Tailored Scalable Dimensionality Reduction

Abstract

Although there is a rich literature on scalable methods for dimensionality reduction, the focus has been on widely applicable approaches which, in certain applications, are far from optimal or not even applicable. Dimensionality reduction can improve scalability of Bayesian computation, but optimal performance needs tailoring to the model. What kind of dimensionality reduction is sensible in data applications varies by the context of the data, resulting in neglect of information contained in data that do not fit general approaches.This dissertation introduces dimensionality reduction methods tailored to specific computational or data applications. Firstly, we scale up posterior computation in Bayesian linear regression using a dimensionality reduction approach enabled by the linearity in the model. It approximately integrates out nuisance parameters from a high-dimensional likelihood. The resulting posterior approximation scheme is competitive with state-of-the-art scalable posterior inference methods while being easier to interpret, understand, and analyze due to the explicit use of dimensionality reduction. Bayesian variable selection is considered as an example of a challenging posterior where the dimensionality reduction speeds up computation greatly and accurately.Secondly, we show how to reduce dimensionality based on data context in varying-domain functional data, where existing methods do not apply. The data of interest are intraoperative blood pressure and heart rate measurements. The first proposed approach extracts multiple different low-dimensional features from the high-dimensional blood pressure data, which are partly predefined and partly learnt from the data. This yields insights regarding blood pressure variability new to the clinical literature since such detailed inference was not possible with existing methods. The concluding case of dimensionality reduction is quantifying coupling of blood pressure and heart rate. This reduces two time series to one measurement of the strength of coupling. The results show the utility for inference methods of dimensionality reduction that is tailored to the challenge at hand.