Piecewise-Linear Manifold Learning: A Heuristic Approach to Nonlinear Dimensionality Reduction

The need to reduce the dimensionality of a dataset whilst retaining its inherent manifold structure is key to many pattern recognition, machine learning, and computer vision problems. This process is often referred to as manifold learning since the structure is preserved during dimensionality reduction by learning the intrinsic low-dimensional manifold that the data lies upon. In this paper a heuristic approach is presented to tackle this problem by approximating the manifold as a set of piecewise linear models. By merging these linear models in an order defined by their global topology a globally stable and locally accurate model of the manifold can be obtained. A detailed analysis of the proposed approach is presented along with comparison with existing manifold learning techniques. Results obtained on both artificial and image based data show that in many cases this heuristic approach to manifold learning is able to out-perform traditional techniques.