Adaptive Locally Linear Embedding
Ali Goli, Mahdieh Alizadeh, Hadi Sadoghi Yazdi
TL;DR
The paper tackles the limitation of fixed distance metrics in Locally Linear Embedding by introducing Adaptive Locally Linear Embedding (ALLE), which learns a data-driven PSD metric $\boldsymbol{M}$ and uses the associated distance $d_{\mathbf{M}}(\cdot,\cdot)$ to define neighborhoods. The method alternates between reconstructing each point from its neighbors, updating the metric to reduce reconstruction error, and computing a low-dimensional embedding via an eigen-decomposition, with a PSD guarantee obtained by updating a factor $\mathbf{L}$ where $\mathbf{M}=\mathbf{L}^T\mathbf{L}$. Empirical results on MNIST, Swiss Roll, and Olivetti demonstrate improved neighborhood preservation (trustworthiness/continuity) and often higher clustering/classification performance, illustrating the practical impact of a topology-aware metric in manifold learning. The work advances dimensionality reduction by tightly integrating metric learning with LLE, enabling embeddings that better reflect intrinsic geometry in complex, high-dimensional data.
Abstract
Manifold learning techniques, such as Locally linear embedding (LLE), are designed to preserve the local neighborhood structures of high-dimensional data during dimensionality reduction. Traditional LLE employs Euclidean distance to define neighborhoods, which can struggle to capture the intrinsic geometric relationships within complex data. A novel approach, Adaptive locally linear embedding(ALLE), is introduced to address this limitation by incorporating a dynamic, data-driven metric that enhances topological preservation. This method redefines the concept of proximity by focusing on topological neighborhood inclusion rather than fixed distances. By adapting the metric based on the local structure of the data, it achieves superior neighborhood preservation, particularly for datasets with complex geometries and high-dimensional structures. Experimental results demonstrate that ALLE significantly improves the alignment between neighborhoods in the input and feature spaces, resulting in more accurate and topologically faithful embeddings. This approach advances manifold learning by tailoring distance metrics to the underlying data, providing a robust solution for capturing intricate relationships in high-dimensional datasets.