Dimensionality Reduction of Collective Motion by Principal Manifolds

TL;DR

This paper introduces a novel method for low-dimensional embedding of collective motion data using principal manifolds, which better preserves structure in noisy, sparse datasets compared to existing methods.

Contribution

It proposes a two-dimensional principal manifold approach using cubic smoothing splines and geodesic distances, avoiding spectral decomposition limitations of traditional methods.

Findings

Retains original data structure in noisy datasets

Effective in analyzing complex collective behaviors

Compared favorably with established nonlinear reduction methods

Abstract

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.