Population-Guided Large Margin Classifier for High-Dimension Low -Sample-Size Problems

Various applications in different fields, such as gene expression analysis or computer vision, suffer from data sets with high-dimensional low-sample-size (HDLSS), which has posed significant challenges for standard statistical and modern machine learning methods. In this paper, we propose a novel linear binary classifier, denoted by population-guided large margin classifier (PGLMC), which is applicable to any sorts of data, including HDLSS. PGLMC is conceived with a projecting direction w given by the comprehensive consideration of local structural information of the hyperplane and the statistics of the training samples. Our proposed model has several advantages compared to those widely used approaches. First, it is not sensitive to the intercept term b. Second, it operates well with imbalanced data. Third, it is relatively simple to be implemented based on Quadratic Programming. Fourth, it is robust to the model specification for various real applications. The theoretical properties of PGLMC are proven. We conduct a series of evaluations on two simulated and six real-world benchmark data sets, including DNA classification, digit recognition, medical image analysis, and face recognition. PGLMC outperforms the state-of-the-art classification methods in most cases, or at least obtains comparable results.Comment: 48 Page

Another Look at DWD: Thrifty Algorithm and Bayes Risk Consistency in RKHS

Distance weighted discrimination (DWD) is a margin-based classifier with an interesting geometric motivation. DWD was originally proposed as a superior alternative to the support vector machine (SVM), however DWD is yet to be popular compared with the SVM. The main reasons are twofold. First, the state-of-the-art algorithm for solving DWD is based on the second-order-cone programming (SOCP), while the SVM is a quadratic programming problem which is much more efficient to solve. Second, the current statistical theory of DWD mainly focuses on the linear DWD for the high-dimension-low-sample-size setting and data-piling, while the learning theory for the SVM mainly focuses on the Bayes risk consistency of the kernel SVM. In fact, the Bayes risk consistency of DWD is presented as an open problem in the original DWD paper. In this work, we advance the current understanding of DWD from both computational and theoretical perspectives. We propose a novel efficient algorithm for solving DWD, and our algorithm can be several hundred times faster than the existing state-of-the-art algorithm based on the SOCP. In addition, our algorithm can handle the generalized DWD, while the SOCP algorithm only works well for a special DWD but not the generalized DWD. Furthermore, we consider a natural kernel DWD in a reproducing kernel Hilbert space and then establish the Bayes risk consistency of the kernel DWD. We compare DWD and the SVM on several benchmark data sets and show that the two have comparable classification accuracy, but DWD equipped with our new algorithm can be much faster to compute than the SVM

Weighted second-order cone programming twin support vector machine for imbalanced data classification

We propose a method of using a Weighted second-order cone programming twin support vector machine (WSOCP-TWSVM) for imbalanced data classification. This method constructs a graph based under-sampling method which is utilized to remove outliers and reduce the dispensable majority samples. Then, appropriate weights are set in order to decrease the impact of samples of the majority class and increase the effect of the minority class in the optimization formula of the classifier. These weights are embedded in the optimization problem of the Second Order Cone Programming (SOCP) Twin Support Vector Machine formulations. This method is tested, and its performance is compared to previous methods on standard datasets. Results of experiments confirm the feasibility and efficiency of the proposed method.Comment: This manuscript is under revision at Pattern Recognition Letter

Optimal arrangements of hyperplanes for multiclass classification

In this paper, we present a novel approach to construct multiclass classifiers by means of arrangements of hyperplanes. We propose different mixed integer (linear and non linear) programming formulations for the problem using extensions of widely used measures for misclassifying observations where the \textit{kernel trick} can be adapted to be applicable. Some dimensionality reductions and variable fixing strategies are also developed for these models. An extensive battery of experiments has been run which reveal the powerfulness of our proposal as compared with other previously proposed methodologies.Comment: 8 Figures, 2 Table

Sparse Distance Weighted Discrimination

Distance weighted discrimination (DWD) was originally proposed to handle the data piling issue in the support vector machine. In this paper, we consider the sparse penalized DWD for high-dimensional classification. The state-of-the-art algorithm for solving the standard DWD is based on second-order cone programming, however such an algorithm does not work well for the sparse penalized DWD with high-dimensional data. In order to overcome the challenging computation difficulty, we develop a very efficient algorithm to compute the solution path of the sparse DWD at a given fine grid of regularization parameters. We implement the algorithm in a publicly available R package sdwd. We conduct extensive numerical experiments to demonstrate the computational efficiency and classification performance of our method.Comment: 16 page

Transductive Optimization of Top k Precision

Consider a binary classification problem in which the learner is given a labeled training set, an unlabeled test set, and is restricted to choosing exactly $k$ test points to output as positive predictions. Problems of this kind---{\it transductive precision@$k$}---arise in information retrieval, digital advertising, and reserve design for endangered species. Previous methods separate the training of the model from its use in scoring the test points. This paper introduces a new approach, Transductive Top K (TTK), that seeks to minimize the hinge loss over all training instances under the constraint that exactly $k$ test instances are predicted as positive. The paper presents two optimization methods for this challenging problem. Experiments and analysis confirm the importance of incorporating the knowledge of $k$ into the learning process. Experimental evaluations of the TTK approach show that the performance of TTK matches or exceeds existing state-of-the-art methods on 7 UCI datasets and 3 reserve design problem instances

Flexible High-dimensional Classification Machines and Their Asymptotic Properties

Classification is an important topic in statistics and machine learning with great potential in many real applications. In this paper, we investigate two popular large margin classification methods, Support Vector Machine (SVM) and Distance Weighted Discrimination (DWD), under two contexts: the high-dimensional, low-sample size data and the imbalanced data. A unified family of classification machines, the FLexible Assortment MachinE (FLAME) is proposed, within which DWD and SVM are special cases. The FLAME family helps to identify the similarities and differences between SVM and DWD. It is well known that many classifiers overfit the data in the high-dimensional setting; and others are sensitive to the imbalanced data, that is, the class with a larger sample size overly influences the classifier and pushes the decision boundary towards the minority class. SVM is resistant to the imbalanced data issue, but it overfits high-dimensional data sets by showing the undesired data-piling phenomena. The DWD method was proposed to improve SVM in the high-dimensional setting, but its decision boundary is sensitive to the imbalanced ratio of sample sizes. Our FLAME family helps to understand an intrinsic connection between SVM and DWD, and improves both methods by providing a better trade-off between sensitivity to the imbalanced data and overfitting the high-dimensional data. Several asymptotic properties of the FLAME classifiers are studied. Simulations and real data applications are investigated to illustrate the usefulness of the FLAME classifiers.Comment: 49 pages, 11 figure

Gaussian Robust Classification

Supervised learning is all about the ability to generalize knowledge. Specifically, the goal of the learning is to train a classifier using training data, in such a way that it will be capable of classifying new unseen data correctly. In order to acheive this goal, it is important to carefully design the learner, so it will not overfit the training data. The later can is done usually by adding a regularization term. The statistical learning theory explains the success of this method by claiming that it restricts the complexity of the learned model. This explanation, however, is rather abstract and does not have a geometric intuition. The generalization error of a classifier may be thought of as correlated with its robustness to perturbations of the data: a classifier that copes with disturbance is expected to generalize well. Indeed, Xu et al. [2009] have shown that the SVM formulation is equivalent to a robust optimization (RO) formulation, in which an adversary displaces the training and testing points within a ball of pre-determined radius. In this work we explore a different kind of robustness, namely changing each data point with a Gaussian cloud centered at the sample. Loss is evaluated as the expectation of an underlying loss function on the cloud. This setup fits the fact that in many applications, the data is sampled along with noise. We develop an RO framework, in which the adversary chooses the covariance of the noise. In our algorithm named GURU, the tuning parameter is a spectral bound on the noise, thus it can be estimated using physical or applicative considerations. Our experiments show that this framework performs as well as SVM and even slightly better in some cases. Generalizations for Mercer kernels and for the multiclass case are presented as well. We also show that our framework may be further generalized, using the technique of convex perspective functions.Comment: Master's dissertation of the first author, carried out under the supervision of the second autho

A Summary Of The Kernel Matrix, And How To Learn It Effectively Using Semidefinite Programming

Kernel-based learning algorithms are widely used in machine learning for problems that make use of the similarity between object pairs. Such algorithms first embed all data points into an alternative space, where the inner product between object pairs specifies their distance in the embedding space. Applying kernel methods to partially labeled datasets is a classical challenge in this regard, requiring that the distances between unlabeled pairs must somehow be learnt using the labeled data. In this independent study, I will summarize the work of G. Lanckriet et al.'s work on "Learning the Kernel Matrix with Semidefinite Programming" used in support vector machines (SVM) algorithms for the transduction problem. Throughout the report, I have provide alternative explanations / derivations / analysis related to this work which is designed to ease the understanding of the original article.Comment: Independent study / summary, 20 pages total (14 pages main body, 6 pages appendices and proofs

Fast SVM-based Feature Elimination Utilizing Data Radius, Hard-Margin, Soft-Margin

Margin maximization in the hard-margin sense, proposed as feature elimination criterion by the MFE-LO method, is combined here with data radius utilization to further aim to lower generalization error, as several published bounds and bound-related formulations pertaining to lowering misclassification risk (or error) pertain to radius e.g. product of squared radius and weight vector squared norm. Additionally, we propose additional novel feature elimination criteria that, while instead being in the soft-margin sense, too can utilize data radius, utilizing previously published bound-related formulations for approaching radius for the soft-margin sense, whereby e.g. a focus was on the principle stated therein as "finding a bound whose minima are in a region with small leave-one-out values may be more important than its tightness". These additional criteria we propose combine radius utilization with a novel and computationally low-cost soft-margin light classifier retraining approach we devise named QP1; QP1 is the soft-margin alternative to the hard-margin LO. We correct an error in the MFE-LO description, find MFE-LO achieves the highest generalization accuracy among the previously published margin-based feature elimination (MFE) methods, discuss some limitations of MFE-LO, and find our novel methods herein outperform MFE-LO, attain lower test set classification error rate. On several datasets that each both have a large number of features and fall into the `large features few samples' dataset category, and on datasets with lower (low-to-intermediate) number of features, our novel methods give promising results. Especially, among our methods the tunable ones, that do not employ (the non-tunable) LO approach, can be tuned more aggressively in the future than herein, to aim to demonstrate for them even higher performance than herein.Comment: Incomplete but good, again. To Apr 28 version, made few misc text and notation improvements including typo corrections, probably mostly in Appendix, but probably best to read in whole again. New results for one of the datasets (Leukemia gene dataset