Classification with Large Sparse Datasets: Convergence Analysis and Scalable Algorithms

Large and sparse datasets, such as user ratings over a large collection of items, are common in the big data era. Many applications need to classify the users or items based on the high-dimensional and sparse data vectors, e.g., to predict the profitability of a product or the age group of a user, etc. Linear classifiers are popular choices for classifying such datasets because of their efficiency. In order to classify the large sparse data more effectively, the following important questions need to be answered. 1. Sparse data and convergence behavior. How different properties of a dataset, such as the sparsity rate and the mechanism of missing data systematically affect convergence behavior of classification? 2. Handling sparse data with non-linear model. How to efficiently learn non-linear data structures when classifying large sparse data? This thesis attempts to address these questions with empirical and theoretical analysis on large and sparse datasets. We begin by studying the convergence behavior of popular classifiers on large and sparse data. It is known that a classifier gains better generalization ability after learning more and more training examples. Eventually, it will converge to the best generalization performance with respect to a given data distribution. In this thesis, we focus on how the sparsity rate and the missing data mechanism systematically affect such convergence behavior. Our study covers different types of classification models, including generative classifier and discriminative linear classifiers. To systematically explore the convergence behaviors, we use synthetic data sampled from statistical models of real-world large sparse datasets. We consider different types of missing data mechanisms that are common in practice. From the experiments, we have several useful observations about the convergence behavior of classifying large sparse data. Based on these observations, we further investigate the theoretical reasons and come to a series of useful conclusions. For better applicability, we provide practical guidelines for applying our results in practice. Our study helps to answer whether obtaining more data or missing values in the data is worthwhile in different situations, which is useful for efficient data collection and preparation. Despite being efficient, linear classifiers cannot learn the non-linear structures such as the low-rankness in a dataset. As a result, its accuracy may suffer. Meanwhile, most non-linear methods such as the kernel machines cannot scale to very large and high-dimensional datasets. The third part of this thesis studies how to efficiently learn non-linear structures in large sparse data. Towards this goal, we develop novel scalable feature mappings that can achieve better accuracy than linear classification. We demonstrate that the proposed methods not only outperform linear classification but is also scalable to large and sparse datasets with moderate memory and computation requirement. The main contribution of this thesis is to answer important questions on classifying large and sparse datasets. On the one hand, we study the convergence behavior of widely used classifiers under different missing data mechanisms; on the other hand, we develop efficient methods to learn the non-linear structures in large sparse data and improve classification accuracy. Overall, the thesis not only provides practical guidance for the convergence behavior of classifying large sparse datasets, but also develops highly efficient algorithms for classifying large sparse datasets in practice

Sparse Bayesian Learning with Diagonal Quasi-Newton Method for Large Scale Classification

Sparse Bayesian Learning (SBL) constructs an extremely sparse probabilistic model with very competitive generalization. However, SBL needs to invert a big covariance matrix with complexity O(M^3 ) (M: feature size) for updating the regularization priors, making it difficult for practical use. There are three issues in SBL: 1) Inverting the covariance matrix may obtain singular solutions in some cases, which hinders SBL from convergence; 2) Poor scalability to problems with high dimensional feature space or large data size; 3) SBL easily suffers from memory overflow for large-scale data. This paper addresses these issues with a newly proposed diagonal Quasi-Newton (DQN) method for SBL called DQN-SBL where the inversion of big covariance matrix is ignored so that the complexity and memory storage are reduced to O(M). The DQN-SBL is thoroughly evaluated on non-linear classifiers and linear feature selection using various benchmark datasets of different sizes. Experimental results verify that DQN-SBL receives competitive generalization with a very sparse model and scales well to large-scale problems.Comment: 11 pages,5 figure

Predicting diabetes-related hospitalizations based on electronic health records

OBJECTIVE: To derive a predictive model to identify patients likely to be hospitalized during the following year due to complications attributed to Type II diabetes. METHODS: A variety of supervised machine learning classification methods were tested and a new method that discovers hidden patient clusters in the positive class (hospitalized) was developed while, at the same time, sparse linear support vector machine classifiers were derived to separate positive samples from the negative ones (non-hospitalized). The convergence of the new method was established and theoretical guarantees were proved on how the classifiers it produces generalize to a test set not seen during training. RESULTS: The methods were tested on a large set of patients from the Boston Medical Center - the largest safety net hospital in New England. It is found that our new joint clustering/classification method achieves an accuracy of 89% (measured in terms of area under the ROC Curve) and yields informative clusters which can help interpret the classification results, thus increasing the trust of physicians to the algorithmic output and providing some guidance towards preventive measures. While it is possible to increase accuracy to 92% with other methods, this comes with increased computational cost and lack of interpretability. The analysis shows that even a modest probability of preventive actions being effective (more than 19%) suffices to generate significant hospital care savings. CONCLUSIONS: Predictive models are proposed that can help avert hospitalizations, improve health outcomes and drastically reduce hospital expenditures. The scope for savings is significant as it has been estimated that in the USA alone, about $5.8 billion are spent each year on diabetes-related hospitalizations that could be prevented.Accepted manuscrip

Binary Linear Classification and Feature Selection via Generalized Approximate Message Passing

For the problem of binary linear classification and feature selection, we propose algorithmic approaches to classifier design based on the generalized approximate message passing (GAMP) algorithm, recently proposed in the context of compressive sensing. We are particularly motivated by problems where the number of features greatly exceeds the number of training examples, but where only a few features suffice for accurate classification. We show that sum-product GAMP can be used to (approximately) minimize the classification error rate and max-sum GAMP can be used to minimize a wide variety of regularized loss functions. Furthermore, we describe an expectation-maximization (EM)-based scheme to learn the associated model parameters online, as an alternative to cross-validation, and we show that GAMP's state-evolution framework can be used to accurately predict the misclassification rate. Finally, we present a detailed numerical study to confirm the accuracy, speed, and flexibility afforded by our GAMP-based approaches to binary linear classification and feature selection

A Direct Estimation Approach to Sparse Linear Discriminant Analysis

This paper considers sparse linear discriminant analysis of high-dimensional data. In contrast to the existing methods which are based on separate estimation of the precision matrix \O and the difference \de of the mean vectors, we introduce a simple and effective classifier by estimating the product \O\de directly through constrained $\ell_1$ minimization. The estimator can be implemented efficiently using linear programming and the resulting classifier is called the linear programming discriminant (LPD) rule. The LPD rule is shown to have desirable theoretical and numerical properties. It exploits the approximate sparsity of \O\de and as a consequence allows cases where it can still perform well even when \O and/or \de cannot be estimated consistently. Asymptotic properties of the LPD rule are investigated and consistency and rate of convergence results are given. The LPD classifier has superior finite sample performance and significant computational advantages over the existing methods that require separate estimation of \O and \de. The LPD rule is also applied to analyze real datasets from lung cancer and leukemia studies. The classifier performs favorably in comparison to existing methods.Comment: 39 pages.To appear in Journal of the American Statistical Associatio