Non-linear dimensionality reduction techniques for classification

This thesis project concerns on dimensionality reduction through manifold learning with a focus on non linear techniques. Dimension Reduction (DR) is the process of reducing high dimension dataset with d feature (dimension) to one with a lower number of feature p (p ≪ d) that preserves the information contained in the original higher dimensional space. More in general, the concept of manifold learning is introduced, a generalized approach that involves algorithm for dimensionality reduction. Manifold learning can be divided in two main categories: Linear and Non Linear method. Although, linear method, such as Principal Component Analysis (PCA) and Multidimensional Scaling (MDS) are widely used and well known, there are plenty of non linear techniques i.e. Isometric Feature Mapping (Isomap), Locally Linear Embedding (LLE), Local Tangent Space Alignment (LTSA), which in recent years have been subject of studies. This project is inspired by the work done by [Bahadur et Al., 2017 ], with the aim to estimate the US market dimensionality using Russell 3000 as a proxy of financial market. Since financial markets are high dimensional and complex environment an approach with non linear techniques among linear is proposed.