A method for dimension reduction with clustering, classification, or discriminant analysis is introduced. This mixture model-based approach is based on fitting generalized hyperbolic mixtures on a reduced subspace within the paradigm of model-based clustering, classification, or discriminant analysis. A reduced subspace of the data is derived by considering the extent to which group means and group covariances vary. The members of the subspace arise through linear combinations of the original data, and are ordered by importance via the associated eigenvalues. The observations can be projected onto the subspace, resulting in a set of variables that captures most of the clustering information available. The use of generalized hyperbolic mixtures gives a robust framework capable of dealing with skewed clusters. Although dimension reduction is increasingly in demand across many application areas, the authors are most familiar with biological applications and so two of the five real data examples are within that sphere. Simulated data are also used for illustration. The approach introduced herein can be considered the most general such approach available, and so we compare results to three special and limiting cases. Comparisons with several well established techniques illustrate its promising performance.
Model-based clustering imposes a finite mixture modelling structure on data for clustering. Finite mixture models assume that the population is a convex combination of a finite number of densities, the distribution within each population is a basic assumption of each particular model. Among all distributions that have been tried, the generalized hyperbolic distribution has the advantage that is a generalization of several other methods, such as the Gaussian distribution, the skew t-distribution, etc. With specific parameters, it can represent either a symmetric or a skewed distribution. While its inherent flexibility is an advantage in many ways, it means the estimation of more parameters than its special and limiting cases. The aim of this work is to propose a mixture of generalized hyperbolic factor analyzers to introduce parsimony and extend the method to high dimensional data. This work can be seen as an extension of the mixture of factor analyzers model to generalized hyperbolic mixtures. The performance of our generalized hyperbolic factor analyzers is illustrated on real data, where it performs favourably compared to its Gaussian analogue.
The Variational Auto-Encoder (VAE) model has become widely popular as a way to learn at once a generative model and embeddings for observations living in a high-dimensional space. In the real world, many such observations may be assumed to be hierarchically structured, such as living organisms data which are related through the evolutionary tree. Also, it has been theoretically and empirically shown that data with hierarchical structure can efficiently be embedded in hyperbolic spaces. We therefore endow the VAE with a hyperbolic geometry and empirically show that it can better generalise to unseen data than its Euclidean counterpart, and can qualitatively recover the hierarchical structure.
Multidimensional scaling (MDS) is a class of projective algorithms traditionally used in Euclidean space to produce two- or three-dimensional visualizations of datasets of multidimensional points or point distances. More recently however, several authors have pointed out that for certain datasets, hyperbolic target space may provide a better fit than Euclidean space. In this paper we develop PD-MDS, a metric MDS algorithm designed specifically for the Poincare disk (PD) model of the hyperbolic plane. Emphasizing the importance of proceeding from first principles in spite of the availability of various black box optimizers, our construction is based on an elementary hyperbolic line search and reveals numerous particulars that need to be carefully addressed when implementing this as well as more sophisticated iterative optimization methods in a hyperbolic space model.
We present a large scale hyperbolic recommender system. We discuss why hyperbolic geometry is a more suitable underlying geometry for many recommendation systems and cover the fundamental milestones and insights that we have gained from its development. In doing so, we demonstrate the viability of hyperbolic geometry for recommender systems, showing that they significantly outperform Euclidean models on datasets with the properties of complex networks. Key to the success of our approach are the novel choice of underlying hyperbolic model and the use of the Einstein midpoint to define an asymmetric recommender system in hyperbolic space. These choices allow us to scale to millions of users and hundreds of thousands of items.