通过主流形进行集体运动的降维
Dimensionality Reduction of Collective Motion by Principal Manifolds
AI总结：
本文提出基于立方平滑样条构建二维主流形的方法，用于降维分析集体运动数据，保留原始结构并优于现有非线性降维方法。
摘要：
尽管已证明集体运动模式中存在低维嵌入流形，但现有非线性降维方法无法有效分析此类流形，主要原因是谱分解步骤限制了高维空间到嵌入空间的映射控制。本文提出一种替代方法，要求二维嵌入以拓扑方式总结高维数据。具体而言，我们直接在高维空间中使用立方平滑样条构建二维主流形，并用测地距离定义嵌入坐标。通过代表性示例，我们展示与现有非线性降维方法相比，主流形在噪声和稀疏数据集上仍能保留原始结构。主流形寻找算法应用于多个代理动态系统模拟复杂机动（捕食者围攻）得到的配置，并将所得二维嵌入与已建立的非线性降维方法进行比较。
英文摘要：
While the existence of low-dimensional embedding manifolds has been shown in patterns of collective motion, the current battery of nonlinear dimensionality reduction methods are not amenable to the analysis of such manifolds. This is mainly due to the necessary spectral decomposition step, which limits control over the mapping from the original high-dimensional space to the embedding space. Here, we propose an alternative approach that demands a two-dimensional embedding which topologically summarizes the high-dimensional data. In this sense, our approach is closely related to the construction of one-dimensional principal curves that minimize orthogonal error to data points subject to smoothness constraints. Specifically, we construct a two-dimensional principal manifold directly in the high-dimensional space using cubic smoothing splines, and define the embedding coordinates in terms of geodesic distances. Thus, the mapping from the high-dimensional data to the manifold is defined in terms of local coordinates. Through representative examples, we show that compared to existing nonlinear dimensionality reduction methods, the principal manifold retains the original structure even in noisy and sparse datasets. The principal manifold finding algorithm is applied to configurations obtained from a dynamical system of multiple agents simulating a complex maneuver called predator mobbing, and the resulting two-dimensional embedding is compared with that of a well-established nonlinear dimensionality reduction method.
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