Expanding the Hyperbolic Kernels: A Curvature-aware Isometric Embedding View

1 Introduction

This paper studies a curvature-aware isometric embedding to map the hyperbolic representations into an intermediate reproducing kernel Hilbert space (RKHS), where a set of kernels can be further developed.

In recent years, the hyperbolic space has shown its superiority to encode the data hierarchy, since the volume of hyperbolic space increases exponentially with radius [Ganea et al., 2018a; Ganea et al., 2018b]. This property enables the hyperbolic space to encode the graph-structured or treelikeness data with arbitrarily low distortion in case of low dimension, and benefits a lot of data embeddings, e.g., language data [Ganea et al., 2018a], visual data [Khrulkov et al., 2020; Fang et al., 2023], graph data [Cho et al., 2019], etc.

To enjoy the representation power of the hyperbolic space, various learning paradigms, including hyperbolic neural networks [Ganea et al., 2018b; Gulcehre et al., 2018; Shimizu et al., 2020] and hyperbolic kernel machines [Cho et al., 2019; Fang et al., 2021], are developed in the hyperbolic space, gaining substantial improvement over many AI applications. In [Ganea et al., 2018b], the hyperbolic geometry is first integrated into the deep neural network as the embedding space, and some essential hyperbolic neural layers are developed to work with the hyperbolic geometry. To make the hyperbolic space benefit from the rich kernel theories, Cho et al. first propose a hyperbolic polynomial kernel, an indefinite kernel defined in the Lorentz model [Cho et al., 2019]. The proposed kernel improves the SVM classification over graph and language data. As the complementary concept of the indefinite kernel, the positive definite (PD) kernels are further studied in [Fang et al., 2021]. Fang et al. leverage the identity tangent space to approximate the hyperbolic space and propose a set of PD kernels in the tangent space. Those kernels show superior performance on visual recognition tasks. However, the fixed curvature of hyperbolic space limits the adaptation capacity of the kernel [Cho et al., 2019], or the employment of the identity tangent space causes the distortion of hyperbolic data [Fang et al., 2021].

∗Corresponding author.

The work in [Arcozzi et al., 2007] proposes a kernel mapping, which realizes the isometry from the unit Poincar´e model to a special RKHS. In other words, such a mapping is able to address the distortion issue of embedding the hyperbolic data in Euclidean space 1 . Also, many existing successful practices of hyperbolic space also show that the Poincar´e model can adapt different data by tuning the curvature [Ganea et al., 2018b; Chami et al., 2019; Khrulkov et al., 2020; Fang et al., 2021]. The analysis arises a question: can we develop some valid kernels that can jointly preserve the data structure in the hyperbolic space, while fitting the data adaptively?

We answer this question by developing a curvature-aware isometric embedding, yielding the isometry to map data from the Poincar´e model to an RKHS. Then this RKHS, with Euclidean structure, allows us to construct a series of new hyperbolic kernel functions, including PD and indefinite kernels. Enjoying the property of the proposed curvature-aware isometric embedding, our kernels can reduce the distortion of working on hyperbolic data, and also improve the adaptability to different data. These advantages generalize our kernels to different practical applications.

1 The RKHS can be understood as a high-dimensional (or even infinite-dimensional) Euclidean space.

The contributions of this paper are summarized as follows:

• We propose a curvature-aware isometric embedding that realizes an isometry between the Poincar´e model and an RKHS. This mapping can preserve the rich structure of the hyperbolic data, and adapt to different data.
• On top of the embedding function, we develop a series of hyperbolic kernels, including PD kernels and an indefinite kernel. The PD kernels have the curvatureaware hyperbolic linear kernel, the curvature-aware hyperbolic polynomial kernel, the curvature-aware hyperbolic RBF kernel, and the curvature-aware hyperbolic Laplacian kernel, while the indefinite kernel is called the curvature-aware hyperbolic Sigmoid kernel.
• Thorough experiments are performed on different learning scenarios (e.g., graph learning and zero-shot learning) to evaluate the superiority of the proposed kernels, and the adaptation capacity to different data.

2 Preliminaries

2.1 Notations

Throughout the paper, we use Cn , Pnc , Dnc to denote the ndimensional complex space, n-dimensional Poincar´e model with curvature -c, n-dimensional open ball with radius 1/√c. It is worthy noted that we omit the n or c when n = 1 or c = 1 for simplicity. We also denote H as an RKHS and K as a reproducing kernel Kre˘ın space (RKKS). The matrices, vectors and scalars are denoted by bold capital letters (e.g., X), bold lower-case letters (e.g., x) and thin letters (e.g., x), respectively.

2.2 Hyperbolic Geometry

The hyperbolic space is a Riemannian manifold with constant negative curvature [Ratcliffe, 1994]. Five isometric models, including the Poincar´e model, the Lorentz model, the Klein model, the Half-space model and the Hemisphere mode, are used to model the hyperbolic space [Cannon et al., 1997]. In this work, we use Poincar´e model, as in many existing works [Nickel and Kiela, 2017; Ganea et al., 2018b; Fang et al., 2021]. The gyrovector space provides an elegant framework to work with hyperbolic space analytically [Ungar, 2022], and the M¨obius gyrovector space comes in handy to work with Poincar´e model [Ungar, 1998].

Poincar´e model. The n-dimensional Poincar´e model can be characterized as

where Cn and -c represent the n-dimensional complex space and curvature of the Poincar´e model, respectively.

M¨obius Gyrovector Space. A M¨obius gyrovector space is a gyrometric space with gyrometric, known as the M¨obius gyrodistance function which obeys the gyrotriangle inequality [Ungar, 2022]. It provides a set of algebraic formalism to work for the Poincar´e model. That said, one can enjoy from the M¨obius gyrovector space to perform the basic vector operations, e.g., addition, subtraction etc. For any two points zi, zj ∈ Pnc , the M¨obius addition is given by

The M¨obius gyrodistance can be deduced as

where ⊖nc is the M¨obius subtraction.

2.3 Isometry from the Unit Poincar´e Model to an RKHS

Isometry is a mapping that preserves the distance (or metric) between two metric spaces [Coxeter and Greitzer, 1967]. Concretely, the equivalence between pseudo-hyperbolic distance of the unit Poincar´e model and the metric, defined on an RKHS, establish the isometry. In this section, we will show the details of such an isometry realized by a kernel mapping.

Pseudo-hyperbolic Distance

The complex open unit ball Dn ∈ Cn , together with the pseudo-hyperbolic distance and the automorphisms 2 , i.e., M¨obius self-mappings, is the Poincar´e model [Duren and Weir, 2007; Rochberg, 2019]. That is, one can define the unit Poincar´e model Pn , with the pseudo-hyperbolic distance, by utilizing the M¨obius self-mappings [Duren and Weir, 2007].

When n = 1, the M¨obius self-mapping on the unit Poincar´e disk is defined as the M¨obius subtraction in the M¨obius gyrovector space [Duren and Weir, 2007]. Specifically, for zi, zj ∈ P, the mapping is given by

When n > 1, for zi, zj ∈ Cn let Pzi (zj) denote the orthogonal projection of zj to the subspace [zi] = {λzi : λ ∈ C} spanned by zi, and Qzi (zj) = zj - Pzi (zj) be the projection onto the orthogonal complement of [zi]. We further denote szi = (1-‖zi‖2 )1/2 . Then the M¨obius self-mapping on Pn is formulated as

Having the M¨obius self-mappings at hand, one can define the pseudo-hyperbolic distance in the unit Poincar´e ball, given by

where zi, zj ∈ Pn . Then the Eq. (6) can be further transformed to

Note that the Eq. (7) holds for both n = 1 and n > 1. In other words, both the pseudo-hyperbolic distance for the unit Poincar´e disk and the unit Poincar´e ball can be obtained by the M¨obius self-mappings.

2 The automorphism can project the ball to itself. Properties of the automorphism are in the supplementary material.

Drury Arveson Hardy Space

For an n-dimensional open unit ball Dn ∈ Cn , a corresponding Drury Arveson Hardy space, denoted by Dn , can be defined as a RKHS on Dn [Arcozzi et al., 2007]. Specifically, this is realized by a kernel function, given by

where zi, zj ∈ Dn and kz ∈ Dn is the kernel for the point z in Dn .

Denote ˆ kz = kz ‖kz‖ as the unit vector in the direction of kz, one can define the metric for zi, zj ∈ Dn as follows:

Since kz is a vector in Dn , and the right side of Eq. (9) fits the definition of a metric [Bellet et al., 2015], this metric can also be understood as a distance between kzi and kzj in Dn .

In this Drury Arveson Hardy space, one can plug the kernel mapping in Eq. (8) into Eq. (9), obtaining the formulation of the DA metric as

By observing the Eq. (7) and Eq. (10), we can find that the DA metric δDn on Dn equals to the pseudo-hyperbolic distance ρn on Pn . In other words, the unit ball Dn with DA metric δDn is the Poincar´e model Pn [Rochberg, 2019]. Since δDn is also a distance metric between kzi and kzj , the kernel mapping in Eq. (8) can be understood as an isometry from the unit Poincar´e model Pn to the Drury Arveson Hardy space Dn , meaning that the data in the unit Poincar´e model Pn can be embedded into the RKHS without distortion.

3 Isometry from the Curvature-aware Poincar´e Model to an RKHS

Section 2 introduces the isometry, realized by a kernel function, which maps the data from the unit Poincar´e model to an RKHS. Since the isometric mapping can preserves the distance metric in two space, the distortion issue of embedding the hierarchical data can be addressed. Also, recent studies [Fang et al., 2021; Chami et al., 2019] have verified that the curvature of the hyperbolic model influences the down-stream tasks significantly. This suggests generalizing the isometry to the Poincar´e model with adaptive curvature.

In this section, we propose an curvature-aware isometric embedding from the Poincar´e model to an RKHS. The embedding is realized by a curvature-aware kernel function. Figure 1 shows the process of this isometric-embedding. Specifically, we first define a pseudo-hyperbolic distance ρnc on the Poincar´e model Pnc , and then construct a new kernel kDn c z on the ball Dnc . Given the kernel kDn c z , a mapping fromDnc to the RKHS Dn c is realized and a metric δDnc on the ball Dnc can be defined. Besides, the metric δDnc can also be viewed as a distance measure in Dn c and we can establish the equivalence between the pseudo-hyperbolic distance ρnc and the metric δDnc . Therefore, using δDnc on Dn c as a bridge, an isometry from the Poincar´e ball to the RKHS can be established.

3.1 Generalization of the Pseudo-hyperbolic Distance in Pnc

The pseudo-hyperbolic distance in the unit Poincar´e model is defined via the norm of M¨obius self-mappings (see Eq. (6)) [Duren and Weir, 2007]. To define such a pseudohyperbolic distance in the curvature-aware Poincar´e ball, one needs to generalize the M¨obius self-mapping from Pn to Pnc .

M¨obius Self-mappings

When n = 1, the M¨obius self-mappings of the Poincar´e disk Pc is defined as the M¨obius subtraction:

where zi, zj ∈ Pc and ¯ zi means the complex conjugate of zi. We further generalize the M¨obius self-mapping from Pc to Pnc and propose the pseudo-hyperbolic distance of Pnc in case of n > 1. For zi, zj ∈ Dnc , let P c zi (zj) be the orthogonal projection of zj onto the subspace [zi] spanned by zi, and let Qczi (zj ) = zj -P c zi (zj ) be the projection onto the orthogonal complement of [zi]. Therefore, it holds that: P c zi (zj) = 0 when zi = 0, and P c zi (zj) = 〈zj,zi〉 〈zi,zi〉 zi if zi ̸= 0. Let sczi = √1 - czi 2 , we define the M¨obius self-mappings as

Note that the Eq. (12) can be simplified to ϕczi (zj) = zi-zj 1-c¯ zizj when n = 1.

Here are the properties of the proposed maps ϕczi .

Theorem 1. For every zi ∈ Dnc , ϕczi has the following properties:

• ϕczi (0) = zi, ϕczi (zi) = 0.
• ϕczi is a involution: ϕczi (ϕczi (zj)) = zj.

Proof. The proof is given in supplementary material.

According to Theorem 1, ϕczi is a one-to-one map, and that ϕczi = (ϕczi )-1 . This proves that the M¨obius self-mappings are automorphisms.

Pseudo-hyperbolic Distance

With the M¨obius self-mappings ϕczi at our disposal, we can come up with the pseudo-hyperbolic distance of the ball Dnc , given by

In this case, when n = 1, the pseudo-hyperbolic distance is defined as

where d is the M¨obius gyrodistance (see Eq. (3)) and |.| represents the absolute value of the number. Therefore, the pseudo-hyperbolic distance in Poincar´e disk with curvature -c can be rescaled by the M¨obius gyrodistance between the gyrovectors. When c = 1, it presents to the pseudohyperbolic distance in the unit Poincar´e disk.

Moreover, when n > 1, for zi, zj ∈ Dnc , ϕczi (zj) has the following property:

It is obvious that when c = 1, it becomes the pseudohyperbolic distance defined in the unit Poincar´e ball. Besides, the Eq. (13) is allowed to predigest to the equation as

which can be obtained by rescaling the M¨obius gyrodistance between zi and zj in Eq. (3).

Given the analysis, we have the following theorem.

Theorem 2. The pseudo-hyperbolic distance of the ball Dnc has the properties:

• ρnc (zi, zj) ≥ 0 and ρnc (zi, zj) = 0 if and only if zi = zj , since ϕzi (zj ) = 0 only for zi = zj ;
• ρnc (zi, zj) = ρnc (zj, zi);
• The triangle inequality takes strong form:

Proof. The proof is given in supplementary material.

The Theorem 2 shows the pseudo-hyperbolic distance is a true metric of the open ball Dnc . Furthermore, endowed with the pseudo-hyperbolic distance, the the open ball Dnc is the Poincar´e model Pnc .

3.2 RKHS with a Curvature-aware Kernel

In this section, we propose a new kernel function whose induced RKHS has a distance corresponds to the pseudohyperbolic distance in Poincar´e model.

For zi, zj ∈ Dnc , we define a curvature-aware kernel in the form of

where Dnc = {z ∈ Cn : c‖z‖2 < 1, c > 0} and Dn c is the RKHS with the kernel kDn c z . The kernel is PD and the proof is shown in the supplementary material.

After proposing the kernel, we define a metric on the ball similar to Eq. (9) in the form of

Concretely, it can be written as

by plugging the kernel in Eq. (17) to Eq. (18). When n = 1, the metric (18) can be simplified as

Through comparing Eq. (20) with Eq. (14), we note δDc (zi, zj) = ρc(zi, zj). Therefore, the metric is identified with the pseudo-hyperbolic distance of the Poincar´e disk with curvature -c.

When n > 1, compare Eq. (19) with Eq. (15), we know the metric δDnc (zi, zj ) also equals with the pseudo-hyperbolic distance ρnc (zi, zj). Therefore, the ball Dnc with the metric δDnc (zi, zj) is a Poincar´e model Pnc .

Furthermore, since (18) can also be viewed as distance in the RKHS Dn c by taking kDn c z as a vector in Dn c , the distancepreserved kernel mapping from the Poincar´e model Pnc to the RKHS with kernel kDn c (zi, zj) = 1/(1 - c〈zj , zi〉) realizes an isometry.

4 Curvature-aware Hyperbolic Kernels

In this section, we propose to construct a set of kernels in hyperbolic spaces by leveraging the proposed curvature-aware isometric embedding in Eq. (17).

4.1 Curvature-aware Hyperbolic Linear Kernel

We first come up with the simplest kernel in this context. The embedding in Eq. (17) establishes the isometry from the Poincar´e model Pnc and the RKHS Dn c . That said, this embedding can realize the linear kernel in Dn c , and we term it as curvature-aware hyperbolic linear (CHL) kernel . For zi, zj ∈ Pnc , the CHL kernel is given by

4.2 Curvature-aware Hyperbolic Polynomial Kernel

One popular kernel in the Euclidean space is the polynomial (Poly) kernel, which can encode the high-order statistics of the data. For xi, xj ∈ Cn , it can be expressed as

From the formulation to Eq. (22), we can simply extend it in the Poincar´e model Pnc . Specifically, the inner product, which can be understood as the linear kernel in the Euclidean space, in Eq. (22) can be replaced by Eq. (21), thus we can obtain the curvature-aware hyperbolic polynomial (CHPoly) kernel, formulated by

The proposed kernel is a PD one, and we have provided the proof in the supplementary material.

4.3 Curvature-aware Hyperbolic RBF Kernel

In the Euclidean space, the radial basis function (RBF) kernel is recognized as a powerful kernel due to it property of universal approximation [Smola and Sch¨olkopf, 1998]. It takes the form of

where τ > 0 is the bandwidth parameter of the RBF kernel. It is difficult to extand the RBF kernel to the Poincar´e model since the RBF kernel derived from the geodesic distance in the Poincar´e model is not a valid PD kernel [Fang et al., 2021; Jayasumana et al., 2015; Cho et al., 2019]. Fang et al. first propose a PD RBF in hyperbolic space in case of using the tangent space. In our work, we extend the RBF kernel in Dn c , called curvature-aware hyperbolic RBF (CHRBF) kernel. It can be formulated as

for τ > 0. It is worth noting that

Plug Eq. (26) into Eq. (25), we can obtain the full fumulation of CHRBF kernel as

Of note, in our formulation, the CHRBF kernel is also a valid PD kernel.

4.4 Curvature-aware Hyperbolic Laplacian Kernel

Another famous exponential-type kernel in the Euclidean space is known as the Laplacian (Lap) kernel. Its formulation is given by

Analogously, we can deduce the curvature-aware hyperbolic Laplacian kernel (CHLap) kernel as

Following the process of develop the CHRBF kernel in Eq. (27), one can yield the formulation of CHLap kernel as:

The PD property of CHLap kernel is also proven in the supplementary material.

4.5 Curvature-aware Hyperbolic Sigmoid Kernel

The former proposed kernels are PD. As a complementary concept of the PD kernel, the indefinite kernel has also been studied in the learning community [Ong et al., 2004]. In contrast to PD kernels, the indefinite kernel maps the data to an RKKS with indefinite inner product [Ong et al., 2004]. In [Cho et al., 2019], an indefinite kernel has been investigated and shows its effectiveness. This inspires us to further develop an indefinite kernel using the proposed embedding( Eq. (17) or Eq. (21)). In this case, we leverage the Sigmoid (Sig) kernel to develop an indefinite kernel. For xi, xj ∈ Cn , the Sig kernel can be expressed as

where γ > 0 and θ < 0. This is an indefinite kernel [Bishop and Nasrabadi, 2006]. Plug the Eq. 21 into Eq. 31, we can obtain the curvature-aware hyperbolic Sigmoid (CHSig) kernel for zi, zj ∈ Pnc , as

where γ > 0 and θ < 0. Since the Sig kernel is an indefinite kernel, the proposed CHSig kernel is also an indefinite kernel, which maps the hyperbolic data from the Poincar´e model to an RKKS. The optimization algorithms in RKKSs can be used to address the tasks in hyperbolic space [Loosli et al., 2015; Xu et al., 2017].

5 Experiments

In this section, we conduct thorough experiments to evaluate the superiority of the proposed kernels. Source code and Appendix are available at https://github.com/MMeiYang/Codeand-Appendix-for-Expanding-the-Hyperbolic-Kernels-ACurvature-aware-Isometric-Embedding-View.

5.1 Graph Learning

Learning on the graphs has gained significant attention in the learning community since the graph-structured data are popular to model the message passing networks including social networks, citation networks and communication networks [Xia et al., 2021]. We first evaluate the proposed kernels over the graph data with existing hyperbolic kernels and Euclidean kernels. The hyperbolic kernels include hyperbolic polynomial (HPoly) kernel proposed by [Cho et al., 2019], hyperbolic tangent (HTang) kernel, hyperbolic RBF

(HRBF) kernel, hyperbolic Laplace (HLap) kernel and hyperbolic Binomial (HBin) kernel in [Fang et al., 2021] and the Euclidean kernels includes Euclidean linear (EL) kernel, Euclidean polynomial (EPoly) kernel, Euclidean RBF (ERBF) kernel, Euclidean Laplacian (ELap) kernel and Euclidean Sigmoid (ESig) kernel.

Specifically, the graph are first mapped into the Poincar´e model for hyperbolic kernels, while the graph are mapped into the Euclidean space for Euclidean kernels, and then the graph nodes are classified by kernel SVMs. The details of the SVM solvers for PD and indefinite kernels are in supplementary material.

Datasets and Evaluation Protocol

Five real-world graph datasets including Facebook [Rozemberczki et al., 2019], Terrorist [Zhao et al., 2006], Wiki [Cucerzan, 2007], Amazon Electronics Computers (AC) [Shchur et al., 2018], Cora ML [Bojchevski and G¨unnemann, 2017] are used in this study. Both the hyperbolic space and the Euclidean space are employed as the embedding space with 2, 5, 10, 25 dimensions. The details of the datasets are provided in the supplementary material.

In addition, an one-vs-all (OVA) strategy is utilized to implement multi-class classification and Platt scaling maps the predicted scores on the hold-out data to probabilities for all categories [Platt and others, 1999]. We use the mean classification accuracy (ACC) to measure the performance of all the methods. Due to space limitation, other two performance metrics including area under the ROC curve (AUC) and macro-averaged area under precision recall curve (AUPR) are reported in the supplementary material.

Experimental Results

We first compare our proposed hyperbolic kernel methods with popular Euclidean kernel methods. The results are shown in Table 1. We observe that each of our hyperbolic kernels shows a great improvement against its Euclidean counterpart. For example, our CHL, CHPoly, CHRBF, CHLap and CHSig kernels improve the mean ACC by 20.3%, 12.3%, 5.8%, 6.3% and 8.1% compared with the Euclidean counterparts including EL, EPloy, ERBF, ELap and ESig kernels respectively. This shows that the hyperbolic space is a better option to encode the graph-structured data, and can improve the representation power of the graphs.

Next, the state-of-the-art hyperbolic kernel methods are employed as baseline to further demonstrate the superiority of our proposed kernels. Table 1 illustrates that our methods also obviously improve classification performance compared with existing hyperbolic kernel methods. For example, even the CHRBF kernel with the lowest performance in our proposed hyperbolic kernels improves the ACC value over HTang, HPoly, HRBF, HLap and HBin kernels by 15.6%, 8.1%, 0.9%, 0.3%, 3.1%, respectively. Compared with the HPoly kernel which fixes the curvature of hyperbolic space as -1, the good performance of our proposed kernels on different datasets demonstrates that curvature-aware hyperbolic kernels can improve the adaptability. In addition, comparisons with HTang, HRBF, HLap and HBin kernels further demonstrate the superiority of our method, which shows that the isometry from Poincar´e model to the RKHS reduces the distortion of mapping hyperbolic data to Euclidean space.

5.2 Zero-shot Learning

Zero-shot Learning (ZSL) portrays the problem of recognizing the unseen object by matching its semantic features [Akata et al., 2016; Xian et al., 2018], and can be modeled as a cross-modality matching task. We follow the practice in [Fang et al., 2021] to train the network and the paradigm is shown in Figure 2. For a batch of images, e.g., X = [X1, . . . ,XN], and all semantic embedding, e.g., E = [E1, . . . ,EM], as input, the visual features (e.g., x = [x1, . . . , xN]) and semantic features (e.g., e = [e1, . . . , eM]) are extracted by two neural networks, as x = f(X) and e = g(E). Having two-modality of data at hand, the network can be optimized by the alignment loss.

In the experiments, we follow the loss function of baseline in [Fang et al., 2021] to optimize the network. More details about the loss functions for ZSL are demonstrated in supplementary material.

Datasets and Evaluation Protocol

We use CUB [Wah et al., 2011], AWA1 [Lampert et al., 2013] and AWA2 [Akata et al., 2016] to evaluate the ZSL task. The statistics of the dataset is provided in the supplemental material. The model performance is calculated by the harmonic mean (HM) score, i.e., HM= 2×S×U U+S , where S and U denote the mean accuracy of seen and unseen classes respectively.

Experimental Results

We first evaluate our methods by comparing them with the baseline. As shown in Table 2, all our kernels show obvious improvements to the baseline. Concretely, compared with the baseline, even proposed simplest CHL kernel improve 23.4%, 20.9% and 21.6% on HM for CUB, AWA1 and AWA2. Besides, the powerful indefinite CHSig kernel further improves the accuracy of ZSL, which demonstrates the superiority of hyperbolic kernel for representation learning.

In addition, to further show the superiority of our method, a series of ZSL methods are applied for comparison, including the non-generative algorithms [Zhang and Shi, 2019; Li et al., 2019]. We note that our proposed CHSig kernel yields the best results on all datasets.

Finally, we compare our methods with the state-of-the-art methods with hyperbolic kernels for ZSL [Fang et al., 2021].

We observe from Table 2 that our methods are still very competitive compared with the existing hyperbolic kernel methods. In particular, our CHSig kernel outperforms existing methods across all 3 datasets, clearly showing the superiority of the proposed kernels.

5.3 Further Studies

In this part, we study the effect of the curvature of the Poincar´e ball. We conduct this study in the graph learning task. Specifically, the Facebook dataset and five kernels, e.g., CHL kernel, CHPoly kernel, CHRBF kernel, CHLap kernel, and CHSig kernel, are used in this analysis. In the experiments, we set c to {2-6 , 2-5 , 2-4 , 2-3 , 2-2 , 2-1 , 20 , 21 , 22 } while the curvature value is represented as -c. Figure 3 shows the results. We can observe that: (1) In the regime of the small c value, e.g., c < 0.5, the overall trend of classification accuracy of the all kernels is leveled off. (2) However, for PD kernels, it also shows that a large value of the c will degrade the classification performance. (3) The indefinite kernel, i.e., is insensitive to the curvature, indicating its superiority and robustness. This study indeed shows the discrimination of the Poincar´e model can benefit from the curvature.

6 Conclusion

This paper proposed a family of hyperbolic kernels with a new curvature-aware isometric embedding function. This embedding was realized by a kernel mapping, which projected the hyperbolic space to an RKHS. Then we developed a set of kernels including several PD kernels and an indefinite kernel. The proposed kernels can adapt to different data regimes and preserve the raw data's hierarchical structure. Extensive experiments on graph learning and zero-shot learning evaluate the effectiveness of the kernels. We believe the proposed isometric-embedding function can construct more powerful kernel machines.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 62076062) and the Social Development Science and Technology Project of Jiangsu Province ( No.BE2022811). Furthermore, the work was also supported by the Collaborative Innovation Center of Wireless Communications Technology and the Big Data Computing Center of Southeast University.