Generative Locally Linear Embedding

TL;DR

This paper introduces Generative LLE (GLLE), a probabilistic extension of Locally Linear Embedding that generates diverse embeddings by incorporating stochastic linear reconstruction, inspired by variational inference and factor analysis.

Contribution

The paper presents two novel stochastic versions of LLE that enable generative modeling of embeddings, expanding LLE's capabilities with probabilistic methods.

Findings

GLLE effectively unfolds and generates data submanifolds.

The methods relate to variational inference and factor analysis.

Simulations demonstrate successful stochastic embedding generation.

Abstract

Locally Linear Embedding (LLE) is a nonlinear spectral dimensionality reduction and manifold learning method. It has two main steps which are linear reconstruction and linear embedding of points in the input space and embedding space, respectively. In this work, we propose two novel generative versions of LLE, named Generative LLE (GLLE), whose linear reconstruction steps are stochastic rather than deterministic. GLLE assumes that every data point is caused by its linear reconstruction weights as latent factors. The proposed GLLE algorithms can generate various LLE embeddings stochastically while all the generated embeddings relate to the original LLE embedding. We propose two versions for stochastic linear reconstruction, one using expectation maximization and another with direct sampling from a derived distribution by optimization. The proposed GLLE methods are closely related to and inspired by variational inference, factor analysis, and probabilistic principal component analysis. Our simulations show that the proposed GLLE methods work effectively in unfolding and generating submanifolds of data.