Core Concepts
A novel nonlinear dynamical system is proposed for dimensionality reduction, inspired by the formation control of mobile agents. The system combines local and global geometric constraints to preserve the intrinsic structure of high-dimensional data.
Abstract
The paper presents a new dimensionality reduction model inspired by the formation control of mobile agents. The key idea is to regard the dimensionality reduction process as the interaction between many bodies, where the bodies (data points) move towards a desired formation by keeping local distances (preserving local geometry) and controlling their distance to remote points (accounting for global structure).
The proposed model consists of two main components:
- Control of neighbor points: This addresses the local structure of the data by minimizing the difference between the Euclidean distance of the low-dimensional representations and the geodesic distance of the high-dimensional data.
- Control of remote points: This accounts for the global structure by introducing a repulsive force between the low-dimensional representations and the remote points, based on an approximate geodesic distance.
The authors analyze the stability of the dynamical system and provide a computational scheme using the forward Euler method. Numerical experiments on both synthetic and real datasets demonstrate the effectiveness of the proposed model in preserving the local and global structures of the data, as evidenced by the generalization performance of 1-nearest neighbor classifiers and the trustworthiness and continuity measures.
The key advantages of the proposed approach are:
- It offers a fresh perspective on dimensionality reduction by drawing inspiration from formation control in multi-agent systems.
- The dynamical system formulation allows for local stability analysis and provides insights into the underlying geometric properties.
- The model is able to capture both local and global structures of the data, outperforming several existing dimensionality reduction techniques on the benchmark datasets.
Stats
The paper reports the following key statistics:
The synthetic datasets (Swiss roll, helix, twin peaks, broken Swiss roll) consist of 5,000 samples each, unless otherwise specified.
The MNIST dataset consists of 60,000 handwritten digits, with 5,000 randomly selected for the experiments.
The COIL20 dataset contains 1,440 images of 20 different objects.
The ORL dataset has 400 grayscale face images.
The HIVA dataset has 3,845 datapoints with dimensionality 1,617.
Quotes
No significant quotes were extracted from the content.