# Generative Locally Linear Embedding

arXiv: https://arxiv.org/abs/2104.01525 · HTML: https://mlsanity.com/paper/2104.01525/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 31 references that are in the map.
- First posted on arXiv 4 Apr 2021.

## Builds on (most cited first, 7 of 7)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Adversarial Autoencoders](https://mlsanity.com/paper/1511.05644/) (2015)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [Eigenvalue and Generalized Eigenvalue Problems: Tutorial](https://mlsanity.com/paper/1903.11240/) (2019)
- [Factor Analysis, Probabilistic Principal Component Analysis, Variational Inference, and Variational Autoencoder: Tutorial and Survey](https://mlsanity.com/paper/2101.00734/) (2021)
- [Feature Selection and Feature Extraction in Pattern Analysis: A Literature Review](https://mlsanity.com/paper/1905.02845/) (2019)
- [Locally Linear Embedding and its Variants: Tutorial and Survey](https://mlsanity.com/paper/2011.10925/) (2020)

## Built on by (most cited first, 1 of 1)

- [Principal Manifold Flows](https://mlsanity.com/paper/2202.07037/) (2022)

## Abstract (arXiv)

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.