基于莫比乌斯陀螺矢量空间的双曲正定核

Hyperbolic Positive Definite Kernels Based on Möbius Gyrovector Space

层次结构数据广泛存在于各类机器学习场景中,双曲空间能够以极低的失真编码层次结构数据,引入核方法后,可进一步提高双曲空间的表征能力.然而,现有的双曲核仍然存在自适应能力较低或数据失真的缺陷.为了解决这些问题,文中提出基于莫比乌斯陀螺矢量空间的双曲正定核方法.利用莫比乌斯陀螺矢量空间与庞加莱模型之间的关系,构造莫比乌斯径向基核.具体使用莫比乌斯陀螺距离代替欧几里得距离,构造莫比乌斯高斯核和莫比乌斯拉普拉斯核,并进一步证明核函数的正定性.另外,将该核函数从复空间转换到实空间上,更适用于大多数机器学习任务.在多组真实的社交网络数据集上的实验验证文中方法的有效性.

Hierarchical data is widely present in various machine learning scenarios and the data can be encoded in hyperbolic spaces with very low distortion.Kernel methods are introduced to further enhance the representation capability of hyperbolic space.However,the existing hyperbolic kernels still have the drawbacks of low adaptive capacity or data distortion.To address these issues,hyperbolic positive definite kernels based on Möbius gyrovector space is proposed in this paper.By leveraging the relationship between the Möbius gyrovector space and the Poincarémodel,a class of hyperbolic kernel functions,the Möbius radial basis kernels,are constructed.Specifically,the Möbius gyrodistance is employed in place of the Euclidean distance to construct the Möbius Gaussian kernel and the Möbius Laplacian kernel,with the positive definiteness of the kernel functions further demonstrated.Moreover,kernel functions are transformed from complex space to real space,and thus they are more suitable for most machine learning tasks.Experiments on several real-world social network datasets validate the effectiveness of the proposed method.