Generative Locally Linear Embedding
This work addresses the need for generative capabilities in manifold learning for researchers in dimensionality reduction, though it is incremental as it builds on existing LLE methods.
The authors tackled the problem of making Locally Linear Embedding (LLE) generative by introducing stochastic linear reconstruction steps, resulting in GLLE methods that can generate various embeddings related to the original LLE embedding, with simulations showing effective unfolding and generation of data submanifolds.
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.