We consider the representation power of siamese-style similarity functions
used in neural network-based graph embedding. The inner product similarity
(IPS) with feature vectors computed via neural networks is commonly used for
representing the strength of association between two nodes. However, only a
little work has been done on the representation capability of IPS. A very
recent work shed light on the nature of IPS and reveals that IPS has the
capability of approximating any positive definite (PD) similarities. However, a
simple example demonstrates the fundamental limitation of IPS to approximate
non-PD similarities. We then propose a novel model named Shifted IPS (SIPS)
that approximates any Conditionally PD (CPD) similarities arbitrary well. CPD
is a generalization of PD with many examples such as negative Poincar\'e
distance and negative Wasserstein distance, thus SIPS has a potential impact to
significantly improve the applicability of graph embedding without taking great
care in configuring the similarity function. Our numerical experiments
demonstrate the SIPS's superiority over IPS. In theory, we further extend SIPS
beyond CPD by considering the inner product in Minkowski space so that it
approximates more general similarities.

本文提出了一种名为 SIPS 的新模型，通过在 Minkowski 空间内考虑内积，SIPS 能够很好地逼近各种条件正定的相似度，从而从理论上得出 SIPS 优于 IPS 的结论。