Tabular Numeric Stretch Transformation

Tabular data presents unique challenges for deep learning due to its heterogeneous nature, where numeric features exhibit diverse distributions, scales, and statistical properties. Although recent advances have improved how models learn from tabular data, how numeric data are transformed into model-friendly representations remains comparatively underexplored. We introduce the stretch transformation framework, which formulates numeric feature preprocessing as an optimization problem to make the target function smoother and thus more learnable. Our framework has two variants: (1) unsupervised stretch, which uniformly redistributes feature density via minimax optimization, and (2) supervised stretch, which optimizes target-aware numeric feature transformations from the perspective of target-function smoothness by minimizing the target function's Dirichlet energy in the transformed space. Our theoretical analysis further connects this framework to several popular transformations: unsupervised stretch is closely related to Piecewise Linear Encoding through a shared piecewise-linear geometry and approaches the empirical CDF transformation as the number of bins grows, while supervised stretch becomes closely related to target encoding in the fine-binning limit. Comprehensive experiments on 38 datasets from the TALENT benchmark demonstrate that supervised stretch consistently outperforms all baselines. These results show that explicitly optimizing for target function smoothness is a powerful and underexplored strategy for tabular deep learning.

Persistent Gaussian Perturbations Prevent Oversmoothing in Recurrent Graph Neural Networks

Oversmoothing is a fundamental limitation of deep graph neural networks (GNNs), where repeated message passing causes node representations to become increasingly similar, eventually collapsing toward a low-dimensional subspace. This phenomenon limits the effective depth of message-passing architectures and motivates the search for mechanisms that preserve representation diversity. In this paper, we study a recurrent graph neural network in which independent Gaussian noise is injected after every propagation step and analyze the resulting architecture as a stochastic dynamical system. Under a standard global contraction assumption on the deterministic update, we prove that the hidden representations form a geometrically ergodic Markov chain admitting a unique invariant probability measure. Our main theoretical result establishes an explicit positive lower bound on the expected stationary Dirichlet energy, proportional to both the noise variance and the spectral gap of the underlying graph. Consequently, the stationary representations cannot collapse onto the constant manifold, providing a rigorous guarantee that asymptotic oversmoothing is prevented in the sense of non-vanishing Dirichlet energy. Our analysis reveals persistent stochastic perturbations as a fundamentally different mechanism for combating oversmoothing, complementing existing deterministic approaches based on residual connections, normalization, and graph rewiring. Finally, numerical experiments on both linear and nonlinear recurrent graph neural networks closely match the theoretical predictions, illustrating the emergence of a stationary distribution and the predicted dependence of the limiting Dirichlet energy on the noise intensity.

From Diffusion to Reaction-Diffusion: A Dynamical-Systems View of Oversmoothing in Hypergraph Neural Networks

Higher-order couplings enhance the expressive power of hypergraph neural networks (HGNNs), but they also intensify representation collapse in deep propagation due to strong multi-way feature mixing. This work investigates hypergraph oversmoothing from a dynamical-systems perspective and develops a reaction--diffusion framework for depth-resistant hypergraph learning. By defining hypergraph gradient and divergence operators, we interpret message passing as an incidence-level diffusion process. The analysis of pure diffusion shows that its continuous semiflow exponentially contracts the null-mode-free component of node representations and drives the Dirichlet energy to zero, revealing hypergraph oversmoothing as an intrinsic transverse-energy dissipation phenomenon. Motivated by this analysis, we propose Hypergraph Neural Reaction--Diffusion (HNRD), which introduces a reaction mechanism acting on the transverse component to compensate diffusion-induced dissipation and stabilize discriminative variations. We establish global well-posedness of the proposed dynamics and prove that the null-mode-free Dirichlet energy remains bounded away from zero. A forward-Euler discretization provides a practical HNRD layer with a stability condition for deep propagation. Experiments on benchmark and synthetic heterophilic hypergraphs demonstrate that HNRD consistently improves over representative hypergraph baselines. Depth, robustness, and efficiency analyses further show that HNRD preserves stable performance and nonzero Dirichlet energy under deep propagation and perturbations. These results provide a principled dynamical framework for designing deep hypergraph architectures that maintain higher-order expressiveness without representation collapse.

PIEFS: Physics-Informed Eigenfunction Features with Learnable Scaling

Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric $A(x)=Λ(x)U(x)$. The diagonal factor $Λ(x)$ controls anisotropic scaling, while the orthogonal factor $U(x)$ is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.

Revisiting Positive Samples in Graph Contrastive Learning: From the Perspective of Message Passing

Graph Contrastive Learning (GCL), which trains graph encoders by maximizing similarity between positive samples and minimizing it between negative ones, has emerged as a mainstream graph pre-training paradigm. It is widely recognized that positive samples are essential in GCLs. Ideally, maximizing the similarity of positive samples enables graph encoders to capture intrinsic semantic and patterns of graph data. However, we discover an interesting phenomenon: GCLs can achieve competitive performance even without positive samples. This motivates us to revisit the fundamental mechanism of positive samples in GCLs. From the perspective of Dirichlet energy, we theoretically finds that message passing, a key mechanism in graph encoders, trivializes the maximization of positive samples, preventing GCLs from effectively learning from positive samples. To address this, we propose SPGCL to mitigate the trivialization caused by message passing and restore the learning efficacy of positive samples. Specifically, we find that high Dirichlet energy features help positive samples provide effective learning signals while low Dirichlet energy features contribute little to positive learning signal but is useful for positive sampling. Based on this, SPGCL propagates only high Dirichlet energy features and uses low energy features to construct a probability matrix for reliable positive sampling. Extensive experiments demonstrate the effectiveness of SPGCL.