings.
Revised: Although the model is built in Q s,tβ , the trainable parameters are all defined in Euclidean space through the diffeomorphic mappings.

Original: Following the standard tangential optimizationstrategy [28], the parameters can be optimized via Euclidean optimization by applying layer-wisediffeomorphic expmap and logmap .
Revised: Following the standard tangential optimizationstrategy [28], the parameters can be optimized via Euclidean optimization by applying layer-wisediffeomorphic expmap and logmap .

Original: One optional strategy is to use dedicated optimization like [21],which we left for our future work.
Revised: One optional strategy is to use dedicated optimization like [21],which we left for our future work.

Original:  Experiments
Revised:  Experiments

Original: We evaluate the effectiveness of Q -GCN on graph reconstruction, node classification and link prediction.
Revised: We evaluate the effectiveness of Q -GCN on graph reconstruction, node classification and link prediction.

Original: Firstly, we study the geometric properties of the used datasets including the graph sectionalcurvature [17] and the δ -hyperbolicity [7].
Revised: Firstly, we study the geometric properties of the used datasets including the graph sectionalcurvature [17] and the δ -hyperbolicity [7].

Original: Fig.
Revised: Fig.

Original: 3 in the Appendix shows the histograms of sectionalcurvature and the mean sectional curvature for all datasets.
Revised: 3 in the Appendix shows the histograms of sectionalcurvature and the mean sectional curvature for all datasets.

Original: It can be seen that all datasets haveboth positive and negative sectional curvatures, showcasing that all graphs contain mixed graphtopologies.
Revised: It can be seen that all datasets haveboth positive and negative sectional curvatures, showcasing that all graphs contain mixed graphtopologies.

Original: To further analyze the degree of the hierarchy, we apply δ -hyperbolicity to identifythe tree-likeness, as shown in Table 9 in the Appendix.
Revised: To further analyze the degree of the hierarchy, we apply δ -hyperbolicity to identifythe tree-likeness, as shown in Table 9 in the Appendix.

Original: We conjecture that the datasets with positive graph sectional curvature or larger δ -hyperbolicity should be suitable for pseudo-hyperboloidwith a smaller time dimension, while datasets with negative graph sectional curvature or smallerδ -hyperbolicity should be aligned well with pseudo-hyperboloid with a larger time dimension.
Revised: We conjecture that the datasets with positive graph sectional curvature or larger δ -hyperbolicity should be suitable for pseudo-hyperboloidwith a smaller time dimension, while datasets with negative graph sectional curvature or smallerδ -hyperbolicity should be aligned well with pseudo-hyperboloid with a larger time dimension.

Original: 4.1 Graph reconstruction
Revised: 4.1 Graph reconstruction

Original: Datasets and baselines.
Revised: Datasets and baselines.

Original: We benchmark graph reconstruction on four real-world graphs including1)
Revised: We benchmark graph reconstruction on four real-world graphs including1)

Original: Web-Edu [29]: a web network consisting of the .edu domain; 2) Power [30]: a power grid distribution network with backbone structure; 3) Bio-Worm [31]: a worms gene network; 4) Facebook[32]: a dense social network from Facebook.
Revised: Web-Edu [29]: a web network consisting of the .edu domain; 2) Power [30]: a power grid distribution network with backbone structure; 3) Bio-Worm [31]: a worms gene network; 4) Facebook[32]: a dense social network from Facebook.

Original: We compare our method with Euclidean GCN [2], hyperbolic GCN (HGCN) [28], spherical GCN, and product manifold GCNs ( κ -GCN) [33] with threesignatures (i.e. H 5 × H 5 , H 5 × S 5 and S 5 × S 5 ).
Revised: We compare our method with Euclidean GCN [2], hyperbolic GCN (HGCN) [28], spherical GCN, and product manifold GCNs ( κ -GCN) [33] with threesignatures (i.e. H 5 × H 5 , H 5 × S 5 and S 5 × S 5 ).

Original: Besides, five variants of our model are implementedwith different time dimension in [1 , 3 , 5 , 7 , 10] for comparison.
Revised: Besides, five variants of our model are implementedwith different time dimension in [1 , 3 , 5 , 7 , 10] for comparison.

Original: [Table]
Revised: [Table]

Original: .94 κ -GCN ( S 5 × S 5 ) 86.
Revised: κ -GCN ( H 5 × H 5 ) 89. 84.94 κ -GCN ( S 5 × S 5 ) 86.

Original: 84.56 κ -GCN ( H 5 × S 5 ) 87.
Revised: 84.56 κ -GCN ( H 5 × S 5 ) 87.

Original: 87.73 Q -GCN ( Q 9 , 1 ) 87.
Revised: 87.73 Q -GCN ( Q 9 , 1 ) 87.

Original: 81.60 Q -GCN ( Q 7 , 3 ) 99.
Revised: 81.60 Q -GCN ( Q 7 , 3 ) 99.

Original: 87.74 Q -GCN ( Q 5 , 5 ) 98.
Revised: 87.74 Q -GCN ( Q 5 , 5 ) 98.

Original: 87.03 Q -GCN ( Q 3 , 7 ) 97.
Revised: 87.03 Q -GCN ( Q 3 , 7 ) 97.

Original: 91.75 Q -GCN ( Q 0 , 10 ) 82.
Revised: 91.75 Q -GCN ( Q 0 , 10 ) 82.

Original: Experimental settings.
Revised: Experimental settings.

Original: Following[17, 21, 33], we use one-hot embeddings as initial node features.
Revised: Following[17, 21, 33], we use one-hot embeddings as initial node features.

Original: To avoidthe time dimensions being 0 , we uniformly perturb each dimension witha small random value in the interval[ − ϵ, ϵ ] , where ϵ = 0 .
Revised: To avoidthe time dimensions being 0 , we uniformly perturb each dimension witha small random value in the interval[ − ϵ, ϵ ] , where ϵ = 0 .

Original: 02 in practice.
Revised: 02 in practice.

Original: In addition, the same 10 -dimensionalembedding and 2 hidden layers areused for all baselines to ensure a faircomparison.
Revised: In addition, the same 10 -dimensionalembedding and 2 hidden layers areused for all baselines to ensure a faircomparison.

Original: The learning rate is set to0 .
Revised: The learning rate is set to0 .

Original: 01 , the learning rate of curvature isset to 0 . 0001 .
Revised: 01 , the learning rate of curvature isset to 0 . 0001 .

Original: Q -GCN is implementedwith the Adam optimizer.
Revised: Q -GCN is implementedwith the Adam optimizer.

Original: We repeatthe experiments 10 times via differentrandom seeds influencing weightinitialization and data batching.
Revised: We repeatthe experiments 10 times via differentrandom seeds influencing weightinitialization and data batching.

Original: [Table]
Revised: [Table]

Original: Results.
Revised: Results.

Original: Table 1 shows the mean average precision (mAP) [17] results of graph reconstruction onfour datasets.
Revised: Table 1 shows the mean average precision (mAP) [17] results of graph reconstruction onfour datasets.

Original: It shows that Q -GCN achieves the best performance across all benchmarks comparedwith both Riemannian space and product manifolds.
Revised: It shows that Q -GCN achieves the best performance across all benchmarks comparedwith both Riemannian space and product manifolds.

Original: We observe that by setting proper signatures,the product spaces perform better than a single geometry.
Revised: We observe that by setting proper signatures,the product spaces perform better than a single geometry.

Original: It is consistent with our statement thatthe expression power of a single view geometry is limited.
Revised: It is consistent with our statement thatthe expression power of a single view geometry is limited.

Original: Specifically, all the top three resultsare achieved by Q -GCN, with one exception on Power where H 5 × S 5 achieved the third-bestperformance.
Revised: Specifically, all the top three resultsare achieved by Q -GCN, with one exception on Power where H 5 × S 5 achieved the third-bestperformance.

Original: More precisely, for datasets that have smaller graph sectional curvature like Web-Edu,
Revised: More precisely, for datasets that have smaller graph sectional curvature like Web-Edu,

Original: Power and Bio-Worm, Q 7 , 3 perform the best, while Q 3 , 7 perform the best on Facebook with positivesectional curvature.
Revised: Power and Bio-Worm, Q 7 , 3 perform the best, while Q 3 , 7 perform the best on Facebook with positivesectional curvature.

Original: We conjecture that the number of time dimensions controls the geometry ofthe pseudo-hyperboloid.
Revised: We conjecture that the number of time dimensions controls the geometry ofthe pseudo-hyperboloid.

Original: We find that the graphs with more hierarchical structures are inclined tobe embedded with fewer time dimensions.
Revised: We find that the graphs with more hierarchical structures are inclined tobe embedded with fewer time dimensions.

Original: By analyzing the sectional curvature in Fig.
Revised: By analyzing the sectional curvature in Fig.

Original: 3, we findthat this makes sense as the mean sectional curvature of Power, Bio-Wormnet and Web-Edu arenegative while it is negative for Facebook.
Revised: 3, we findthat this makes sense as the mean sectional curvature of Power, Bio-Wormnet and Web-Edu arenegative while it is negative for Facebook.

Original: Such results give us an intuition to determine the besttime dimension based on the geometric properties of graphs.
Revised: Such results give us an intuition to determine the besttime dimension based on the geometric properties of graphs.

Original: 4.2 Node classification and link prediction
Revised: 4.2 Node classification and link prediction

Original: Datasets and baselines.
Revised: Datasets and baselines.

Original: We consider four benchmark datasets: Airport, Pubmed, Citeseer and
Revised: We consider four benchmark datasets: Airport, Pubmed, Citeseer and

Original: Cora, where Airport is airline networks, Pubmed, Citeseer and Cora are three citation networks.
Revised: Cora, where Airport is airline networks, Pubmed, Citeseer and Cora are three citation networks.

Original: Weobserve that the graph sectional curvatures of the four datasets are consistently negative withoutsignificant differences in Fig.
Revised: Weobserve that the graph sectional curvatures of the four datasets are consistently negative withoutsignificant differences in Fig.

Original: 3, hence we report the additional δ -hyperbolicity for comparison in
Revised: 3, hence we report the additional δ -hyperbolicity for comparison in

Original: Table 2.
Revised: Table 2.

Original: GCN [2], GAT [3], SAGE [34] and SGC [4] are used as Euclidean GCN counterparts.
Revised: GCN [2], GAT [3], SAGE [34] and SGC [4] are used as Euclidean GCN counterparts.

Original: Fornon-Euclidean GCN baselines
Revised: Fornon-Euclidean GCN baselines

Original: , we compare HGCN [28] and κ -GCN [33] with its three variants asexplained before.
Revised: , we compare HGCN [28] and κ -GCN [33] with its three variants asexplained before.

Original: For Q -GCN, we empirically set the time dimension as [1 , 2 , 3 , 14 , 15 , 16] as sixvariants since these settings best reflect the geometric properties of hyperbolic and spherical space,respectively.
Revised: For Q -GCN, we empirically set the time dimension as [1 , 2 , 3 , 14 , 15 , 16] as sixvariants since these settings best reflect the geometric properties of hyperbolic and spherical space,respectively.

Original: Experimental settings.
Revised: Experimental settings.

Original: For node classification, we use the same dataset split as [35] for citationdatasets, where 20 nodes per class are used for training, and 500 nodes are used for validation and1000 nodes are used for testing.
Revised: For node classification, we use the same dataset split as [35] for citationdatasets, where 20 nodes per class are used for training, and 500 nodes are used for validation and1000 nodes are used for testing.

Original: For Airport, we split the dataset into 70 / 15 / 15 .
Revised: For Airport, we split the dataset into 70 / 15 / 15 .

Original: For link prediction,the edges are split into 85 / 5 / 10 percent for training, validation and testing for all datasets.
Revised: For link prediction,the edges are split into 85 / 5 / 10 percent for training, validation and testing for all datasets.

Original: To ensurea fair comparison, we set the same 16 -dimension hidden embedding, 0 .
Revised: To ensurea fair comparison, we set the same 16 -dimension hidden embedding, 0 .

Original: 01 initial learning rate and0 .
Revised: 01 initial learning rate and0 .

Original: 0001 learning rate for curvature.
Revised: 0001 learning rate for curvature.

Original: The optimal regularization with weight decay, dropout rate, thenumber of layers and activation functions are obtained by grid search for each method.
Revised: The optimal regularization with weight decay, dropout rate, thenumber of layers and activation functions are obtained by grid search for each method.

Original: We reportthe mean accuracy over 10 random seeds influencing weight initialization and batching sequence.
Revised: We reportthe mean accuracy over 10 random seeds influencing weight initialization and batching sequence.

Original: Results.
Revised: Results.

Original: Table 2 shows the averaged ROC AUC for link prediction, and F1 score for node classification.
Revised: Table 2 shows the averaged ROC AUC for link prediction, and F1 score for node classification.

Original: As we can see from the δ -hyperbolicity, Airport and Pubmed are more hierarchical than CiteSeer and Cora.
Revised: As we can see from the δ -hyperbolicity, Airport and Pubmed are more hierarchical than CiteSeer and Cora.

Original: For Airport and Pubmed with dominating hierarchical properties (lower δ ), Q -GCNswith fewer time dimensions achieve the results on par with hyperbolic space based methods suchas HGCN [28], κ -GCN ( H 16 ).
Revised: For Airport and Pubmed with dominating hierarchical properties (lower δ ), Q -GCNswith fewer time dimensions achieve the results on par with hyperbolic space based methods suchas HGCN [28], κ -GCN ( H 16 ).

Original: While for CiteSeer and Cora with less tree-like properties (higherδ ), Q -GCNs achieve the state-of-the-art results, showcasing the flexibility of our model to embedcomplex graphs with different curvatures.
Revised: While for CiteSeer and Cora with less tree-like properties (higherδ ), Q -GCNs achieve the state-of-the-art results, showcasing the flexibility of our model to embedcomplex graphs with different curvatures.

Original: More specifically, Q -GCN with more time dimensionsconsistently performs best on Cora.
Revised: More specifically, Q -GCN with more time dimensionsconsistently performs best on Cora.

Original: While for CiteSeer, albeit Q -GCN achieves the best results, thecorresponding best variants are not consistent on both tasks.
Revised: While for CiteSeer, albeit Q -GCN achieves the best results, thecorresponding best variants are not consistent on both tasks.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: 4.3 Parameter sensitivity and analysis
Revised: 4.3 Parameter sensitivity and analysis

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: Time dimension.
Revised: Time dimension.

Original: We study the influence of timedimension for graph reconstruction by setting varying number of time dimensions under the conditionof s + t = 10 .
Revised: We study the influence of timedimension for graph reconstruction by setting varying number of time dimensions under the conditionof s + t = 10 .

Original: Fig.
Revised: Fig.

Original: 2 shows that the time dimensiont acts as a knob for controlling the geometric properties of Q s,tβ .
Revised: 2 shows that the time dimensiont acts as a knob for controlling the geometric properties of Q s,tβ .

Original: The best performance are achievedby neither hyperboloid ( t = 1 ) nor sphere cases( t = 10 ), showcasing the advantages of Q s,tβ on representing graphs of mixed topologies.
Revised: The best performance are achievedby neither hyperboloid ( t = 1 ) nor sphere cases( t = 10 ), showcasing the advantages of Q s,tβ on representing graphs of mixed topologies.

Original: It shows thaton Web-Edu, Power and Bio-Worm with smallermean sectional curvature, after the optimal value isreached at a lower t , the performance decrease ast increases.
Revised: It shows thaton Web-Edu, Power and Bio-Worm with smallermean sectional curvature, after the optimal value isreached at a lower t , the performance decrease ast increases.

Original: While on Facebook with larger (positive) mean sectional curvature, as t rises, the effectgradually increases until it reaches a peak at a higher t .
Revised: While on Facebook with larger (positive) mean sectional curvature, as t rises, the effectgradually increases until it reaches a peak at a higher t .

Original: It is consistent with our hypothesis thatgraphs with more hierarchical structure are inclined to be embedded in Q s,tβ with smaller t , whilecyclical data is aligned well with larger t .
Revised: It is consistent with our hypothesis thatgraphs with more hierarchical structure are inclined to be embedded in Q s,tβ with smaller t , whilecyclical data is aligned well with larger t .

Original: The results give us an intuition to determine the best timedimension based on the geometric properties of graphs.
Revised: The results give us an intuition to determine the best timedimension based on the geometric properties of graphs.

Original: Q -NN VS Q -GCN.
Revised: Q -NN VS Q -GCN.

Original: We also introduce Q -NN, a generalization of MLP into pseudo-Riemannianmanifold, defined as multiple layers of f ( x ) = σ ⊗ ( W ⊗ β x ⊕ β b ) , where σ ⊗ is the tangentialactivation.
Revised: We also introduce Q -NN, a generalization of MLP into pseudo-Riemannianmanifold, defined as multiple layers of f ( x ) = σ ⊗ ( W ⊗ β x ⊕ β b ) , where σ ⊗ is the tangentialactivation.

Original: Table 3 shows that Q -NN with appropriate time dimension outperforms MLP and HNNon node classification, showcasing the expression power of pseudo-hyperboloid.
Revised: Table 3 shows that Q -NN with appropriate time dimension outperforms MLP and HNNon node classification, showcasing the expression power of pseudo-hyperboloid.

Original: Furthermore, compared with the results of Q -GCN in Table 2, Q -GCN performs better than Q -NN, suggesting thatthe benefits of the neighborhood aggregation equipped with the proposed GCN operations.
Revised: Furthermore, compared with the results of Q -GCN in Table 2, Q -GCN performs better than Q -NN, suggesting thatthe benefits of the neighborhood aggregation equipped with the proposed GCN operations.

Original: Computation efficiency.
Revised: Computation efficiency.

Original: We compare the running time of Q -GCN, GCN and Prod-GCN per epoch.
Revised: We compare the running time of Q -GCN, GCN and Prod-GCN per epoch.

Original: Table 4 shows that Q -GCN achieves higher efficiency than Prod-GCN ( H 5 × S 5 ).
Revised: Table 4 shows that Q -GCN achieves higher efficiency than Prod-GCN ( H 5 × S 5 ).

Original: This is mainlyowing to that the component R in our diffeomorphic manifold ( S × R ) runs faster than non-Euclideancomponents in H 5 × S 5 .
Revised: This is mainlyowing to that the component R in our diffeomorphic manifold ( S × R ) runs faster than non-Euclideancomponents in H 5 × S 5 .

Original: The additional running time mainly comes from the mapping operations andthe projection from the time dimensions to S .
Revised: The additional running time mainly comes from the mapping operations andthe projection from the time dimensions to S .

Original: The running time grows when increasing the numberof time dimensions.
Revised: The running time grows when increasing the numberof time dimensions.

Original: Overall, albeit slower than Euclidean GCN, the running time of all variants of
Revised: Overall, albeit slower than Euclidean GCN, the running time of all variants of

Original: Q -GCN is smaller than the twice of time in Euclidean GCN, which is within the acceptable limits.
Revised: Q -GCN is smaller than the twice of time in Euclidean GCN, which is within the acceptable limits.

Original:  Conclusion
Revised:  Conclusion

Original: In this paper, we generalize GCNs to pseudo-Riemannian manifolds of constant nonzero curvaturewith elegant theories of diffeomorphic geometry tools.
Revised: In this paper, we generalize GCNs to pseudo-Riemannian manifolds of constant nonzero curvaturewith elegant theories of diffeomorphic geometry tools.

Original: The proposed Q -GCN have the flexibility tofit complex graphs with mixed curvatures and have shown promising results on graph reconstruction,node classification and link prediction.
Revised: The proposed Q -GCN have the flexibility tofit complex graphs with mixed curvatures and have shown promising results on graph reconstruction,node classification and link prediction.

Original: One limitation might be the choice of time dimension, weprovide some insights to decide the best time dimension but this could still be improved, whichwe left for our future work.
Revised: One limitation might be the choice of time dimension, weprovide some insights to decide the best time dimension but this could still be improved, whichwe left for our future work.

Original: The developed geodesic tools are application-agnostic and could beextended to more deep learning methods.
Revised: The developed geodesic tools are application-agnostic and could beextended to more deep learning methods.

Original: We foresee our work would shed light on the direction ofnon-Euclidean geometric deep learning.
Revised: We foresee our work would shed light on the direction ofnon-Euclidean geometric deep learning.

Original: Checklist
Revised: Checklist

Original: For all authors...
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
Revised: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s

Original: contributions and scope?
Revised: contributions and scope?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes]
Revised: [Yes]

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A]
Revised: [N/A]

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to

Original: them?
Revised: them?

Original: [Yes]
Revised: [Yes]

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] All proofs are pro-
Revised: [Yes] All proofs are pro-

Original: vided in the Appendix.3.
Revised: vided in the Appendix.

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi-
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main experi-

Original: [Table]

Original: mental results (either in the supplemental material or as a URL)?
Revised: mental results (either in the supplemental material or as a URL)?

Original: [Yes] All code, data
Revised: [Yes] All code, data

Original: [Table]

Original: and instructions to reproduce the results are submitted in https://anonymous.4open.
Revised: and instructions to reproduce the results are submitted in https://anonymous.4open.

Original: [Table]

Original: science/r/PseudoRiemannianGCN.
Revised: science/r/PseudoRiemannianGCN.

Original: [Table] (b) Did you specify all the training details (e.g., data splits, hyperparameters,
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters,

Original: how they
Revised: how they

Original: [Table]

Original: were chosen)?
Revised: were chosen)?

Original: [Yes] All training details, hyperparameters are described in the Ap-[Table]
Revised: [Yes] All training details, hyperparameters are described in the Ap-

Original: pendix.
Revised: pendix.

Original: [Table] (c) Did you report error bars (e.g., with respect to the random seed after running experi-[Table]
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experi-

Original: ments multiple times)?
Revised: ments multiple times)?

Original: [Yes] We run all experiments 10 times with random seeds and
Revised: [Yes] We run all experiments 10 times with random seeds and

Original: [Table] report the averaged results with standard deviations (for graph reconstruction, we omit[Table]
Revised: report the averaged results with standard deviations (for graph reconstruction, we omit

Original: them
Revised: them

Original: because the values are very small).
Revised: because the values are very small).

Original: [Table]

Original: (d) Did you include the total amount of compute and the type of resources used (e.g.,
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g.,

Original: [Table]

Original: type of GPUs, internal cluster, or cloud provider)?
Revised: type of GPUs, internal cluster, or cloud provider)?

Original: [Yes] The computing resources are
Revised: [Yes] The computing resources are

Original: [Table]

Original: described in the Appendix.
Revised: described in the Appendix.

Original: [Table]

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...

Original: [Table]

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes]
Revised: [Yes]

Original: [Table]

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes]
Revised: [Yes]

Original: [Table]

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [N/A]
Revised: [N/A]

Original: [Table]

Original: (d) Did you discuss whether and how consent was obtained from people whose data
Revised: (d) Did you discuss whether and how consent was obtained from people whose data

Original: [Table]

Original: you’re using/curating?
Revised: you’re using/curating?

Original: [N/A]
Revised: [N/A]

Original: [Table]

Original: (e) Did you discuss whether the data you are using/curating contains personally identifi-
Revised: (e) Did you discuss whether the data you are using/curating contains personally identifi-

Original: [Table]

Original: able information or offensive content?
Revised: able information or offensive content?

Original: [N/A]
Revised: [N/A]

Original: [Table]

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: [Table] (a) Did you include the full text of instructions given to participants and screenshots, if[Table]
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?
Revised: applicable?

Original: [N/A]
Revised: [N/A]

Original: [Table] Did [Table] describe any potential 
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: [Table]
Revised: [Table]

Original: , with

Original: [Table]

Original: to

Original: [Table]

Original: [Table]

Original: IRB) approvals, if applicable?
Revised: (IRB) approvals, if applicable?

Original: [N/A
Revised: [N/A

Original: [Table]

Original: [Table]
Revised: [Table]

Original: ) Did you include the estimated hourly wage paid to participants and the total amount
Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A
Revised: [N/A]

Original: [Table]

Original: Appendices
Revised: Appendices

Original: Supplemental Material of "Pseudo-Riemannian Graph Convolution
Revised: Supplemental Material of "Pseudo-Riemannian Graph Convolution

Original: Networks"
Revised: Networks"

Original: Broader Impact
Revised: Broader Impact

Original: Going beyond Riemannian manifolds, we extend GCNs to the pseudo-Riemannian manifolds, whichis a smooth manifold furnished with indefinite metric.
Revised: Going beyond Riemannian manifolds, we extend GCNs to the pseudo-Riemannian manifolds, whichis a smooth manifold furnished with indefinite metric.

Original: Our contribution is both theoretical and practical.
Revised: Our contribution is both theoretical and practical.

Original: We introduce necessary geodesic tools in geodesically disconnected pseudo-Riemannian manifolds to help machine learning researchers to define vector operations like matrix multiplicationand vector addition.
Revised: We introduce necessary geodesic tools in geodesically disconnected pseudo-Riemannian manifolds to help machine learning researchers to define vector operations like matrix multiplicationand vector addition.

Original: These operations are application agnostic and could be applied in the context of machine learning algorithms.
Revised: These operations are application agnostic and could be applied in the context of machine learning algorithms.

Original: Q -GCN is a flexible framework and is especially powerful onmodeling real-world graphs such as social networks and molecular networks which usually exhibitheterogeneous topological structures.
Revised: Q -GCN is a flexible framework and is especially powerful onmodeling real-world graphs such as social networks and molecular networks which usually exhibitheterogeneous topological structures.

Original: We foresee our method can be applied to more practical settings such as recommender systems and drug discovery, and also stimulate more applications ofpseudo-Riemannian geometry in machine learning and related communities.
Revised: We foresee our method can be applied to more practical settings such as recommender systems and drug discovery, and also stimulate more applications ofpseudo-Riemannian geometry in machine learning and related communities.

Original: However, we emphasize that current models might suffer from the same problems as other deepmodels such as limited interpretability, and thus, are not capable of replacing human expertise.
Revised: However, we emphasize that current models might suffer from the same problems as other deepmodels such as limited interpretability, and thus, are not capable of replacing human expertise.

Original: Weadvocate researchers to focus more on the interpretability of non-Euclidean graph embeddings.
Revised: Weadvocate researchers to focus more on the interpretability of non-Euclidean graph embeddings.

Original: We structurize our supplementary materials as follows:
Revised: We structurize our supplementary materials as follows:

Original: • Sec.
Revised: • Sec.

Original: A summarizes the frequently used notations and the conventions.
Revised: A summarizes the frequently used notations and the conventions.

Original: • Sec.
Revised: • Sec.

Original: B reviews some related works on Riemannian embeddings and the application of
Revised: B reviews some related works on Riemannian embeddings and the application of

Original: pseudo-Riemannian geometry in machine learning.
Revised: pseudo-Riemannian geometry in machine learning.

Original: • Sec.
Revised: • Sec.

Original: C reviews pseudo-Riemannian geometry and provides some standard geodesic tools
Revised: C reviews pseudo-Riemannian geometry and provides some standard geodesic tools

Original: in pseudo-hyperboloid.
Revised: in pseudo-hyperboloid.

Original: • Sec.
Revised: • Sec.

Original: D provides necessary proofs of the theorems mentioned in the paper.
Revised: D provides necessary proofs of the theorems mentioned in the paper.

Original: • Sec.
Revised: • Sec.

Original: E provides more details about the experimental setup and analysis.
Revised: E provides more details about the experimental setup and analysis.

Original: A Notation
Revised: A Notation

Original: In this paper, we denote the points on manifolds by boldface Roman characters.
Revised: In this paper, we denote the points on manifolds by boldface Roman characters.

Original: The tangent vectorson tangent spaces are denoted by boldface Greek fonts.
Revised: The tangent vectorson tangent spaces are denoted by boldface Greek fonts.

Original: The notations are summarized in Table 5.
Revised: The notations are summarized in Table 5.

Original: B Related Works
Revised: B Related Works

Original: Riemannian embedding.
Revised: Riemannian embedding.

Original: Non-Euclidean Riemannian spaces have recently gained extensive attention in learning representation for non-euclidean data.
Revised: Non-Euclidean Riemannian spaces have recently gained extensive attention in learning representation for non-euclidean data.

Original: [36] was the first work to learn hierarchicalembeddings in hyperbolic space for link prediction.
Revised: [36] was the first work to learn hierarchicalembeddings in hyperbolic space for link prediction.

Original: Following this work, [37] applied hyperbolicembedding in word embedding.
Revised: Following this work, [37] applied hyperbolicembedding in word embedding.

Original: [38] proposed a translation model in hyperbolic space for multirelational graph embedding, and [39] proposed a hyperbolic embedding for answering complex logical queries.
Revised: [38] proposed a translation model in hyperbolic space for multirelational graph embedding, and [39] proposed a hyperbolic embedding for answering complex logical queries.

Original: On the other hand, spherical space offers benefits for embedding spherical or cyclicaldata [11, 40, 41].
Revised: On the other hand, spherical space offers benefits for embedding spherical or cyclicaldata [11, 40, 41].

Original: For mixed curvature space, product manifolds [17, 18] on Euclidean, hyperbolicand spherical space has been proposed to embed graph and knowledge graphs [42] with mixedstructures.
Revised: For mixed curvature space, product manifolds [17, 18] on Euclidean, hyperbolicand spherical space has been proposed to embed graph and knowledge graphs [42] with mixedstructures.

Original: HNN [12, 43] extended neural networks into the hyperbolic space and defined somefundamental neural operations.
Revised: HNN [12, 43] extended neural networks into the hyperbolic space and defined somefundamental neural operations.

Original: Riemannian GCN.
Revised: Riemannian GCN.

Original: Going beyond GCNs in Euclidean space [1, 2], HGNN [13] and HGCN [28]first extended GCNs into hyperbolic space, achieving state-of-the-art performance on learning graphembedding for scale-free networks.
Revised: Going beyond GCNs in Euclidean space [1, 2], HGNN [13] and HGCN [28]first extended GCNs into hyperbolic space, achieving state-of-the-art performance on learning graphembedding for scale-free networks.

Original: The main solution is to move the graph convolutional operations (e.g. feature transformation , aggregation ) into the tangent space of manifolds and perform
Revised: The main solution is to move the graph convolutional operations (e.g. feature transformation , aggregation ) into the tangent space of manifolds and perform

Original: Euclidean operations on the space.
Revised: Euclidean operations on the space.

Original: To model graphs of mixed topologies, κ -GCN [33] unifies curvatures in a κ -stereographic model and extends GCNs into the products of Riemannian projection
Revised: To model graphs of mixed topologies, κ -GCN [33] unifies curvatures in a κ -stereographic model and extends GCNs into the products of Riemannian projection

Original: [Table]
Revised: [Table]

Original: geodesic
Revised: geodesic

Original: [Table]
Revised: [Table]

Original: mapping
Revised: mapping

Original: [Table]
Revised: [Table]

Original: real value
Revised: real value

Original: [Table]
Revised: [Table]

Original: manifolds.
Revised: manifolds.

Original: GIL [14] proposed graph geometric interaction learning that models interaction betweenhyperbolic and Euclidean space.
Revised: GIL [14] proposed graph geometric interaction learning that models interaction betweenhyperbolic and Euclidean space.

Original: Different from these works, Q -GCN learns graph embeddings in asingle pseudo-Riemannian space with indefinite metrics while capturing mixed graph structures.
Revised: Different from these works, Q -GCN learns graph embeddings in asingle pseudo-Riemannian space with indefinite metrics while capturing mixed graph structures.

Original: Pseudo-Riemannian machine learning.
Revised: Pseudo-Riemannian machine learning.

Original: Few works [5, 27, 21, 22, 20] have explored the application of pseudo-Riemannian geometry in machine learning.
Revised: Few works [5, 27, 21, 22, 20] have explored the application of pseudo-Riemannian geometry in machine learning.

Original: [5] first applied pseudo-Riemanniangeometry (or pseudo-Euclidean space ) to embed non-metric data that preserves local information.[20] exploited Lorentzian spacetime on embedding directed acyclic graphs, [27] developed a manifold learning framework on the pseudo-Riemannian manifold and its submanifolds.
Revised: [5] first applied pseudo-Riemanniangeometry (or pseudo-Euclidean space ) to embed non-metric data that preserves local information.[20] exploited Lorentzian spacetime on embedding directed acyclic graphs, [27] developed a manifold learning framework on the pseudo-Riemannian manifold and its submanifolds.

Original: More recently,[21] proposed learning graph embeddings on pseudo-hyperboloid and provided some geodesic tools.[22] further extended it into directed graph embedding with novel tools.
Revised: More recently,[21] proposed learning graph embeddings on pseudo-hyperboloid and provided some geodesic tools.[22] further extended it into directed graph embedding with novel tools.

Original: [44] first studied the pseudoRiemannian geometry in the setting of deep learning.
Revised: [44] first studied the pseudoRiemannian geometry in the setting of deep learning.

Original: However, their embedding is learned in a quotient manifold.
Revised: However, their embedding is learned in a quotient manifold.

Original: Different from [44], our embedding is directly learned from the pseudo-hyperboloid.
Revised: Different from [44], our embedding is directly learned from the pseudo-hyperboloid.

Original: C Review of Pseudo-Riemannian GeometryC.1 Pseudo-Riemannian manifold
Revised: C Review of Pseudo-Riemannian GeometryC.1 Pseudo-Riemannian manifold

Original: The pseudo-Riemannian manifold M is equipped with a pseudo-Riemannian metric g x : T x M ×
Revised: The pseudo-Riemannian manifold M is equipped with a pseudo-Riemannian metric g x : T x M ×

Original: T x M at point x .
Revised: T x M at point x .

Original: The metric g x is a nondegenerate symmetric bilinear form, which means that if fora given ξ ∈ T x M , for any ζ ∈ T
Revised: The metric g x is a nondegenerate symmetric bilinear form, which means that if fora given ξ ∈ T x M , for any ζ ∈ T

Original: x M we have g x ( ξ , ζ ) = 0 , then ξ = 0 .
Revised: x M we have g x ( ξ , ζ ) = 0 , then ξ = 0 .

Original: The M manifold has theconstant sectional curvature of 1 /β and constant mean curvature κ = | β | − 1 / 2 .
Revised: The M manifold has theconstant sectional curvature of 1 /β and constant mean curvature κ = | β | − 1 / 2 .

Original: We recommend thereader to refer [23, 45] for details.
Revised: We recommend thereader to refer [23, 45] for details.

Original: C.2 Geodesic tools of pseudo-hyperboloid
Revised: C.2 Geodesic tools of pseudo-hyperboloid

Original: Although pseudo-hyperboloid is a geodescially disconnected manifold, previous works [21, 27] havedefined some standard geodesic tools that do not consider the broken cases which we recall below.
Revised: Although pseudo-hyperboloid is a geodescially disconnected manifold, previous works [21, 27] havedefined some standard geodesic tools that do not consider the broken cases which we recall below.

Original: Geodesic.
Revised: Geodesic.

Original: The geodesics of pseudo-hyperboloid are a combination of the hyperbolic, flat and spherical cases, depending on the sign of ⟨ ξ , ξ ⟩ t .
Revised: The geodesics of pseudo-hyperboloid are a combination of the hyperbolic, flat and spherical cases, depending on the sign of ⟨ ξ , ξ ⟩ t .

Original: Formally, the geodesic γ x → ξ ( τ ) of Q s,tβ with β < 0 is defined as Eq.
Revised: Formally, the geodesic γ x → ξ ( τ ) of Q s,tβ with β < 0 is defined as Eq.

Original: [Equation]
Revised: [Equation]

Original: Exponential and logarithmic maps.
Revised: Exponential and logarithmic maps.

Original: The exponential and logarithmic maps in the pseudohyperboloid manifold are defined as exp x : T x Q s,tβ → Q s,tβ and log x : Q s,tβ → T x Q s,tβ , respectively.
Revised: The exponential and logarithmic maps in the pseudo- hyperboloid manifold are defined as exp x : T x Q s,tβ → Q s,tβ and log x : Q s,tβ → T x Q s,tβ , respec- tively.

Original: We have the following closed form expressions of the exponential and the logarithmic maps[21], which allow us to perform operations on points on the pseudo-hyperboloid or pseudo-spheremanifolds by mapping them to tangent spaces and vice-versa, formally given by
Revised: We have the following closed form expressions of the exponential and the logarithmic maps [21], which allow us to perform operations on points on the pseudo hyperboloid or pseudo-spheremanifolds by mapping them to tangent spaces and vice-versa formally given by

Original: [Equation]
Revised: [Equation]

Original:  
Revised:  

Original: [Equation]
Revised: [Equation]

Original: It is worth mentioning that not all points on the pseudo-hyperboloid manifold are connected by ageodesic.
Revised: It is worth mentioning that not all points on the pseudo-hyperboloid manifold are connected by ageodesic.

Original: For these broken points, there does not exist a tangent vector ξ such that y = exp x ( ξ ) .
Revised: For these broken points, there does not exist a tangent vector ξ such that y = exp x ( ξ ) .

Original: Namely, the logarithmic map is not defined if ⟨ x , y ⟩ ≥ | β | .
Revised: Namely, the logarithmic map is not defined if ⟨ x , y ⟩ ≥ | β | .

Original: Distance.
Revised: Distance.

Original: By the means of the geodesic, the distance between x and y in pseudo-Riemannian manifolds Q s,tβ is defined as the arc length of geodesic γ ( τ ) , given by Eq.
Revised: By the means of the geodesic, the distance between x and y in pseudo-Riemannian manifolds Q s,tβ is defined as the arc length of geodesic γ ( τ ) , given by Eq.

Original: [Equation]
Revised: [Equation]

Original: Parallel transport.
Revised: Parallel transport.

Original: Given the geodesic γ ( τ ) on Q s,tβ passing through x ∈ Q s,tβ with the tangentdirection ξ ∈ T x Q s,tβ , the parallel transport of ζ ∈ T x Q s,tβ is defined as Eq.
Revised: Given the geodesic γ ( τ ) on Q s,tβ passing through x ∈ Q s,tβ with the tangentdirection ξ ∈ T x Q s,tβ , the parallel transport of ζ ∈ T x Q s,tβ is defined as Eq.

Original: Pβ
Revised: P

Original: [Equation]
Revised: [Equation]

Original: otherwise
Revised: otherwise

Original: [Equation]
Revised: [Equation]

Original: 12) where ξ = log x ( y ) .
Revised: 12) where ξ = log x ( y ) .

Original: D Proof of Theorems
Revised: D Proof of Theorems

Original: D.1 Proof of Theorem 1
Revised: D.1 Proof of Theorem 1

Original: Theorem 1 (Theorem 4.1 in [21]) .
Revised: Theorem 1 (Theorem 4.1 in [21]) .

Original: For any point x ∈ Q s,tβ , there exists a diffeomorphism ψ :
Revised: For any point x ∈ Q s,tβ , there exists a diffeomorphism ψ :

Original: Q s,tβ → S t 1 × R s that maps x into the product manifolds of an unit sphere and the Euclidean space,the mapping and its inverse are given by,
Revised: Q s,tβ → S t 1 × R s that maps x into the product manifolds of an unit sphere and the Euclidean space,the mapping and its inverse are given by,

Original: [Equation]
Revised: [Equation]

Original: where x = (cid:18) ts (cid:19) ∈ Q s,tβ with t ∈ R t +1 ∗ and s ∈ R s .
Revised: where x = (cid:18) ts (cid:19) ∈ Q s,tβ with t ∈ R t +1 ∗ and s ∈ R s .

Original: z = (cid:18) uv
Revised: z = (cid:18) uv

Original: (cid:19) ∈ S t 1 × R s with u ∈ S t 1 andv ∈ R s .
Revised: (cid:19) ∈ S t 1 × R s with u ∈ S t 1 andv ∈ R s .

Original: Please refer Appendix C.5 in [21] for proof of this theorem.
Revised: Please refer Appendix C.5 in [21] for proof of this theorem.

Original: D.2 Proof of Theorem 2
Revised: D.2 Proof of Theorem 2

Original: Theorem 2.
Revised: Theorem 2.

Original: For any point x ∈ Q s,tβ , there exists a diffeomorphism ψ : Q s,tβ → S t − β × R s that mapsx into the product manifolds of a sphere and the Euclidean space, the mapping and its inverse aregiven by,
Revised: For any point x ∈ Q s,tβ , there exists a diffeomorphism ψ : Q s,tβ → S t − β × R s that mapsx into the product manifolds of a sphere and the Euclidean space, the mapping and its inverse aregiven by,

Original: [Equation]
Revised: [Equation]

Original: where x = (cid:18) ts (cid:19) ∈ Q s,tβ with t ∈ R t ∗ and s ∈ R s .
Revised: where x = (cid:18) ts (cid:19) ∈ Q s,tβ with t ∈ R t ∗ and s ∈ R s .

Original: z = (cid:18) uv (cid:19) ∈ S t − β ×
Revised: z = (cid:18) uv (cid:19) ∈ S t − β ×

Original: R s with u ∈ S t − β andv ∈ R s .
Revised: R s with u ∈ S t − β andv ∈ R s .

Original: Proof.
Revised: Proof.

Original: It is easy to show that the ψ and ψ − 1 are smooth functions as they only involve with a linearmapping with a constant scaling vector.
Revised: It is easy to show that the ψ and ψ − 1 are smooth functions as they only involve with a linearmapping with a constant scaling vector.

Original: Hence, we only need to show that ψ (cid:0) ψ − 1 ( z ) (cid:1) =
Revised: Hence, we only need to show that ψ (cid:0) ψ − 1 ( z ) (cid:1) =

Original: z andψ − 1 ( ψ ( x )) =
Revised: z andψ − 1 ( ψ ( x )) =

Original: x .
Revised: x .

Original: Here, we consider space dimensions and time dimensions separately.
Revised: Here, we consider space dimensions and time dimensions separately.

Original: For space dimensions, the mapping of the space dimensions of x to the space dimensions of z is anidentity function (i.e. v = s , s = v ).
Revised: For space dimensions, the mapping of the space dimensions of x to the space dimensions of z is anidentity function (i.e. v = s , s = v ).

Original: Thus, we only need to show the invertibility of the mappingstaking time dimensions as inputs.
Revised: Thus, we only need to show the invertibility of the mappingstaking time dimensions as inputs.

Original: For time dimensions, we first show that:
Revised: For time dimensions, we first show that:

Original: [Equation]
Revised: [Equation]

Original: Note that the last equality can be inferred by the fact that x ∈ Q s,tβ and β < 0 .
Revised: Note that the last equality can be inferred by the fact that x ∈ Q s,tβ and β < 0 .

Original: We then show that:
Revised: We then show that:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Note that the last equality can be inferred by the fact that u ∈ S t − β and β < 0 .
Revised: Note that the last equality can be inferred by the fact that u ∈ S t − β and β < 0 .

Original: D.3 Proof of Theorem
Revised: D.3 Proof of Theorem

Original: Theorem 3.
Revised: Theorem 3.

Original: For any reference point x = (cid:18) ts (cid:19) ∈ Q s,tβ with space dimension s = 0 , the inducedtangent space of Q s,tβ is equal to the tangent space of its diffeomorphic manifold S t − β ×
Revised: For any reference point x = (cid:18) ts (cid:19) ∈ Q s,tβ with space dimension s = 0 , the inducedtangent space of Q s,tβ is equal to the tangent space of its diffeomorphic manifold S t − β ×

Original: R s , namely,
Revised: R s , namely,

Original: T ψ ( x ) ( S t − β × R s ) =
Revised: T ψ ( x ) ( S t − β × R s ) =

Original: T x Q s,tβ .
Revised: T x Q s,tβ .

Original: Proof.
Revised: Proof.

Original: For any point x = (cid:18) ts (cid:19) ∈ Q s,tβ with s = 0 , the corresponding point in the diffeomorphicmanifold is ψ ( x ) =
Revised: For any point x = (cid:18) ts (cid:19) ∈ Q s,tβ with s = 0 , the corresponding point in the diffeomorphicmanifold is ψ ( x ) =

Original: (cid:18) (cid:112) | β | t ∥ t ∥ (cid:19) .
Revised: (cid:18) (cid:112) | β | t ∥ t ∥ (cid:19) .

Original: Based on the definition of tangent space, for any tangent vectorξ ∈ T x Q s,tβ , ⟨ x , ξ ⟩ t = 0 implies − t ξ t + 0 ξ s = 0 , which means t ξ t = 0 .
Revised: Based on the definition of tangent space, for any tangent vectorξ ∈ T x Q s,tβ , ⟨ x , ξ ⟩ t = 0 implies − t ξ t + 0 ξ s = 0 , which means t ξ t = 0 .

Original: Thus, ⟨ ψ ( x ) , ξ ⟩ t = 0 .
Revised: Thus, ⟨ ψ ( x ) , ξ ⟩ t = 0 .

Original: Based on the definition of tangent space, ξ ∈ T ψ ( x ) ( S t − β × R s ) .
Revised: Based on the definition of tangent space, ξ ∈ T ψ ( x ) ( S t − β × R s ) .

Original: D.4 Proof of Theorem4
Revised: D.4 Proof of Theorem

Original: Theorem 4.
Revised: Theorem 4.

Original: For any point x ∈ Q s,t β , the union of the normal neighborhood of x and the normalneighborhood of its antipodal point − x cover the entire manifold.
Revised: For any point x ∈ Q s,t β , the union of the normal neighborhood of x and the normalneighborhood of its antipodal point − x cover the entire manifold.

Original: Namely, U x ∪ U − x = Q s,tβ .
Revised: Namely, U x ∪ U − x = Q s,tβ .

Original: Proof.
Revised: Proof.

Original: For any point x ∈ Q s,tβ and y / ∈ U x .
Revised: For any point x ∈ Q s,tβ and y / ∈ U x .

Original: Based on the definition of normal neighborhood,⟨ x , y ⟩ t ≥ | β | → ⟨− x , y ⟩ t ≤ | β | .
Revised: Based on the definition of normal neighborhood,⟨ x , y ⟩ t ≥ | β | → ⟨− x , y ⟩ t ≤ | β | .

Original: Thus, y ∈ U − x , and U x ∪ U − x
Revised: Thus, y ∈ U − x , and U x ∪ U − x

Original: = Q s,tβ .
Revised: = Q s,tβ .

Original: E Supplementary experimental details
Revised: E Supplementary experimental details

Original: In this section, we present the details of our experiments and implementations, as well as someadditional analyses.
Revised: In this section, we present the details of our experiments and implementations, as well as someadditional analyses.

Revised:  E.1 Graph Reconstruction

Revised: For graph reconstruction, one straightforward method is to minimize the graph distortion [33, 17]given by

Original: Implementation details.
Revised: [Equation] where d ( u , v ) is the distance function in the embedding space and d G ( u, v ) is the graph distance(the length of shortest path) between node u and v , | V | is the number of nodes in graph. The objectivefunction is to preserve all pairwise graph distances.

Revised: Motivated by the fact that most of graphs arepartially observable, we minimize an alternative loss function [21, 36] that preserves local graphdistance, given by,

Revised: [Equation]

Revised: [Equation]

Revised: where D is the connected relations in the graph, E ( u ) = { v | ( u, v ) / ∈ D ∪ u } is the set of negativeexamples for node u , d ( u , v ) is the distance function in the embedding space.

Revised: For evaluation, weapply the mean average precision (mAP) to evaluate the graph reconstruction task.

Revised: mAP is a localmetric that measures the average proportion of the nearest points of a node which are actually itsneighbors in the original graph.

Revised: The mAP is defined as Eq.

Revised: [Equation]

Revised: where f is the embedding function, | V | is the number of nodes in graph, deg( u ) is the degree ofnode u , N u is the one-hop neighborhoods in the graph, R u,v denotes whether two nodes u and v areconnected.

Revised: Implementation details.

Original: We implement our models by Pytorch and Geoopt tool 2 .
Revised: We implement our models by Pytorch and Geoopt tool 2 .

Original: Our models andbaselines are built upon the codes 3 .
Revised: Our models andbaselines are built upon the codes 3 .

Original: For baselines of product space and κ -GCN, we use the code 4 .
Revised: For baselines of product space and κ -GCN, we use the code 4 .

Original: Our models are trained on NVIDIA A100 with 4 GPU cards.
Revised: Our models are trained on NVIDIA A100 with 4 GPU cards.

Original: Similar to [28], we use the pre-traninedembeddings on link prediction as the initialization of node classification, to preserve graph structuresinformation.
Revised: Similar to [28], we use the pre-traninedembeddings on link prediction as the initialization of node classification, to preserve graph structuresinformation.

Original: All non-Euclidean methods do not use attention mechanism for a fair comparison.
Revised: All non-Euclidean methods do not use attention mechanism for a fair comparison.

Original: Fornode classification and link prediction
Revised: Fornode classification and link prediction

Original: , the optimal hyper-parameters for all methods are obtainedby the same grid search strategy.
Revised: , the optimal hyper-parameters for all methods are obtainedby the same grid search strategy.

Original: The ranges of grid search are summarized in Table 6.
Revised: The ranges of grid search are summarized in Table 6.

Original: [Table]
Revised: [Table]

Original: Numerical instability.
Revised: Numerical instability.

Original: In practice, we find that the numerical computation in pseudo-hyperboloidis not stable due to the limited machine precision and rounding.
Revised: In practice, we find that the numerical computation in pseudo-hyperboloidis not stable due to the limited machine precision and rounding.

Original: To ensure the learned embeddingsand tangent vectors to remain on the manifold and tangent spaces, we conduct projection operationsvia Eq.
Revised: To ensure the learned embeddings and tangent vectors to remain on the manifold and tangent spaces, we conduct projection operationsvia Eq.

Original: (17) and Eq.
Revised: (20) and Eq.

Original: (18) after each vector operations.
Revised: (21) after each vector operations.

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: (17) where ψ can be instantiated by Theorem 2.
Revised: where ψ can be instantiated by Theorem 2.

Original: Note that ψ ( · ) is an identity function when x is alreadylocalted in Q s,tβ .
Revised: Note that ψ ( · ) is an identity function when x is alreadylocalted in Q s,tβ .

Original: For input x / ∈ Q s,tβ , applying ψ ( · ) can project the x back to Q s,tβ , from which ψ ( · )is no longer an identity.
Revised: For input x / ∈ Q s,tβ , applying ψ ( · ) can project the x back to Q s,tβ , from which ψ ( · )is no longer an identity.

Original: [Equation]
Revised: [Equation]

Original: (18) where z is the tangent vector to be projected.
Revised: where z is the tangent vector to be projected.

Original: Trainable curvature.
Revised: Trainable curvature.

Original: Similar to the trainable curvature in the hyperbolic space [28], we give ananalogue of Theorem 4.1 in [28] as following corollary.
Revised: Similar to the trainable curvature in the hyperbolic space [28], we give ananalogue of Theorem 4.1 in [28] as following corollary.

Original: Corollary 1.
Revised: Corollary 1.

Original: For any two points x , y ∈ Q s,tβ , there exists a mapping ϕ : Q s,tβ → Q s,tβ ′ that maps x , yinto x ′ ,
Revised: For any two points x , y ∈ Q s,tβ , there exists a mapping ϕ : Q s,tβ → Q s,tβ ′ that maps x , yinto x ′ ,

Original: y ′ ∈
Revised: y ′ ∈

Original: Q s,tβ ′ , such that the pseudo-Riemannian inner product is scaled by a constant factor β ′ β .
Revised: Q s,tβ ′ , such that the pseudo-Riemannian inner product is scaled by a constant factor β ′ β .

Original: Namely ⟨ x , y ⟩ t = β ′ β ⟨ x ′ , y ′ ⟩ t .
Revised: Namely ⟨ x , y ⟩ t = β ′ β ⟨ x ′ , y ′ ⟩ t .

Original: The mapping is given by ϕ ( x )
Revised: The mapping is given by ϕ ( x )

Original: = (cid:113) β
Revised: = (cid:113) β

Original: ′ β x .
Revised: ′ β x .

Original: Corollary 1 shows that assuming infinite machine precision, pseudo-hyperboloids of varying curvatures should have the same expressive power.
Revised: Corollary 1 shows that assuming infinite machine precision, pseudo-hyperboloids of varying curvatures should have the same expressive power.

Original: However, due to the limited machine precision [28],we set the curvature 1 /β as a trainable parameter to capture the best scale of the embeddings.
Revised: However, due to the limited machine precision [28],we set the curvature 1 /β as a trainable parameter to capture the best scale of the embeddings.

Original: Skip connection.
Revised: Skip connection.

Original: In preliminary experiments of graph reconstruction, we found that the performance of GCN, HGCN, κ -GCN and Q -GCN could not compete with the method in [17] that directlyoptimizes the distance function.
Revised: In preliminary experiments of graph reconstruction, we found that the performance of GCN, HGCN, κ -GCN and Q -GCN could not compete with the method in [17] that directlyoptimizes the distance function.

Original: We hypothesized that this was probably caused by over-smoothingof graph convolution kernel.
Revised: We hypothesized that this was probably caused by over-smoothingof graph convolution kernel.

Original: Therefore, we decided to apply skip-connection strategy [46, 47, 48]to avoid oversmoothing.
Revised: Therefore, we decided to apply skip-connection strategy [46, 47, 48]to avoid oversmoothing.

Original: Specifically, we take the mean of the hidden output of each layer to the decoder.
Revised: Specifically, we take the mean of the hidden output of each layer to the decoder.

Original: In the paper, we report the graph reconstruction results of GCN, HGCN, κ -GCN and Q -GCNapplied with skip-connection.
Revised: In the paper, we report the graph reconstruction results of GCN, HGCN, κ -GCN and Q -GCNapplied with skip-connection.

Original: Dataset analysis.
Revised: Dataset analysis.

Original: Table 7 and Table 8 summarize the statistics of the datasets we used for ourexperiments.
Revised: Table 7 and Table 8 summarize the statistics of the datasets we used for ourexperiments.

Original: For each dataset, we report the number of nodes, the number of edges, the number [-of 1.0 Sectional curvature2 # N o d e s power mean 1. 1.0 Sectional curvature  # N o d e s bio-wormnet mean 1. 1.0 Sectional curvature
-]
Revised: For each dataset, we report the number of nodes, the number of edges, the number [-of 1.0 Sectional curvature2 # N o d e s power mean 1. 1.0 Sectional curvature  # N o d e s bio-wormnet mean 1. 1.0 Sectional curvature
-]ofclasses and the feature dimensions.

Original: # N o d e s web-edu mean 1.

Original: 1.0 Sectional curvature  # N o d

Original: e

Original: s facebook mean 1.

Original: 1.0 Sectional curvature 0.

Original: # N o d e s airport mean 1.

Original: 1.0 Sectional curvature 0.

Original: # N o d e s pubmed mean 1.

Original: 1.0 Sectional curvature # N o d e s citeseer mean 1.

Original: 1.0 Sectional curvature # N o d e s cora mean classes and the feature dimensions.

Original: Besides, we measure the graph sectional curvature to identifythe dominating geometry of each dataset.
Revised: Besides, we measure the graph sectional curvature to identifythe dominating geometry of each dataset.

Original: Fig.
Revised: Fig.

Original: 3 shows the histograms of sectional curvature andthe mean sectional curvature for all datasets.
Revised: 3 shows the histograms of sectional curvature andthe mean sectional curvature for all datasets.

Original: It can be seen that all datasets have both positiveand negative sectional curvatures, showcasing that all graphs contain mixed geometric structures.
Revised: It can be seen that all datasets have both positiveand negative sectional curvatures, showcasing that all graphs contain mixed geometric structures.

Original: However, the datasets possess a preference for the negative half sectional curvature except the caseof Facebook that has dominating positive sectional curvature, showcasing that most of real-worldgraphs exhibit more hierarchical structures than cyclic structures.
Revised: However, the datasets possess a preference for the negative half sectional curvature except the caseof Facebook that has dominating positive sectional curvature, showcasing that most of real-worldgraphs exhibit more hierarchical structures than cyclic structures.

Original: To further analyze the degree of hierarchy of each graph, we apply δ -hyperbolicity to identify thetree-likeness.
Revised: To further analyze the degree of hierarchy of each graph, we apply δ -hyperbolicity to identify thetree-likeness.

Original: Table 9 shows the δ -hyperbolicity of different datasets.
Revised: Table 9 shows the δ -hyperbolicity of different datasets.

Original: It shows that although mostof the datasets have negative mean sectional curvature, their degrees of hierarchy vary significantly.
Revised: It shows that although mostof the datasets have negative mean sectional curvature, their degrees of hierarchy vary significantly.

Original: For example, Airport and Pubmed are more hierarchical than CiteSeer and Cora.
Revised: For example, Airport and Pubmed are more hierarchical than CiteSeer and Cora.

Original: [Table]
Revised: [Table]

Original: [Table]

Original: [Table]

Original: Complexity analysis.
Revised: Complexity analysis.

Original: The time complexity is the same as a vanilla GCN given by O ( | V | dd ′ + | E | d ′ ) ,where | V | and | E | are the number of nodes and edges, d and d ′ are the dimension of input and hiddenfeatures.
Revised: The time complexity is the same as a vanilla GCN given by O ( | V | dd ′ + | E | d ′ ) ,where | V | and | E | are the number of nodes and edges, d and d ′ are the dimension of input and hiddenfeatures.

Original: The computation can be parallelized across all nodes.
Revised: The computation can be parallelized across all nodes.

Original: Similar to other non-Euclidean GCNs[28, 33, 14], the mapping from manifolds to the tangent space consume additional computationresources, compared with Euclidean GCNs, which is within the acceptable limits. 
Revised: Similar to other non-Euclidean GCNs[28, 33, 14], the mapping from manifolds to the tangent space consume additional computationresources, compared with Euclidean GCNs, which is within the acceptable limits. Visualization.

Revised: To visualize the embeddings learned by Q -GCN, we use UMAP tool 5 to project thelearned embeddings for Link Prediction on Pubmed and Cora into low-dimensional spherical and

Revised: [Table]

Revised: [Table]

Revised: 1.0 Sectional curvature 0.

Revised: # N o d e s airport mean 1.

Revised: 1.0 Sectional curvature 0.

Revised: # N o d e s pubmed mean 1.

Revised: 1.0 Sectional curvature # N o d e s citeseer mean 1.

Revised: 1.0 Sectional curvature # N o d e s cora mean

Revised: [Figure]

Revised: hyperbolic spaces.

Revised: The projections include spherical projection into 3D sphere, hyperbolic projection into 3D plane, and 2D Poincar ´ e disk.

Revised: As shown in Fig.

Revised: 4 (a,b), for Pubmed with more hyperbolicstructures, the class separability is more significant in hyperbolic projection than that is in sphericalprojection.

Revised: While the corresponding result is opposite for less tree-like Cora.

Revised: Furthermore, Fig. 4 (c)provides a more clear insight of the hierarchy.

Revised: It shows that there are more hub nodes near the originof Poincar ´ e disk in Pubmed than in Cora, showcasing the dominating tree-likeness of Pubmed.

Revised: Algorithm 1 describes the training process of the proposed method.

Revised: Input:

Revised: The input graph G =

Revised: ( V , E ) , the input node features x , the time dimension t , the space dimension s , the initial curvature β , the number of epoches E , the number of layers L , the initial learning rate, the activation function σ , the layer-wise trainable weight W and bias term b .

Revised: Output:

Revised: The trained model, i.e., the parameters W , b and β .

Revised: The resulting embeddings h of nodes.

Original: {+1: Feature initialization: map the input Euclidean features x into Pseudo-Riemannian space by the double projection h 
+}
Revised: {+1: Feature initialization: map the input Euclidean features x into Pseudo-Riemannian space by the double projection h 
+}

Revised: =

Revised: ( ψ − 1 ◦ ψ )( x ) where ψ and ψ − 1 are given by ▷ Eq.

Revised: 2: for e ∈ [0 , E ) do 3: for l ∈ [0 , L ) do 4: for i ∈ V do 5:

Revised: [Equation]

Revised: (cid:0)

Revised: [Equation]

Revised: ( Feature transformation ) 6:

Revised: [Equation]

Revised: (cid:68)

Revised: [Equation]

Revised: (cid:16)

Revised: [Equation]

Revised: (cid:0)

Revised: [Equation]

Revised: , if (cid:68)

Revised: [Equation]

Revised: (cid:101)

Revised: [Equation]

Revised: (cid:69)

Revised: [Equation]

Revised: ≥

Revised: [Equation]

Revised: β | ( Bias translation ) 7: h ℓ +

Revised: [Equation]

Revised: i

Revised: [Equation]

Revised: (cid:18)

Revised: [Equation]

Revised: (cid:18)(cid:80)

Revised: [Equation]

Revised: [Table]

Revised: 17: return