In contrast to regularizationbased approaches, we formulate the adversarially robust learning problem as one of loss minimization with a Lipschitz constraint, and show that the saddle point of the associated Lagrangian is characterized by a Poisson equation with weighted Laplace operator. Further, the weighting for the Laplace operator is given by the Lagrange multiplier for the Lipschitz constraint, which modulates the sensitivity of the minimizer to perturbations.

Our novelty has also been affirmed by R1, R2 and R4. However, we should clarify that (1) our work differs completely from MMD-GANs, and (2) although Ref [4] Our supplementary material includes the s.o.t.a. Below we discuss the reviewers' comments and will address all of them in the revision. Lipschitz constraint is not a necessity in our RCF-GAN. Please refer to our proof. Fig.4 in the paper shows the image reconstruction and interpolation, validating our superior performances on clear We will elaborate more upon this in the revision.

Lipschitz constraints under L2 norm on deep neural networks are useful for provable adversarial robustness bounds, stable training, and Wasserstein distance estimation. While heuristic approaches such as the gradient penalty have seen much practical success, it is challenging to achieve similar practical performance while provably enforcing a Lipschitz constraint. In principle, one can design Lipschitz constrained architectures using the composition property of Lipschitz functions, but Anil et al. recently identified a key obstacle to this approach: gradient norm attenuation. They showed how to circumvent this problem in the case of fully connected networks by designing each layer to be gradient norm preserving. We extend their approach to train scalable, expressive, provably Lipschitz convolutional networks. In particular, we present the Block Convolution Orthogonal Parameterization (BCOP), an expressive parameterization of orthogonal convolution operations. We show that even though the space of orthogonal convolutions is disconnected, the largest connected component of BCOP with 2n channels can represent arbitrary BCOP convolutions over n channels. Our BCOP parameterization allows us to train large convolutional networks with provable Lipschitz bounds. Empirically, we find that it is competitive with existing approaches to provable adversarial robustness and Wasserstein distance estimation.

In this work we propose a graph-based learning framework to train models with provable robustness to adversarial perturbations. In contrast to regularization-based approaches, we formulate the adversarially robust learning problem as one of loss minimization with a Lipschitz constraint, and show that the saddle point of the associated Lagrangian is characterized by a Poisson equation with weighted Laplace operator. Further, the weighting for the Laplace operator is given by the Lagrange multiplier for the Lipschitz constraint, which modulates the sensitivity of the minimizer to perturbations.

Observational data is widely utilized when randomized experiments are infeasible or fail to adequately represent target populations. A key challenge in observational analysis is the lack of overlap between treatment and control groups. Even when a nominally large dataset is collected, the effective sample size may be prohibitively small when there is a region with little overlap between treated and control populations. As an example, if the treatment of interest is rarely observed among older citizens, estimating their counterfactual (treated) outcome becomes inherently unreliable. This challenge is further exacerbated in modern operational contexts, where high-dimensional covariate representations [15] increase data sparsity, making causal identification particularly difficult in regions of the covariate space with small effective sample size.

Currently, they are often characterized via intensity function which limits model's expressiveness due to unrealistic assumptions on