Adaptive Locally Linear Embedding

Ali Goli 1, Mahdieh Alizadeh 1, and Hadi Sadoghi Yazdi 1,2 1 Department of Computer Engineering, Ferdowsi University of Mashhad, Mashhad, Iran 2 Center of Excellence in Soft Computing and Intelligent Information Processing, Ferdowsi University of Mashhad, Mashhad, Iran April 10, 2025 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. Keywords-- Manifold Learning, Adaptive Locally Linear Embedding, Dimensionality Reduction, Topological Preservation, Complex Geometries, High-Dimensional Data, Topological Neighborhood Inclusion, Intrinsic Geometric Relationships 1 Introduction Locally linear embedding(LLE) is a prominent manifold learning technique designed to reduce the dimensionality of high-dimensional datasets while preserving their intrinsic geometric structure. Proposed by Roweis and Saul, LLE operates through a systematic process that includes identifying the K-nearest neighbors for each data point, calculating reconstruction weights to express each point as a linear combination of its neighbors, and ultimately generating a low-dimensional representation that retains local relationships [14]. However, LLE traditionally relies on fixed distance metrics, such as Euclidean distance, which may inadequately represent complex data distributions and fail to capture nuanced topological relationships. In response to these limitations, we introduce a novel approach termed Adaptive LLE(ALLE), which integrates a flexible, data-driven metric into the LLE framework.