Expressive 1-Lipschitz Neural Networks for Robust Multiple Graph Learning against Adversarial Attacks

Recent findings have shown multiple graph learning models, such as graph classification and graph matching, are highly vulnerable to adversarial attacks, i.e. small input perturbations in graph structures and node attributes can cause the model failures. Existing defense techniques often defend specific attacks on particular multiple graph learning tasks. This paper proposes an attack-agnostic graph-adaptive 1-Lipschitz neural network, ERNN, for improving the robustness of deep multiple graph learning while achieving remarkable expressive power. A K-l-Lipschitz Weibull activation function (f) over bar is designed to enforce the gradient norm parallel to del(f) over bar (x)parallel to as K-l at layer l. The nearest matrix orthogonalization and polar decomposition techniques are utilized to constraint the weight norm parallel to(W) over bar (l)parallel to as 1/K-l and make (W) over bar (l) close to the original weight (W) over bar (l). The theoretical analysis is conducted to derive lower and upper bounds of feasible K-l under the 1-Lipschitz constraint. The combination of norm-constrained (f ) over bar and (W) over bar (l) leads to the 1-Lipschitz neural network for expressive and robust multiple graph learning.