Out of all the local Nash equilibria of HCC games, there exists a special subclass, the vectors(θθθ,φφφ) that implement the von Neumann solution of the convex-concave game.

In this appendix, further preliminary information and relevant discussions are provided. According to [15], we list some well-known GNN models that fall inside this framework in Table 5. (l, 1) (l, 2) (Theorem 1).

However, none of these solutions eliminates the linear dependence ongraph size.

Please refer to the Appendix 5 for details. Besides, in order to balance the classification loss and the transformation loss we set the scalar factor to beλ = 5 throughoutthetrainingphase. Here the first rows are the untransformed and the transformed images, while the second rows are the corresponding interpretations. This is a supplement toFigure 1 in the main body of the paper. And also there are perturbation methods such as randomized input sampling (RISE) [8] and extremal perturbation (EP) [2].

Withthe proliferation ofmachine learning applications inthe real world, the demand for explaining machine learning predictions continues to grow especially in high-stakes fields. Recent studies havefound that interpretation methods can be sensitive and unreliable, where the interpretations can be disturbed by perturbations or transformations of input data. To address this issue, we propose to learn robust interpretations through transformation equivariant regularization in a self-interpretable model.