Non-linear kernel methods can be approximated by fast linear ones using suitable explicit feature maps allowing their application to large scale problems. We investigate how convolution kernels for structured data are composed from base kernels and construct corresponding feature maps. On this basis we propose exact and approximative feature maps for widely used graph kernels based on the kernel trick. We analyze for which kernels and graph properties computation by explicit feature maps is feasible and actually more efficient. In particular, we derive approximative, explicit feature maps for state-of-the-art kernels supporting real-valued attributes including the GraphHopper and graph invariant kernels. In extensive experiments we show that our approaches often achieve a classification accuracy close to the exact methods based on the kernel trick, but require only a fraction of their running time. Moreover, we propose and analyze algorithms for computing random walk, shortest-path and subgraph matching kernels by explicit and implicit feature maps. Our theoretical results are confirmed experimentally by observing a phase transition when comparing running time with respect to label diversity, walk lengths and subgraph size, respectively.

中文翻译：

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图内核的显式和隐式特征图的统一视图

非线性核方法可以使用合适的显式特征图通过快速线性方法近似，从而允许将其应用于大规模问题。我们研究如何从基本内核组成用于结构化数据的卷积内核，并构造相应的特征图。在此基础上，我们基于核技巧为广泛使用的图核提出了精确和近似的特征图。我们分析了通过显式特征图计算哪些内核和图形属性是可行的，实际上更有效。特别是，我们为支持包括GraphHopper和图不变核的实值属性的最新内核推导了近似的显式特征图。在广泛的实验中，我们证明了我们的方法通常可以达到接近基于核技巧的精确方法的分类精度，但只需要其运行时间的一小部分。此外，我们提出并分析了通过显式和隐式特征图计算随机游动，最短路径和子图匹配内核的算法。当比较运行时间与标签多样性，步长和子图大小时的相变，通过实验证实了我们的理论结果。