论文标题
几何测量的高阶相互作用模型
Higher-order interaction model from geometric measurements
论文作者
论文摘要
我们引入了对线性共识模型的更高的简单概括，该模型共享几个共同的特征。众所周知的线性共识模型是梯度流，每对点之间具有一定距离的正方形。我们新建议的模型还表示为梯度流，配备了总$ n $维量功能，该功能由$ n+1 $点组成，作为潜力。通过这种方式，线性共识模型与$ n = 1 $的情况一致，其中距离被理解为1维体积。从简单的数学分析中，可以很容易地表明线性共识模型（具有1维体积功能的梯度流）崩溃到一个点，可以将其视为0-复合物。通过扩展此结果，我们证明了我们的模型的解决方案将收敛到$（N-1）$ - 维仿射子空间。我们还使用有效的算法进行了几个数值模拟，从而降低了计算成本。
We introduce a higher simplicial generalization of the linear consensus model which shares several common features. The well-known linear consensus model is a gradient flow with a sum of squares of distances between each pair of points. Our newly suggested model is also represented as a gradient flow equipped with total $n$-dimensional volume functional consisting of $n+1$ points as a potential. In this manner, the linear consensus model coincides with the case of $n=1$ where distance is understood as the 1-dimensional volume. From a simple mathematical analysis, one can easily show that the linear consensus model (a gradient flow with 1-dimensional volume functional) collapses to one single point, which can be considered as a 0-complex. By extending this result, we show that a solution to our model converges to an $(n-1)$-dimensional affine subspace. We also perform several numerical simulations with an efficient algorithm that reduces the computational cost.
