The proof is an induction on k. Consider the general case p2k+1. It is easy to see that g (x) = ex p2k(x) and g (x) = ex p2k 1(x). By the induction hypothesis, g 0 and therefore g is convex. Thus, the minimum of g is given by its stationary points. It is easy to observe that x = 0 is indeed a stationary point. Thus, minx R g(x) = g(0) = 0, which finishes the proof.

This section provides basic theoretical details on the log-Sinkhorn operator and its convergence results. Then, we define the functional F(Al): C2(M)/R R, as follows: F(Al) = Z Sl A(x)dµθt(x)+ Z Hϵµ[Sl A](y)dνt(y). The log-Sinkhorn iteration S has the a point in C2(M)/R. This fixed point is determined up to an additive constant, and minimizes the functional F uniformly: F(S Al) F(Al+1) F(Al). Then, the function Al L is approximated to the d2/2-Legendre transformation (11) of the function Bm M. [Al L] If A is a fixed point of the log-Sinkhorn operator S on C2(M)/R, 36th Conference on Neural Information Processing Systems (NeurIPS 2022).

We incorporate an exogenous variable in the SCM that is learned and utilised for reasoning about interventionsintheobservation.

Hypergraphs effectively model higher-order relationships in natural phenomena, capturing complex interactions beyond pairwise connections. We introduce a novel hypergraph message passing framework inspired by interacting particle systems, where hyperedges act as fields inducing shared node dynamics. By incorporating attraction, repulsion, and Allen-Cahn forcing terms, particles of varying classes and features achieve class-dependent equilibrium, enabling separability through the particle-driven message passing. We investigate both first-order and second-order particle system equations for modeling these dynamics, which mitigate over-smoothing and heterophily thus can capture complete interactions. The more stable second-order system permits deeper message passing. Furthermore, we enhance deterministic message passing with stochastic element to account for interaction uncertainties. We prove theoretically that our approach mitigates over-smoothing by maintaining a positive lower bound on the hypergraph Dirichlet energy during propagation and thus to enable hypergraph message passing to go deep. Empirically, our models demonstrate competitive performance on diverse real-world hypergraph node classification tasks, excelling on both homophilic and heterophilic datasets.

Heterophilic Graph Neural Networks (HGNNs) have shown promising results for semi-supervised learning tasks on graphs. Notably, most real-world heterophilic graphs are composed of a mixture of nodes with different neighbor patterns, exhibiting local node-level homophilic and heterophilic structures. However, existing works are only devoted to designing better HGNN backbones or architectures for node classification tasks on heterophilic and homophilic graph benchmarks simultaneously, and their analyses of HGNN performance with respect to nodes are only based on the determined data distribution without exploring the effect caused by this structural difference between training and testing nodes. How to learn invariant node representations on heterophilic graphs to handle this structure difference or distribution shifts remains unexplored. In this paper, we first discuss the limitations of previous graph-based invariant learning methods from the perspective of data augmentation. Then, we propose \textbf{HEI}, a framework capable of generating invariant node representations through incorporating heterophily information to infer latent environments without augmentation, which are then used for invariant prediction, under heterophilic graph structure distribution shifts. We theoretically show that our proposed method can achieve guaranteed performance under heterophilic graph structure distribution shifts. Extensive experiments on various benchmarks and backbones can also demonstrate the effectiveness of our method compared with existing state-of-the-art baselines.