Formation-Controlled Dimensionality Reduction

TL;DR

This work reframes dimensionality reduction as a formation-control problem, introducing a nonlinear gradient-flow dynamical system that aligns local neighborhood distances with manifold geodesic distances while optionally enforcing global geometry via a nonlocal term. By restricting attention to neighbor relations and employing a mollified gradient flow, the model yields invariant, bounded trajectories and converges to a stationary embedding under mild conditions, with theoretical support from a Lyapunov-based analysis and Łojasiewicz inequality. Empirically, the approach demonstrates competitive generalization accuracy and favorable trustworthiness/continuity on synthetic manifolds and real datasets (e.g., MNIST, COIL20, ORL, HIVA) relative to classic convex DR methods. The combination of local distance control and optional global regularization provides a scalable, geometry-preserving alternative to traditional dimensionality reduction techniques.

Abstract

Dimensionality reduction represents the process of generating a low dimensional representation of high dimensional data. Motivated by the formation control of mobile agents, we propose a nonlinear dynamical system for dimensionality reduction. The system consists of two parts; the control of neighbor points, addressing local structures, and the control of remote points, accounting for global structures.We also include a brief mathematical analysis of the model and its numerical procedure. Numerical experiments are performed on both synthetic and real datasets and comparisons with existing models demonstrate the soundness and effectiveness of the proposed model.

Figures (3)

• Figure 1: 4 synthetic data sets
• Figure 2: Computation of Swiss roll data using kDRFC
• Figure 3: Computation of Helix data using sDRFC

Theorems & Definitions (16)

• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Proposition 4.4
• proof
• Lemma 4.5
• Theorem 4.6