We study the implications of the modeling choice to use a graph, instead of a
hypergraph, to represent real-world interconnected systems whose constituent
relationships are of higher order by nature. Such a modeling choice typically
involves an underlying projection process that maps the original hypergraph
onto a graph, and is common in graph-based analysis. While hypergraph
projection can potentially lead to loss of higher-order relations, there exists
very limited studies on the consequences of doing so, as well as its
remediation. This work fills this gap by doing two things: (1) we develop
analysis based on graph and set theory, showing two ubiquitous patterns of
hyperedges that are root to structural information loss in all hypergraph
projections; we also quantify the combinatorial impossibility of recovering the
lost higher-order structures if no extra help is provided; (2) we still seek to
recover the lost higher-order structures in hypergraph projection, and in light
of (1)'s findings we propose to relax the problem into a learning-based
setting. Under this setting, we develop a learning-based hypergraph
reconstruction method based on an important statistic of hyperedge
distributions that we find. Our reconstruction method is evaluated on 8
real-world datasets under different settings, and exhibits consistently good
performance. We also demonstrate benefits of the reconstructed hypergraphs via
use cases of protein rankings and link predictions.

本研究探讨了将图而非超图作为表示实际世界互联系统的建模选择的影响，提出了超图投影导致高阶关系丢失的两个普遍模式，并量化了在无额外帮助下恢复丢失高阶结构的组合不可能性。此外，基于学习的超图重建方法通过使用超边分布的重要统计量，在不同设置下在 8 个真实世界数据集上展示了良好的性能，并通过蛋白质排名和链接预测的用例展示了重建超图的优势。

从图到超图：超图投影及其修复

From Graphs to Hypergraphs: Hypergraph Projection and its Remediation

The aim of this paper is to present a novel general framework for the
identification of possibly interconnected systems, while preserving their
physical properties and providing accuracy in multi-step prediction. An
analytical and recursive algorithm for the gradient computation of the
multi-step loss function based on backpropagation is introduced, providing
physical and structural insight directly into the learning algorithm. As a case
study, the proposed approach is tested for estimating the inertia matrix of a
space debris starting from state observations.

提出了一个新颖的通用框架用于识别可能相互关联的系统，并保留其物理特性并提供多步预测的准确性，引入了基于反向传播的多步损失函数梯度计算的分析递归算法，直接为学习算法提供物理和结构洞察力。作为案例研究，测试了该方法用于从状态观测开始估计太空碎片的惯性矩阵。