Linear Manifold Clustering for High-Dimensional Data: A New Approach to Unsupervised Learning

Clustering is a fundamental unsupervised learning problem, essential for understanding the intrinsic structure of high-dimensional data. Traditional clustering methods such as K-means assume that data clusters are globular and can be wellapproximated by Euclidean distance. However, in high-dimensional settings, many real-world datasets lie on low-dimensional manifolds, and clustering these datasets requires methods that respect the underlying manifold structure. This paper introduces a novel approach, Linear Manifold Clustering (LMC), which assumes that data points reside on a linear submanifold of the higher-dimensional space. By leveraging techniques from manifold learning and linear algebra, LMC enhances clustering performance by incorporating the geometric properties of the data. Our approach outperforms traditional clustering algorithms in both clustering accuracy and computational efficiency on high-dimensional datasets, as demonstrated in experiments on synthetic and real-world datasets.