# Locally Linear Embedding and its Variants: Tutorial and Survey

arXiv: https://arxiv.org/abs/2011.10925 · HTML: https://mlsanity.com/paper/2011.10925/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Nov 2020.

## Built on by (most cited first, 15 of 15)

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- [Learning Canonical Embeddings for Unsupervised Shape Correspondence with Locally Linear Transformations](https://mlsanity.com/paper/2209.02152/) (2022)
- [Generative Locally Linear Embedding](https://mlsanity.com/paper/2104.01525/) (2021)
- [Unified Framework for Spectral Dimensionality Reduction, Maximum Variance Unfolding, and Kernel Learning By Semidefinite Programming: Tutorial and Survey](https://mlsanity.com/paper/2106.15379/) (2021)
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- [Unfolding Projection-free SDP Relaxation of Binary Graph Classifier via GDPA Linearization](https://mlsanity.com/paper/2109.04697/) (2021)
- [Theoretical Connection between Locally Linear Embedding, Factor Analysis, and Probabilistic PCA](https://mlsanity.com/paper/2203.13911/) (2022)
- [ImplicitSLIM and How it Improves Embedding-based Collaborative Filtering](https://mlsanity.com/paper/2406.00198/) (2024)
- [Enhancing Medical Data Analysis through AI-Enhanced Locally Linear Embedding: Applications in Medical Point Location and Imagery](https://mlsanity.com/paper/2512.22182/) (2025)
- [a-TMFG: Scalable Triangulated Maximally Filtered Graphs via Approximate Nearest Neighbors](https://mlsanity.com/paper/2603.09564/) (2026)
- [Structured Representation Learning with Locally Linear Embeddings and Adaptive Feature Fusion](https://mlsanity.com/paper/2606.18469/) (2026)

## Abstract (arXiv)

This is a tutorial and survey paper for Locally Linear Embedding (LLE) and its variants. The idea of LLE is fitting the local structure of manifold in the embedding space. In this paper, we first cover LLE, kernel LLE, inverse LLE, and feature fusion with LLE. Then, we cover out-of-sample embedding using linear reconstruction, eigenfunctions, and kernel mapping. Incremental LLE is explained for embedding streaming data. Landmark LLE methods using the Nystrom approximation and locally linear landmarks are explained for big data embedding. We introduce the methods for parameter selection of number of neighbors using residual variance, Procrustes statistics, preservation neighborhood error, and local neighborhood selection. Afterwards, Supervised LLE (SLLE), enhanced SLLE, SLLE projection, probabilistic SLLE, supervised guided LLE (using Hilbert-Schmidt independence criterion), and semi-supervised LLE are explained for supervised and semi-supervised embedding. Robust LLE methods using least squares problem and penalty functions are also introduced for embedding in the presence of outliers and noise. Then, we introduce fusion of LLE with other manifold learning methods including Isomap (i.e., ISOLLE), principal component analysis, Fisher discriminant analysis, discriminant LLE, and Isotop. Finally, we explain weighted LLE in which the distances, reconstruction weights, or the embeddings are adjusted for better embedding; we cover weighted LLE for deformed distributed data, weighted LLE using probability of occurrence, SLLE by adjusting weights, modified LLE, and iterative LLE.