From a fresh data science perspective, this thesis discusses the prediction of coronary artery disease based on genetic variations at the DNA base pair level, called Single-Nucleotide Polymorphisms (SNPs), collected from the Ontario Heart Genomics Study (OHGS). First, the thesis explains two commonly used supervised learning algorithms, the k-Nearest Neighbour (k-NN) and Random Forest classifiers, and includes a complete proof that the k-NN classifier is universally consistent in any finite dimensional normed vector space. Second, the thesis introduces two dimensionality reduction steps, Random Projections, a known feature extraction technique based on the Johnson-Lindenstrauss lemma, and a new method termed Mass Transportation Distance (MTD) Feature Selection for discrete domains. Then, this thesis compares the performance of Random Projections with the k-NN classifier against MTD Feature Selection and Random Forest, for predicting artery disease based on accuracy, the F-Measure, and area under the Receiver Operating Characteristic (ROC) curve. The comparative results demonstrate that MTD Feature Selection with Random Forest is vastly superior to Random Projections and k-NN. The Random Forest classifier is able to obtain an accuracy of 0.6660 and an area under the ROC curve of 0.8562 on the OHGS genetic dataset, when 3335 SNPs are selected by MTD Feature Selection for classification. This area is considerably better than the previous high score of 0.608 obtained by Davies et al. in 2010 on the same dataset.

Much of human knowledge sits in large databases of unstructured text. Leveraging this knowledge requires algorithms that extract and record metadata on unstructured text documents. Assigning topics to documents will enable intelligent search, statistical characterization, and meaningful classification. Latent Dirichlet allocation (LDA) is the state-of-the-art in topic classification. Here, we perform a systematic theoretical and numerical analysis that demonstrates that current optimization techniques for LDA often yield results which are not accurate in inferring the most suitable model parameters. Adapting approaches for community detection in networks, we propose a new algorithm which displays high-reproducibility and high-accuracy, and also has high computational efficiency. We apply it to a large set of documents in the English Wikipedia and reveal its hierarchical structure. Our algorithm promises to make "big data" text analysis systems more reliable.

Given a hypothesis space, the large volume principle by Vladimir Vapnik prioritizes equivalence classes according to their volume in the hypothesis space. The volume approximation has hitherto been successfully applied to binary learning problems. In this paper, we extend it naturally to a more general definition which can be applied to several transductive problem settings, such as multi-class, multi-label and serendipitous learning. Even though the resultant learning method involves a non-convex optimization problem, the globally optimal solution is almost surely unique and can be obtained in O(n^3) time. We theoretically provide stability and error analyses for the proposed method, and then experimentally show that it is promising.

Thompson Sampling is one of the oldest heuristics for multi-armed bandit problems. It is a randomized algorithm based on Bayesian ideas, and has recently generated significant interest after several studies demonstrated it to have better empirical performance compared to the state-of-the-art methods. However, many questions regarding its theoretical performance remained open. In this paper, we design and analyze a generalization of Thompson Sampling algorithm for the stochastic contextual multi-armed bandit problem with linear payoff functions, when the contexts are provided by an adaptive adversary. This is among the most important and widely studied versions of the contextual bandits problem. We provide the first theoretical guarantees for the contextual version of Thompson Sampling. We prove a high probability regret bound of $\tilde{O}(d^{3/2}\sqrt{T})$ (or $\tilde{O}(d\sqrt{T \log(N)})$), which is the best regret bound achieved by any computationally efficient algorithm available for this problem in the current literature, and is within a factor of $\sqrt{d}$ (or $\sqrt{\log(N)}$) of the information-theoretic lower bound for this problem.

For a certain class of distributions, we prove that the linear programming relaxation of $k$-medoids clustering---a variant of $k$-means clustering where means are replaced by exemplars from within the dataset---distinguishes points drawn from nonoverlapping balls with high probability once the number of points drawn and the separation distance between any two balls are sufficiently large. Our results hold in the nontrivial regime where the separation distance is small enough that points drawn from different balls may be closer to each other than points drawn from the same ball; in this case, clustering by thresholding pairwise distances between points can fail. We also exhibit numerical evidence of high-probability recovery in a substantially more permissive regime.

The challenge of object categorization in images is largely due to arbitrary translations and scales of the foreground objects. To attack this difficulty, we propose a new approach called collaborative receptive field learning to extract specific receptive fields (RF's) or regions from multiple images, and the selected RF's are supposed to focus on the foreground objects of a common category. To this end, we solve the problem by maximizing a submodular function over a similarity graph constructed by a pool of RF candidates. However, measuring pairwise distance of RF's for building the similarity graph is a nontrivial problem. Hence, we introduce a similarity metric called pyramid-error distance (PED) to measure their pairwise distances through summing up pyramid-like matching errors over a set of low-level features. Besides, in consistent with the proposed PED, we construct a simple nonparametric classifier for classification. Experimental results show that our method effectively discovers the foreground objects in images, and improves classification performance.

This paper explores combinatorial optimization for problems of max-weight graph matching on multi-partite graphs, which arise in integrating multiple data sources. Entity resolution-the data integration problem of performing noisy joins on structured data-typically proceeds by first hashing each record into zero or more blocks, scoring pairs of records that are co-blocked for similarity, and then matching pairs of sufficient similarity. In the most common case of matching two sources, it is often desirable for the final matching to be one-to-one (a record may be matched with at most one other); members of the database and statistical record linkage communities accomplish such matchings in the final stage by weighted bipartite graph matching on similarity scores. Such matchings are intuitively appealing: they leverage a natural global property of many real-world entity stores-that of being nearly deduped-and are known to provide significant improvements to precision and recall. Unfortunately unlike the bipartite case, exact max-weight matching on multi-partite graphs is known to be NP-hard. Our two-fold algorithmic contributions approximate multi-partite max-weight matching: our first algorithm borrows optimization techniques common to Bayesian probabilistic inference; our second is a greedy approximation algorithm. In addition to a theoretical guarantee on the latter, we present comparisons on a real-world ER problem from Bing significantly larger than typically found in the literature, publication data, and on a series of synthetic problems. Our results quantify significant improvements due to exploiting multiple sources, which are made possible by global one-to-one constraints linking otherwise independent matching sub-problems. We also discover that our algorithms are complementary: one being much more robust under noise, and the other being simple to implement and very fast to run.

The Kaczmarz algorithm is popular for iteratively solving an overdetermined system of linear equations. The traditional Kaczmarz algorithm can approximate the solution in few sweeps through the equations but a randomized version of the Kaczmarz algorithm was shown to converge exponentially and independent of number of equations. Recently an algorithm for finding sparse solution to a linear system of equations has been proposed based on weighted randomized Kaczmarz algorithm. These algorithms solves single measurement vector problem; however there are applications were multiple-measurements are available. In this work, the objective is to solve a multiple measurement vector problem with common sparse support by modifying the randomized Kaczmarz algorithm. We have also modeled the problem of face recognition from video as the multiple measurement vector problem and solved using our proposed technique. We have compared the proposed algorithm with state-of-art spectral projected gradient algorithm for multiple measurement vectors on both real and synthetic datasets. The Monte Carlo simulations confirms that our proposed algorithm have better recovery and convergence rate than the MMV version of spectral projected gradient algorithm under fairness constraints.

In this paper, we propose a theory which unifies kernel learning and symbolic algebraic methods. We show that both worlds are inherently dual to each other, and we use this duality to combine the structure-awareness of algebraic methods with the efficiency and generality of kernels. The main idea lies in relating polynomial rings to feature space, and ideals to manifolds, then exploiting this generative-discriminative duality on kernel matrices. We illustrate this by proposing two algorithms, IPCA and AVICA, for simultaneous manifold and feature learning, and test their accuracy on synthetic and real world data.

Infinite Tucker Decomposition (InfTucker) and random function prior models, as nonparametric Bayesian models on infinite exchangeable arrays, are more powerful models than widely-used multilinear factorization methods including Tucker and PARAFAC decomposition, (partly) due to their capability of modeling nonlinear relationships between array elements. Despite their great predictive performance and sound theoretical foundations, they cannot handle massive data due to a prohibitively high training time. To overcome this limitation, we present Distributed Infinite Tucker (DINTUCKER), a large-scale nonlinear tensor decomposition algorithm on MAPREDUCE. While maintaining the predictive accuracy of InfTucker, it is scalable on massive data. DINTUCKER is based on a new hierarchical Bayesian model that enables local training of InfTucker on subarrays and information integration from all local training results. We use distributed stochastic gradient descent, coupled with variational inference, to train this model. We apply DINTUCKER to multidimensional arrays with billions of elements from applications in the "Read the Web" project (Carlson et al., 2010) and in information security and compare it with the state-of-the-art large-scale tensor decomposition method, GigaTensor. On both datasets, DINTUCKER achieves significantly higher prediction accuracy with less computational time.