LLE with low-dimensional neighborhood representation

TL;DR

This paper identifies limitations of the standard LLE algorithm related to noise sensitivity and linear projection convergence, and proposes a low-dimensional neighborhood representation to improve robustness and embedding quality.

Contribution

The authors introduce a simple modification to LLE that computes weights using low-dimensional neighborhoods, reducing noise sensitivity and eliminating regularization needs.

Findings

Reduced noise sensitivity in LLE with the new method

Elimination of regularization when neighbors exceed input dimension

Improved embedding quality demonstrated through numerical examples

Abstract

The local linear embedding algorithm (LLE) is a non-linear dimension-reducing technique, widely used due to its computational simplicity and intuitive approach. LLE first linearly reconstructs each input point from its nearest neighbors and then preserves these neighborhood relations in the low-dimensional embedding. We show that the reconstruction weights computed by LLE capture the high-dimensional structure of the neighborhoods, and not the low-dimensional manifold structure. Consequently, the weight vectors are highly sensitive to noise. Moreover, this causes LLE to converge to a linear projection of the input, as opposed to its non-linear embedding goal. To overcome both of these problems, we propose to compute the weight vectors using a low-dimensional neighborhood representation. We prove theoretically that this straightforward and computationally simple modification of LLE reduces LLE's sensitivity to noise. This modification also removes the need for regularization when the number of neighbors is larger than the dimension of the input. We present numerical examples demonstrating both the perturbation and linear projection problems, and the improved outputs using the low-dimensional neighborhood representation.