Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.

SEQUENTIAL DECISIONS AND PREDICTIONS IN NATURAL LANGUAGE PROCESSING

Natural language processing has achieved great success in a wide range of ap- plications, producing both commercial language services and open-source language tools. However, most methods take a static or batch approach, assuming that the model has all information it needs and makes a one-time prediction. In this disser- tation, we study dynamic problems where the input comes in a sequence instead of all at once, and the output must be produced while the input is arriving. In these problems, predictions are often made based only on partial information. We see this dynamic setting in many real-time, interactive applications. These problems usually involve a trade-off between the amount of input received (cost) and the quality of the output prediction (accuracy). Therefore, the evaluation considers both objectives (e.g., plotting a Pareto curve). Our goal is to develop a formal understanding of sequential prediction and decision-making problems in natural language processing and to propose efficient solutions. Toward this end, we present meta-algorithms that take an existent batch model and produce a dynamic model to handle sequential inputs and outputs. Webuild our framework upon theories of Markov Decision Process (MDP), which allows learning to trade off competing objectives in a principled way. The main machine learning techniques we use are from imitation learning and reinforcement learning, and we advance current techniques to tackle problems arising in our settings. We evaluate our algorithm on a variety of applications, including dependency parsing, machine translation, and question answering. We show that our approach achieves a better cost-accuracy trade-off than the batch approach and heuristic-based decision- making approaches. We first propose a general framework for cost-sensitive prediction, where dif- ferent parts of the input come at different costs. We formulate a decision-making process that selects pieces of the input sequentially, and the selection is adaptive to each instance. Our approach is evaluated on both standard classification tasks and a structured prediction task (dependency parsing). We show that it achieves similar prediction quality to methods that use all input, while inducing a much smaller cost. Next, we extend the framework to problems where the input is revealed incremen- tally in a fixed order. We study two applications: simultaneous machine translation and quiz bowl (incremental text classification). We discuss challenges in this set- ting and show that adding domain knowledge eases the decision-making problem. A central theme throughout the chapters is an MDP formulation of a challenging problem with sequential input/output and trade-off decisions, accompanied by a learning algorithm that solves the MDP.