Piecewise-Linear Manifold Learning

The need to reduce the dimensionality of a dataset whilst retaining inherent manifold structure is key in many pattern recognition, machine learning and computer vision tasks. 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 on. Since the inception of manifold learning much research has been carried out into the most effective way of tackling this problem. Two main streams emerged to tackle the task: local and global methods. Each aim to preserve either local or global properties of the data. However, in recent years a third stream of research has come forth: global alignment of local models, which aims to preserve local properties over a global scale. We present a framework to tackle this local/global problem that approximates the manifold as a set of piecewise linear models (Piecewise Linear Manifold Learning). By merging these linear models in an order defined by their global topology, we can obtain a globally stable, and locally accurate model of the manifold. Examining the local properties of the data also allows us to present a generalisation to one of the main open problems in manifold learning — the out-of-sample extension. This problem is concerned with embedding new samples into a previously learnt low-dimensional embedding. Our solution — GOoSE — exploits the local geometry of the manifold to project novel samples into a low-dimensional embedding independent of what manifold learning algorithm was initially used. The results obtained for both Piecewise Linear Manifold Learning and GOoSE are significantly improved over existing state of the art algorithms.