Understanding High Dimensional Spaces through Visual Means Employing Multidimensional Projections

Abstract

Data visualisation helps understanding data represented by multiple variables, also called features, stored in a large matrix where individuals are stored in lines and variable values in columns. These data structures are frequently called multidimensional spaces. A large set of mathematical tools, named frequently as multidimensional projections, aim to map such large spaces into 'visual spaces',  that is, to 2 or 3 dimensions, where the aspect of that space can be visualised. While the final product is intuitive in that proximity between points - or iconic representation of points - indicate similarity relationships in the original space, understanding the formulation of the projection methods many times escapes researchers. In this paper, we illustrate ways of employing the visual results of multidimensional projection algorithms to understand and fine-tune the parameters of their mathematical framework. Some of the common mathematical common to these approaches are Laplacian matrices, Euclidian distance, Cosine distance, and statistical methods such as Kullback-Leibler divergence, employed to fit probability distributions and reduce dimensions. Two of the relevant algorithms in the data visualisation field are t-distributed stochastic neighbourhood embedding (t-SNE) and Least-Square Projection (LSP). These algorithms can be used to understand several ranges of mathematical functions including their impact on datasets. In this article, mathematical parameters of underlying techniques such as Principal Component Analysis (PCA) behind t-SNE and mesh reconstruction methods behind LSP are adjusted to reflect the properties afforded by the mathematical formulation. The results, supported by illustrative methods of the processes of LSP and t-SNE, are meant to inspire students in understanding the mathematics behind such methods, in order to apply them in effective data analysis tasks in multiple applications.