Graph matching, also known as network alignment, is the problem of finding a correspondence between the vertices of two separate graphs with strong applications in image correspondence and functional inference in protein networks. One class of successful techniques is based on tensor Kronecker products and tensor eigenvectors. A challenge with these techniques are memory and computational demands that are quadratic (or worse) in terms of problem size. In this manuscript we present and apply a theory of tensor Kronecker products to tensor based graph alignment algorithms to reduce their runtime complexity from quadratic to linear with no appreciable loss of quality. In terms of theory, we show that many matrix Kronecker product identities generalize to straightforward tensor counterparts, which is rare in tensor literature. Improved computation codes for two existing algorithms that utilize this new theory achieve a minimum 10 fold runtime improvement.

中文翻译：

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使用 Tensor Kronecker 乘积结构解决高阶图匹配中的计算瓶颈

图匹配，也称为网络对齐，是寻找两个独立图的顶点之间的对应关系的问题，在蛋白质网络中的图像对应和功能推理中具有很强的应用。一类成功的技术是基于张量 Kronecker 乘积和张量特征向量。这些技术的一个挑战是内存和计算需求在问题规模方面是二次的（或更糟的）。在这份手稿中，我们介绍了张量 Kronecker 乘积理论并将其应用于基于张量的图形对齐算法，以将其运行时复杂度从二次方降低到线性，而不会造成明显的质量损失。在理论方面，我们表明许多矩阵 Kronecker 乘积恒等式可以推广到直接的张量对应物，这在张量文献中很少见。