Learning High Dimensional Demonstrations Using Laplacian Eigenmaps

TL;DR

This paper introduces a method using Laplacian Eigenmaps to learn stable, high-dimensional robot control dynamics from a single demonstration, enabling accurate modeling of complex nonlinear behaviors.

Contribution

The novel approach leverages graph Laplacian eigendecomposition to embed high-dimensional nonlinear dynamics into a quasi-linear latent space, improving stability and efficiency.

Findings

Embeddings become linear with increasing data density.

Method outperforms existing techniques in accuracy and parameter efficiency.

Successfully applied to real robot control tasks.

Abstract

This article proposes a novel methodology to learn a stable robot control law driven by dynamical systems. The methodology requires a single demonstration and can deduce a stable dynamics in arbitrary high dimensions. The method relies on the idea that there exists a latent space in which the nonlinear dynamics appears quasi linear. The original nonlinear dynamics is mapped into a stable linear DS, by leveraging on the properties of graph embeddings. We show that the eigendecomposition of the Graph Laplacian results in linear embeddings in two dimensions and quasi-linear in higher dimensions. The nonlinear terms vanish, exponentially as the number of datapoints increase, and for large density of points, the embedding appears linear. We show that this new embedding enables to model highly nonlinear dynamics in high dimension and overcomes alternative techniques in both precision of reconstruction and number of parameters required for the embedding. We demonstrate its applicability to control real robot tasked to perform complex free motion in space.