The emergence of a new paradigm in machine learning known as semi-supervised learning (SSL) has seen benefits to many applications where labeled data is expensive to obtain. However, unlike supervised learning (SL), which enjoys a rich and deep theoretical foundation, semi-supervised learning, which uses additional unlabeled data for training, still remains a theoretical mystery lacking a sound fundamental understanding. The purpose of this research thesis is to take a first step towards bridging this theory-practice gap. We focus on investigating the inherent limitations of the benefits SSL can provide over SL. We develop a framework under which one can analyze the potential benefits, as measured by the sample complexity of SSL. Our framework is utopian in the sense that a SSL algorithm trains on a labeled sample and an unlabeled distribution, as opposed to an unlabeled sample in the usual SSL model. Thus, any lower bound on the sample complexity of SSL in this model implies lower bounds in the usual model. Roughly, our conclusion is that unless the learner is absolutely certain there is some non-trivial relationship between labels and the unlabeled distribution (``SSL type assumption''), SSL cannot provide significant advantages over SL. Technically speaking, we show that the sample complexity of SSL is no more than a constant factor better than SL for any unlabeled distribution, under a no-prior-knowledge setting (i.e. without SSL type assumptions). We prove that for the class of thresholds in the realizable setting the sample complexity of SL is at most twice that of SSL. Also, we prove that in the agnostic setting for the classes of thresholds and union of intervals the sample complexity of SL is at most a constant factor larger than that of SSL. We conjecture this to be a general phenomenon applying to any hypothesis class. We also discuss issues regarding SSL type assumptions, and in particular the popular cluster assumption. We give examples that show even in the most accommodating circumstances, learning under the cluster assumption can be hazardous and lead to prediction performance much worse than simply ignoring the unlabeled data and doing supervised learning. We conclude with a look into future research directions that build on our investigation.

This thesis studies two problems in theoretical machine learning. The first part of the thesis investigates the statistical stability of clustering algorithms. In the second part, we study the relative advantage of having unlabeled data in classification problems. Clustering stability was proposed and used as a model selection method in clustering tasks. The main idea of the method is that from a given data set two independent samples are taken. Each sample individually is clustered with the same clustering algorithm, with the same setting of its parameters. If the two resulting clusterings turn out to be close in some metric, it is concluded that the clustering algorithm and the setting of its parameters match the data set, and that clusterings obtained are meaningful. We study asymptotic properties of this method for certain types of cost minimizing clustering algorithms and relate their asymptotic stability to the number of optimal solutions of the underlying optimization problem. In classification problems, it is often expensive to obtain labeled data, but on the other hand, unlabeled data are often plentiful and cheap. We study how the access to unlabeled data can decrease the amount of labeled data needed in the worst-case sense. We propose an extension of the probably approximately correct (PAC) model in which this question can be naturally studied. We show that for certain basic tasks the access to unlabeled data might, at best, halve the amount of labeled data needed.

This thesis focuses on two fundamental machine learning problems:unsupervised learning, where no label information is available, and semi-supervised learning, where a small amount of labels are given in addition to unlabeled data. These problems arise in many real word applications, such as Web analysis and bioinformatics,where a large amount of data is available, but no or only a small amount of labeled data exists. Obtaining classification labels in these domains is usually quite difficult because it involves either manual labeling or physical experimentation. This thesis approaches these problems from two perspectives: graph based and distribution based. First, I investigate a series of graph based learning algorithms that are able to exploit information embedded in different types of graph structures. These algorithms allow label information to be shared between nodes in the graph---ultimately communicating information globally to yield effective unsupervised and semi-supervised learning. In particular, I extend existing graph based learning algorithms, currently based on undirected graphs, to more general graph types, including directed graphs, hypergraphs and complex networks. These richer graph representations allow one to more naturally capture the intrinsic data relationships that exist, for example, in Web data, relational data, bioinformatics and social networks. For each of these generalized graph structures I show how information propagation can be characterized by distinct random walk models, and then use this characterization to develop new unsupervised and semi-supervised learning algorithms. Second, I investigate a more statistically oriented approach that explicitly models a learning scenario where the training and test examples come from different distributions. This is a difficult situation for standard statistical learning approaches, since they typically incorporate an assumption that the distributions for training and test sets are similar, if not identical. To achieve good performance in this scenario, I utilize unlabeled data to correct the bias between the training and test distributions. A key idea is to produce resampling weights for bias correction by working directly in a feature space and bypassing the problem of explicit density estimation. The technique can be easily applied to many different supervised learning algorithms, automatically adapting their behavior to cope with distribution shifting between training and test data.

The emergence of a new paradigm in machine learning known as semi-supervised learning (SSL) has seen benefits to many applications where labeled data is expensive to obtain. However, unlike supervised learning (SL), which enjoys a rich and deep theoretical foundation, semi-supervised learning, which uses additional unlabeled data for training, still remains a theoretical mystery lacking a sound fundamental understanding. The purpose of this research thesis is to take a first step towards bridging this theory-practice gap. We focus on investigating the inherent limitations of the benefits SSL can provide over SL. We develop a framework under which one can analyze the potential benefits, as measured by the sample complexity of SSL. Our framework is utopian in the sense that a SSL algorithm trains on a labeled sample and an unlabeled distribution, as opposed to an unlabeled sample in the usual SSL model. Thus, any lower bound on the sample complexity of SSL in this model implies lower bounds in the usual model. Roughly, our conclusion is that unless the learner is absolutely certain there is some non-trivial relationship between labels and the unlabeled distribution (``SSL type assumption''), SSL cannot provide significant advantages over SL. Technically speaking, we show that the sample complexity of SSL is no more than a constant factor better than SL for any unlabeled distribution, under a no-prior-knowledge setting (i.e. without SSL type assumptions). We prove that for the class of thresholds in the realizable setting the sample complexity of SL is at most twice that of SSL. Also, we prove that in the agnostic setting for the classes of thresholds and union of intervals the sample complexity of SL is at most a constant factor larger than that of SSL. We conjecture this to be a general phenomenon applying to any hypothesis class. We also discuss issues regarding SSL type assumptions, and in particular the popular cluster assumption. We give examples that show even in the most accommodating circumstances, learning under the cluster assumption can be hazardous and lead to prediction performance much worse than simply ignoring the unlabeled data and doing supervised learning. We conclude with a look into future research directions that build on our investigation.

This thesis studies two problems in theoretical machine learning. The first part of the thesis investigates the statistical stability of clustering algorithms. In the second part, we study the relative advantage of having unlabeled data in classification problems. Clustering stability was proposed and used as a model selection method in clustering tasks. The main idea of the method is that from a given data set two independent samples are taken. Each sample individually is clustered with the same clustering algorithm, with the same setting of its parameters. If the two resulting clusterings turn out to be close in some metric, it is concluded that the clustering algorithm and the setting of its parameters match the data set, and that clusterings obtained are meaningful. We study asymptotic properties of this method for certain types of cost minimizing clustering algorithms and relate their asymptotic stability to the number of optimal solutions of the underlying optimization problem. In classification problems, it is often expensive to obtain labeled data, but on the other hand, unlabeled data are often plentiful and cheap. We study how the access to unlabeled data can decrease the amount of labeled data needed in the worst-case sense. We propose an extension of the probably approximately correct (PAC) model in which this question can be naturally studied. We show that for certain basic tasks the access to unlabeled data might, at best, halve the amount of labeled data needed.

Unsupervised models can provide supplementary soft constraints to help classify new, "target" data since similar instances in the target set are more likely to share the same class label. Such models can also help detect possible differences between training and target distributions, which is useful in applications where concept drift may take place, as in transfer learning settings. This contribution describes two general frameworks that take as input class membership estimates from existing classifiers learnt on previously encountered "source" data, as well as a set of cluster labels from a cluster ensemble operating solely on the target data to be classified, and yield a consensus labeling of the target data. One of the proposed frameworks admits a wide range of loss functions and classification/clustering methods and exploits properties of Bregman divergences in conjunction with Legendre duality to yield a principled and scalable approach. The other approach is built on probabilistic mixture models and provides additional flexibility of distributed computation that is useful when the target data cannot be gathered in a single place for privacy or security concerns. A variety of experiments show that the proposed frameworks can yield results substantially superior to those provided by popular transductive learning techniques or by naively applying classifiers learnt on the original task to the target data. / text

Traditional supervised machine learning algorithms are expected to have access to a large corpus of labeled examples, but the massive amount of data available in the modern world has made unlabeled data much easier to acquire than accompanying labels. Active learning is an extension of the classical paradigm intended to lessen the expense of the labeling process by allowing the learning algorithm to intelligently choose which examples should be labeled. In this dissertation, we demonstrate that the power to make adaptive label queries has benefits beyond reducing labeling effort over passive learning. We develop and explore several novel methods for active learning that exemplify these new capabilities. Some of these methods use active learning for a non-standard purpose, such as computational speedup, structure discovery, and domain adaptation. Others successfully apply active learning in situations where prior results have given evidence of its ineffectiveness. Specifically, we first give an active algorithm for learning disjunctions that is able to overcome a computational intractability present in the semi-supervised version of the same problem. This is the first known example of the computational advantages of active learning. Next, we investigate using active learning to determine structural properties (margins) of the data-generating distribution that can further improve learning rates. This is in contrast to most active learning algorithms which either assume or ignore structure rather than seeking to identify and exploit it. We then give an active nearest neighbors algorithm for domain adaptation, the task of learning a predictor for some target domain using mostly examples from a different source domain. This is the first formal analysis of the generalization and query behavior of an active domain adaptation algorithm. Finally, we show a situation where active learning can outperform passive learning on very noisy data, circumventing prior results that active learning cannot have a significant advantage over passive learning in high-noise regimes.

Many real-world datasets have spatial components that provide valuable information about characteristics of the data. In this dissertation, a novel framework for adaptive models that exploit spatial information in data is proposed. The proposed framework is mainly based on development and applications of Gaussian processes. First, a supervised learning method is proposed for the classification of hyperspectral data with spatially adaptive model parameters. The proposed algorithm models spatially varying means of each spectral band of a given class using a Gaussian process regression model. For a given location, the predictive distribution of a given class is modeled by a multivariate Gaussian distribution with spatially adjusted parameters obtained from the proposed algorithm. The Gaussian process model is generally regarded as a good tool for interpolation, but not for extrapolation. Moreover, the uncertainty of the predictive distribution increases as the distance from the training instances increases. To overcome this problem, a semi-supervised learning algorithm is presented for the classification of hyperspectral data with spatially adaptive model parameters. This algorithm fits the test data with a spatially adaptive mixture-of-Gaussians model, where the spatially varying parameters of each component are obtained by Gaussian process regressions with soft memberships using the mixture-of-Gaussian-processes model. The proposed semi-supervised algorithm assumes a transductive setting, where the unlabeled data is considered to be similar to the training data. This is not true in general, however, since one may not know how many classes may existin the unexplored regions. A spatially adaptive nonparametric Bayesian framework is therefore proposed by applying spatially adaptive mechanisms to the mixture model with infinitely many components. In this method, each component in the mixture has spatially adapted parameters estimated by Gaussian process regressions, and spatial correlations between indicator variables are also considered. In addition to land cover and land use classification applications based on hyperspectral imagery, the Gaussian process-based spatio-temporal model is also applied to predict ground-based aerosol optical depth measurements from satellite multispectral images, and to select the most informative ground-based sites by active learning. In this application, heterogeneous features with spatial and temporal information are incorporated together by employing a set of covariance functions, and it is shown that the spatio-temporal information exploited in this manner substantially improves the regression model. The conventional meaning of spatial information usually refers to actual spatio-temporal locations in the physical world. In the final chapter of this dissertation, the meaning of spatial information is generalized to the parametrized low-dimensional representation of data in feature space, and a corresponding spatial modeling technique is exploited to develop a nearest-manifold classification algorithm. / text

<p>In this dissertation we study combinatorial Hodge Laplacians on simplicial com-</p><p>plexes using tools generalized from spectral graph theory. Specifically, we consider</p><p>generalizations of graph Cheeger numbers and graph random walks. The results in</p><p>this dissertation can be thought of as the beginnings of a new spectral theory for</p><p>simplicial complexes and a new theory of high-dimensional expansion.</p><p>We first consider new high-dimensional isoperimetric constants. A new Cheeger-</p><p>type inequality is proved, under certain conditions, between an isoperimetric constant</p><p>and the smallest eigenvalue of the Laplacian in codimension 0. The proof is similar</p><p>to the proof of the Cheeger inequality for graphs. Furthermore, a negative result is</p><p>proved, using the new Cheeger-type inequality and special examples, showing that</p><p>certain Cheeger-type inequalities cannot hold in codimension 1.</p><p>Second, we consider new random walks with killing on the set of oriented sim-</p><p>plexes of a certain dimension. We show that there is a systematic way of relating</p><p>these walks to combinatorial Laplacians such that a certain notion of mixing time</p><p>is bounded by a spectral gap and such that distributions that are stationary in a</p><p>certain sense relate to the harmonics of the Laplacian. In addition, we consider the</p><p>possibility of using these new random walks for semi-supervised learning. An algo-</p><p>rithm is devised which generalizes a classic label-propagation algorithm on graphs to</p><p>simplicial complexes. This new algorithm applies to a new semi-supervised learning</p><p>problem, one in which the underlying structure to be learned is flow-like.</p> / Dissertation

In this thesis, semantic mapping is understood to be the process of putting a tag or label on objects or regions in a map. This label should be interpretable by and have a meaning for a human. The use of semantic information has several application areas in mobile robotics. The largest area is in human-robot interaction where the semantics is necessary for a common understanding between robot and human of the operational environment. Other areas include localization through connection of human spatial concepts to particular locations, improving 3D models of indoor and outdoor environments, and model validation. This thesis investigates the extraction of semantic information for mobile robots in outdoor environments and the use of semantic information to link ground-level occupancy maps and aerial images. The thesis concentrates on three related issues: i) recognition of human spatial concepts in a scene, ii) the ability to incorporate semantic knowledge in a map, and iii) the ability to connect information collected by a mobile robot with information extracted from an aerial image. The first issue deals with a vision-based virtual sensor for classification of views (images). The images are fed into a set of learned virtual sensors, where each virtual sensor is trained for classification of a particular type of human spatial concept. The virtual sensors are evaluated with images from both ordinary cameras and an omni-directional camera, showing robust properties that can cope with variations such as changing season. In the second part a probabilistic semantic map is computed based on an occupancy grid map and the output from a virtual sensor. A local semantic map is built around the robot for each position where images have been acquired. This map is a grid map augmented with semantic information in the form of probabilities that the occupied grid cells belong to a particular class. The local maps are fused into a global probabilistic semantic map covering the area along the trajectory of the mobile robot. In the third part information extracted from an aerial image is used to improve the mapping process. Region and object boundaries taken from the probabilistic semantic map are used to initialize segmentation of the aerial image. Algorithms for both local segmentation related to the borders and global segmentation of the entire aerial image, exemplified with the two classes ground and buildings, are presented. Ground-level semantic information allows focusing of the segmentation of the aerial image to desired classes and generation of a semantic map that covers a larger area than can be built using only the onboard sensors.