Inner product-based decoders are among the most influential frameworks used to extract meaningful data from latent embeddings. However, such decoders have shown limitations in representation capacity in numerous works within the literature, which have been particularly notable in graph reconstruction problems. In this paper, we provide the first theoretical elucidation of this pervasive phenomenon in graph data, and suggest straightforward modifications to circumvent this issue without deviating from the inner product framework.

Recent works have shown that extending the message passing paradigm to subgraphs communicating with other subgraphs, especially via higher order messages, can boost the expressivity of graph neural networks. In such architectures, to faithfully account for local structure such as cycles, the local operations must be equivariant to the automorphism group of the local environment. However, enumerating the automorphism groups of all subgraphs of interest and finding appropriate equivariant operations for each one of them separately is generally not feasible. In this paper we propose a solution to this problem based on spectral graph theory that bypasses having to determine the automorphism group entirely and constructs a basis for equivariant operations directly from the graph Laplacian. We show that this approach can boost the performance of GNNs on some standard benchmarks.

Data augmentation has been proven effective for training high-accuracy convolutional neural network classifiers by preventing overfitting. However, building deep neural networks in real-world scenarios requires not only high accuracy on clean data but also robustness when data distributions shift. While prior methods have proposed that there is a trade-off between accuracy and robustness, we propose IPMix, a simple data augmentation approach to improve robustness without hurting clean accuracy. IPMix integrates three levels of data augmentation (image-level, patch-level, and pixel-level) into a coherent and label-preserving technique to increase the diversity of training data with limited computational overhead. To further improve the robustness, IPMix introduces structural complexity at different levels to generate more diverse images and adopts the random mixing method for multi-scale information fusion. Experiments demonstrate that IPMix outperforms state-of-the-art corruption robustness on CIFAR-C and ImageNet-C. In addition, we show that IPMix also significantly improves the other safety measures, including robustness to adversarial perturbations, calibration, prediction consistency, and anomaly detection, achieving state-of-the-art or comparable results on several benchmarks, including ImageNet-R, ImageNet-A, and ImageNet-O.