Sensing Structured Signals with Active and Ensemble Methods
John J. Lipor · Deep Blue (University of Michigan) · 2017
Modern problems in signal processing and machine learning involve the analysis of data that is high-volume, high-dimensional, or both. In one example, scientists studying the environment must choose their set of measurements from an infinite set of possible sample locations. In another, performing inference on high-resolution images involves operating on vectors whose dimensionality is on the order of tens of thousands. To combat the challenges presented by these and other applications, researchers rely on two key features intrinsic to many large datasets. First, large volumes of data can often be accurately represented by a few key points, allowing for efficient processing, summary, and collection of data. Second, high-dimensional data often has low-dimensional intrinsic structure that can be leveraged for processing and storage. This thesis leverages these facts to develop and analyze algorithms capable of handling the challenges presented by modern data. The first scenario considered in this thesis is that of monitoring regions of low oxygen concentration (hypoxia) in lakes via an autonomous robot. Tracking the spatial extent of such hypoxic regions is of great interest and importance to scientists studying the Great Lakes, but current systems rely heavily on hydrodynamic models and a very small number of measurements at predefined sample locations. Existing active learning algorithms minimize the samples required to determine the spatial extent but do not consider the distance traveled during the estimation procedure. We propose a novel active learning algorithm for tracking such regions that balances both the number of measurements taken and the distance traveled in estimating the boundary of the hypoxic zone. The second scenario considered is learning a union of subspaces (UoS) model that best fits a given collection of points. This model can be viewed as a generalization of principal components analysis (PCA) in which data vectors are drawn from one of several low-dimensional linear subspaces of the ambient space and has applications in image segmentation and object recognition. The problem of automatically sorting the data according to nearest subspace is known as subspace clustering, and existing unsupervised algorithms perform this task well in many situations. However, state-of-the-art algorithms do not fully leverage the problem geometry, and the resulting clustering errors are far from the best possible using the UoS model. We present two novel means of bridging this gap. We first present a method of incorporating semi-supervised information into existing unsupervised subspace clustering algorithms in the form of pairwise constraints between items. We next study an ensemble algorithm for unsupervised subspace clustering that functions by combining the outputs from many efficient but inaccurate base clusterings to achieve state-of- the-art performance. Finally, we perform the first principled study of model selection for subspace clustering, in which we define clustering quality metrics that do not rely on the ground truth and evaluate their ability to reliably predict clustering accuracy. The contributions of this thesis demonstrate the applicability of tools from signal processing and machine learning to problems ranging from scientific exploration to computer vision. By utilizing inherent structure in the data, we develop algorithms that are efficient in terms of computational complexity and other realistic costs, making them truly practical for modern problems in data science.