Dimensionality Reduction of Collective Motion by Principal Manifolds

by   Kelum Gajamannage, et al.

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.


page 1

page 2

page 3

page 4


A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics

Existing dimensionality reduction methods are adept at revealing hidden ...

A Tangent Distance Preserving Dimensionality Reduction Algorithm

This paper considers the problem of nonlinear dimensionality reduction. ...

Manifold Learning via Manifold Deflation

Nonlinear dimensionality reduction methods provide a valuable means to v...

Identifying manifolds underlying group motion in Vicsek agents

Collective motion of animal groups often undergoes changes due to pertur...

Principal Manifolds of Middles: A Framework and Estimation Procedure Using Mixture Densities

Principal manifolds are used to represent high-dimensional data in a low...

Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning

In this paper, we propose Wasserstein Isometric Mapping (Wassmap), a par...

A Theoretical Analysis of Joint Manifolds

The emergence of low-cost sensor architectures for diverse modalities ha...

Please sign up or login with your details

Forgot password? Click here to reset