Online Matrix Completion with Side Information

This thesis considers the problem of binary matrix completion with side information in the online setting and the applications thereof. The side information provides additional information on the rows and columns and can yield improved results compared to when such information is not available. We present efficient and general algorithms in transductive and inductive models. The performance guarantees that we prove are with respect to the matrix complexity measures of the max-norm and the margin complexity. We apply our bounds to the hypothesis class of biclustered matrices. Such matrices can be permuted through the rows and columns into homogeneous latent blocks. This class is a natural choice for our problem since the margin complexity and max-norm of these matrices have an upper bound that is easy to interpret in terms of the latent dimensions. We also apply our algorithms to a novel online multitask setting with RKHS hypothesis classes. In this setting, each task is partitioned in a sequence of segments, where a hypothesis is associated with each segment. Our algorithms are designed to exploit the scenario where the number of associated hypotheses is much smaller than the number of segments. We prove performance guarantees that hold for any segmentation of the tasks and any association of hypotheses to the segments. In the single-task setting, this is analogous to switching with long-term memory in the sense of [Bousquet and Warmuth; 2003]

Exploiting structure defined by data in machine learning: some new analyses

This thesis offers some new analyses and presents some new methods for learning in the context of exploiting structure defined by data – for example, when a data distribution has a submanifold support, exhibits cluster structure or exists as an object such as a graph. 1. We present a new PAC-Bayes analysis of learning in this context, which is sharp and in some ways presents a better solution than uniform convergence methods. The PAC-Bayes prior over a hypothesis class is defined in terms of the unknown true risk and smoothness of hypotheses w.r.t. the unknown data-generating distribution. The analysis is “localized” in the sense that complexity of the model enters not as the complexity of an entire hypothesis class, but focused on functions of ultimate interest. Such bounds are derived for various algorithms including SVMs. 2. We consider an idea similar to the p-norm Perceptron for building classifiers on graphs. We define p-norms on the space of functions over graph vertices and consider interpolation using the pnorm as a smoothness measure. The method exploits cluster structure and attains a mistake bound logarithmic in the diameter, compared to a linear lower bound for standard methods. 3. Rademacher complexity is related to cluster structure in data, quantifying the notion that when data clusters we can learn well with fewer examples. In particular we relate transductive learning to cluster structure in the empirical resistance metric. 4. Typical methods for learning over a graph do not scale well in the number of data points – often a graph Laplacian must be inverted which becomes computationally intractable for large data sets. We present online algorithms which, by simplifying the graph in principled way, are able to exploit the structure while remaining computationally tractable for large datasets. We prove state-of-the-art performance guarantees

Multiview semi-supervised learning with consensus

Ministry of Education, Singapore under its Academic Research Funding Tier 1; Tier

Learning with Multiple Similarities

The notion of similarities between data points is central to many classification and clustering algorithms. We often encounter situations when there are more than one set of pairwise similarity graphs between objects, either arising from different measures of similarity between objects or from a single similarity measure defined on multiple data representations, or a combination of these. Such examples can be found in various applications in computer vision, natural language processing and computational biology. Combining information from these multiple sources is often beneficial in learning meaningful concepts from data. This dissertation proposes novel methods to effectively fuse information from these multiple similarity graphs, targeted towards two fundamental tasks in machine learning - classification and clustering. In particular, I propose two models for learning spectral embedding from multiple similarity graphs using ideas from co-training and co-regularization. Further, I propose a novel approach to the problem of multiple kernel learning (MKL), converting it to a more familiar problem of binary classification in a transformed space. The proposed MKL approach learns a ``good'' linear combination of base kernels by optimizing a quality criterion that is justified both empirically and theoretically. The ideas of the proposed MKL method are also extended to learning nonlinear combinations of kernels, in particular, polynomial kernel combination and more general nonlinear kernel combination using random forests

Adaptive Graph-Based Algorithms for Conditional Anomaly Detection and Semi-Supervised Learning

We develop graph-based methods for conditional anomaly detection and semi-supervised learning based on label propagation on a data similarity graph. When data is abundant or arrive in a stream, the problems of computation and data storage arise for any graph-based method. We propose a fast approximate online algorithm that solves for the harmonic solution on an approximate graph. We show, both empirically and theoretically, that good behavior can be achieved by collapsing nearby points into a set of local representative points that minimize distortion. Moreover, we regularize the harmonic solution to achieve better stability properties. Anomaly detection techniques are used to identify anomalous (unusual) patterns in data. In clinical settings, these may concern identifications of unusual patient--state outcomes or unusual patient-management decisions. Therefore, we also present graph-based methods for detecting conditional anomalies and apply it to the identification of unusual clinical actions in hospitals. Our hypothesis is that patient-management actions that are unusual with respect to the past patients may be due to errors and that it is worthwhile to raise an alert if such a condition is encountered. Conditional anomaly detection extends standard unconditional anomaly framework but also faces new problems known as fringe and isolated points. We devise novel nonparametric graph-based methods to tackle these problems. Our methods rely on graph connectivity analysis and soft harmonic solution. Finally, we conduct an extensive human evaluation study of our conditional anomaly methods by 15 experts in critical care