Models that rely solely on pairwise relationships often fail to capture the complete statistical structure of the complex multivariate data found in diverse domains, such as socio-economic, ecological, or biomedical systems. Non-trivial dependencies between groups of more than two variables can play a significant role in the analysis and modelling of such systems, yet extracting such high-order interactions from data remains challenging. Here, we introduce a hierarchy of d-order interaction measures, increasingly inclusive of possible factorisations of the joint probability distribution, and define non-parametric, kernel-based tests to establish systematically the statistical significance of d-order interactions. We also establish mathematical links with lattice theory, which elucidate the derivation of the interaction measures and their composite permutation tests; clarify the connection of simplicial complexes with kernel matrix centring; and provide a means to enhance computational efficiency. We illustrate our results numerically with validations on synthetic data, and through an application to neuroimaging data.

In this paper, we evaluate the quality of knowledge representations encoded in deep neural networks (DNNs) for 3D point cloud processing. We propose a method to disentangle the overall model vulnerability into the sensitivity to the rotation, the translation, the scale, and local 3D structures. Besides, we also propose metrics to evaluate the spatial smoothness of encoding 3D structures, and the representation complexity of the DNN. Based on such analysis, experiments expose representation problems with classic DNNs, and explain the utility of the adversarial training. The code will be released when this paper is accepted.

In this section, in order to help readers understand the metric in the paper, we first revisit the definition of the Shapley value [14], which is widely considered as an unbiased estimation of the numerical importance w.r.t. each input variable. In game theory, the complex system is usually represented as a game, where each input variable is taken as a player, and the output of this system is regarded as the total reward of all players. Given a game with multiple players (input variables) N = {1,2,,n}, some players cooperate to pursue a high reward. Thus, the task is to divide the total reward, and fairly assign the divided elementary reward to each individual player. In this way, the elementary reward can be considered as the numerical importance of the corresponding variable to the complex system. Let 2N def= {S|S N}indicate all potential subsets of N. The game v: 2N R is a function, which estimates the overall reward v(S) earned by each specific subset of players S N. In this way, the Shapley value, denoted by φ(i), represents the numerical importance of the player ito the game v. φ(i) = X Using Shapley values to explain DNNs.

This paper provides a unified view to explain different adversarial attacks and defensemethods,i.e.

The line, triangular and square shapes represent the joint distribution of two, three and four variables respectively.

This paper focuses on the intersection of the above two directions,i.e.

Models that rely solely on pairwise relationships often fail to capture the complete statistical structure of the complex multivariate data found in diverse domains, such as socio-economic, ecological, or biomedical systems. Non-trivial dependencies between groups of more than two variables can play a significant role in the analysis and modelling of such systems, yet extracting such high-order interactions from data remains challenging. Here, we introduce a hierarchy of $d$-order ($d \geq 2$) interaction measures, increasingly inclusive of possible factorisations of the joint probability distribution, and define non-parametric, kernel-based tests to establish systematically the statistical significance of $d$-order interactions. We also establish mathematical links with lattice theory, which elucidate the derivation of the interaction measures and their composite permutation tests; clarify the connection of simplicial complexes with kernel matrix centring; and provide a means to enhance computational efficiency. We illustrate our results numerically with validations on synthetic data, and through an application to neuroimaging data.

Pedestrian trajectory prediction is critical for ensuring safety in autonomous driving, surveillance systems, and urban planning applications. While early approaches primarily focus on one-hop pairwise relationships, recent studies attempt to capture high-order interactions by stacking multiple Graph Neural Network (GNN) layers. However, these approaches face a fundamental trade-off: insufficient layers may lead to under-reaching problems that limit the model's receptive field, while excessive depth can result in prohibitive computational costs. We argue that an effective model should be capable of adaptively modeling both explicit one-hop interactions and implicit high-order dependencies, rather than relying solely on architectural depth. To this end, we propose ViTE (Virtual graph Trajectory Expert router), a novel framework for pedestrian trajectory prediction. ViTE consists of two key modules: a Virtual Graph that introduces dynamic virtual nodes to model long-range and high-order interactions without deep GNN stacks, and an Expert Router that adaptively selects interaction experts based on social context using a Mixture-of-Experts design. This combination enables flexible and scalable reasoning across varying interaction patterns. Experiments on three benchmarks (ETH/UCY, NBA, and SDD) demonstrate that our method consistently achieves state-of-the-art performance, validating both its effectiveness and practical efficiency.