Bayesian Inference on Matrix Manifolds for Linear Dimensionality Reduction

We reframe linear dimensionality reduction as a problem of Bayesian inference on matrix manifolds. This natural paradigm extends the Bayesian framework to dimensionality reduction tasks in higher dimensions with simpler models at greater speeds. Here an orthogonal basis is treated as a single point on a manifold and is associated with a linear subspace on which observations vary maximally. Throughout this paper, we employ the Grassmann and Stiefel manifolds for various dimensionality reduction problems, explore the connection between the two manifolds, and use Hybrid Monte Carlo for posterior sampling on the Grassmannian for the first time. We delineate in which situations either manifold should be considered. Further, matrix manifold models are used to yield scientific insight in the context of cognitive neuroscience, and we conclude that our methods are suitable for basic inference as well as accurate prediction.Comment: All datasets and computer programs are publicly available at http://www.ics.uci.edu/~babaks/Site/Codes.htm

Interpretable Models Capable of Handling Systematic Missingness in Imbalanced Classes and Heterogeneous Datasets

Application of interpretable machine learning techniques on medical datasets facilitate early and fast diagnoses, along with getting deeper insight into the data. Furthermore, the transparency of these models increase trust among application domain experts. Medical datasets face common issues such as heterogeneous measurements, imbalanced classes with limited sample size, and missing data, which hinder the straightforward application of machine learning techniques. In this paper we present a family of prototype-based (PB) interpretable models which are capable of handling these issues. The models introduced in this contribution show comparable or superior performance to alternative techniques applicable in such situations. However, unlike ensemble based models, which have to compromise on easy interpretation, the PB models here do not. Moreover we propose a strategy of harnessing the power of ensembles while maintaining the intrinsic interpretability of the PB models, by averaging the model parameter manifolds. All the models were evaluated on a synthetic (publicly available dataset) in addition to detailed analyses of two real-world medical datasets (one publicly available). Results indicated that the models and strategies we introduced addressed the challenges of real-world medical data, while remaining computationally inexpensive and transparent, as well as similar or superior in performance compared to their alternatives