We study low-rank tensor-product B-spline (TPBS) models for regression tasks and investigate Dirichlet energy as a measure of smoothness. We show that TPBS models admit a closed-form expression for the Dirichlet energy, and reveal scenarios where perfect interpolation is possible with exponentially small Dirichlet energy. This renders global Dirichlet energy-based regularization ineffective. To address this limitation, we propose a novel regularization strategy based on local Dirichlet energies defined on small hypercubes centered at the training points. Leveraging pretrained TPBS models, we also introduce two estimators for inference from incomplete samples. Comparative experiments with neural networks demonstrate that TPBS models outperform neural networks in the overfitting regime for most datasets, and maintain competitive performance otherwise. Overall, TPBS models exhibit greater robustness to overfitting and consistently benefit from regularization, while neural networks are more sensitive to overfitting and less effective in leveraging regularization.

Over-smoothing and over-squashing have been extensively studied in the literature on Graph Neural Networks (GNNs) over the past years. We challenge this prevailing focus in GNN research, arguing that these phenomena are less critical for practical applications than assumed. We suggest that performance decreases often stem from uninformative receptive fields rather than over-smoothing. We support this position with extensive experiments on several standard benchmark datasets, demonstrating that accuracy and over-smoothing are mostly uncorrelated and that optimal model depths remain small even with mitigation techniques, thus highlighting the negligible role of over-smoothing. Similarly, we challenge that over-squashing is always detrimental in practical applications. Instead, we posit that the distribution of relevant information over the graph frequently factorises and is often localised within a small k-hop neighbourhood, questioning the necessity of jointly observing entire receptive fields or engaging in an extensive search for long-range interactions. The results of our experiments show that architectural interventions designed to mitigate over-squashing fail to yield significant performance gains. This position paper advocates for a paradigm shift in theoretical research, urging a diligent analysis of learning tasks and datasets using statistics that measure the underlying distribution of label-relevant information to better understand their localisation and factorisation.

Graph neural networks (GNNs) integrate deep architectures and topological structure modeling in an effective way. However, the performance of existing GNNs would decrease significantly when they stack many layers, because of the over-smoothing issue. Node embeddings tend to converge to similar vectors when GNNs keep recursively aggregating the representations of neighbors. To enable deep GNNs, several methods have been explored recently. But they are developed from either techniques in convolutional neural networks or heuristic strategies.

Graph neural networks (GNNs) have shown state-of-the-art performances in various applications. However, GNNs often struggle to capture long-range dependencies in graphs due to oversmoothing. In this paper, we generalize the concept of oversmoothing from undirected to directed graphs. To this aim, we extend the notion of Dirichlet energy by considering a directed symmetrically normalized Laplacian. As vanilla graph convolutional networks are prone to oversmooth, we adopt a neural graph ODE framework. Specifically, we propose fractional graph Laplacian neural ODEs, which describe non-local dynamics. We prove that our approach allows propagating information between distant nodes while maintaining a low probability of long-distance jumps. Moreover, we show that our method is more flexible with respect to the convergence of the graph's Dirichlet energy, thereby mitigating oversmoothing. We conduct extensive experiments on synthetic and real-world graphs, both directed and undirected, demonstrating our method's versatility across diverse graph homophily levels.

One of the most analyzed problems in Graph Neural Networks (GNNs) is over-smoothing, that is usually described as the exponential convergence of node embeddings to a common vector through the GNN layers. One of the more frequently used metrics to analyze both theoretically and empirically over-smoothing is the Dirichlet energy, that is induced by the graph Laplacian, with different possibilities as analyzed in the next section. A formal axiomatic definition of over-smoothing, based on the definition of a "total over-smoothing" state where all node embeddings are identical, has been proposed in [1]. A key axiom of the proposal is that a smoothness measure should be zero if and only if this state is reached. In this paper, we point out that the widely-used Dirichlet Energy induced by the normalized graph Laplacian does not satisfy this axiom. Recently, some other issues in adopting Dirichlet energies in order to measure over-smoothing were pointed out in [2], where it is explained that Dirichlet energy induced by the normalized Laplacian tends to zero when node embeddings tend to its dominant eigenvector v s.t.

While Dirichlet energy serves as a prevalent metric for quantifying over-smoothing, it is inherently restricted to capturing first-order feature derivatives. To address this limitation, we propose a generalized family of node similarity measures based on the energy of higher-order feature derivatives. Through a rigorous theoretical analysis of the relationships among these measures, we establish the decay rates of Dirichlet energy under both continuous heat diffusion and discrete aggregation operators. Furthermore, our analysis reveals an intrinsic connection between the over-smoothing decay rate and the spectral gap of the graph Laplacian. Finally, empirical results demonstrate that attention-based Graph Neural Networks (GNNs) suffer from over-smoothing when evaluated under these proposed metrics.

Graph neural networks (GNNs) integrate deep architectures and topological structure modeling in an effective way. However, the performance of existing GNNs would decrease significantly when they stack many layers, because of the over-smoothing issue. Node embeddings tend to converge to similar vectors when GNNs keep recursively aggregating the representations of neighbors.

Early MIL methods treated the instances in a bag independently. Recent methods account for global and local dependencies among instances.

Another intriguing phenomenon is the existence of adversarial examples -- imperceptible perturbations of inputs that alter classification results.