Yuh-Jye Lee

SSVM: A Smooth Support Vector Machine for Classification

Smoothing methods, extensively used for solving important mathematical programming problems and applications, are applied here to generate and solve an unconstrained smooth reformulation of the support vector machine for pattern classification using a completely arbitrary kernel. We term such reformulation a smooth support vector machine (SSVM). A fast Newton–Armijo algorithm for solving the SSVM converges globally and quadratically. Numerical results and comparisons are given to demonstrate the effectiveness and speed of the algorithm. On six publicly available datasets, tenfold cross validation correctness of SSVM was the highest compared with four other methods as well as the fastest. On larger problems, SSVM was comparable or faster than SVMlight (T. Joachims, in Advances in Kernel Methods—Support Vector Learning, MIT Press: Cambridge, MA, 1999), SOR (O.L. Mangasarian and David R. Musicant, IEEE Transactions on Neural Networks, vol. 10, pp. 1032–1037, 1999) and SMO (J. Platt, in Advances in Kernel Methods—Support Vector Learning, MIT Press: Cambridge, MA, 1999). SSVM can also generate a highly nonlinear separating surface such as a checkerboard.

Reduced Support Vector Machines: A Statistical Theory

In dealing with large data sets, the reduced support vector machine (RSVM) was proposed for the practical objective to overcome some computational difficulties as well as to reduce the model complexity. In this paper, we study the RSVM from the viewpoint of sampling design, its robustness, and the spectral analysis of the reduced kernel. We consider the nonlinear separating surface as a mixture of kernels. Instead of a full model, the RSVM uses a reduced mixture with kernels sampled from certain candidate set. Our main results center on two major themes. One is the robustness of the random subset mixture model. The other is the spectral analysis of the reduced kernel. The robustness is judged by a few criteria as follows: 1) model variation measure; 2) model bias (deviation) between the reduced model and the full model; and 3) test power in distinguishing the reduced model from the full one. For the spectral analysis, we compare the eigenstructures of the full kernel matrix and the approximation kernel matrix. The approximation kernels are generated by uniform random subsets. The small discrepancies between them indicate that the approximation kernels can retain most of the relevant information for learning tasks in the full kernel. We focus on some statistical theory of the reduced set method mainly in the context of the RSVM. The use of a uniform random subset is not limited to the RSVM. This approach can act as a supplemental algorithm on top of a basic optimization algorithm, wherein the actual optimization takes place on the subset-approximated data. The statistical properties discussed in this paper are still valid

Model selection for support vector machines via uniform design

The problem of choosing a good parameter setting for a better generalization performance in a learning task is the so-called model selection. A nested uniform design (UD) methodology is proposed for efficient, robust and automatic model selection for support vector machines (SVMs). The proposed method is applied to select the candidate set of parameter combinations and carry out a k-fold cross-validation to evaluate the generalization performance of each parameter combination. In contrast to conventional exhaustive grid search, this method can be treated as a deterministic analog of random search. It can dramatically cut down the number of parameter trials and also provide the flexibility to adjust the candidate set size under computational time constraint. The key theoretic advantage of the UD model selection over the grid search is that the UD points are “far more uniform”and “far more space filling” than lattice grid points. The better uniformity and space-filling phenomena make the UD selection scheme more efficient by avoiding wasteful function evaluations of close-by patterns. The proposed method is evaluated on different learning tasks, different data sets as well as different SVM algorithms.

-SSVR: A Smooth Support Vector Machine for -Insensitive Regression

A new smoothing strategy for solving /spl epsi/-support vector regression (/spl epsi/-SVR), tolerating a small error in fitting a given data set linearly or nonlinearly, is proposed in this paper. Conventionally, /spl epsi/-SVR is formulated as a constrained minimization problem, namely, a convex quadratic programming problem. We apply the smoothing techniques that have been used for solving the support vector machine for classification, to replace the /spl epsi/-insensitive loss function by an accurate smooth approximation. This will allow us to solve /spl epsi/-SVR as an unconstrained minimization problem directly. We term this reformulated problem as /spl epsi/-smooth support vector regression (/spl epsi/-SSVR). We also prescribe a Newton-Armijo algorithm that has been shown to be convergent globally and quadratically to solve our /spl epsi/-SSVR. In order to handle the case of nonlinear regression with a massive data set, we also introduce the reduced kernel technique in this paper to avoid the computational difficulties in dealing with a huge and fully dense kernel matrix. Numerical results and comparisons are given to demonstrate the effectiveness and speed of the algorithm.

Anomaly Detection via Online Oversampling Principal Component Analysis

Anomaly detection has been an important research topic in data mining and machine learning. Many real-world applications such as intrusion or credit card fraud detection require an effective and efficient framework to identify deviated data instances. However, most anomaly detection methods are typically implemented in batch mode, and thus cannot be easily extended to large-scale problems without sacrificing computation and memory requirements. In this paper, we propose an online oversampling principal component analysis (osPCA) algorithm to address this problem, and we aim at detecting the presence of outliers from a large amount of data via an online updating technique. Unlike prior principal component analysis (PCA)-based approaches, we do not store the entire data matrix or covariance matrix, and thus our approach is especially of interest in online or large-scale problems. By oversampling the target instance and extracting the principal direction of the data, the proposed osPCA allows us to determine the anomaly of the target instance according to the variation of the resulting dominant eigenvector. Since our osPCA need not perform eigen analysis explicitly, the proposed framework is favored for online applications which have computation or memory limitations. Compared with the well-known power method for PCA and other popular anomaly detection algorithms, our experimental results verify the feasibility of our proposed method in terms of both accuracy and efficiency.

/spl epsi/-SSVR: a smooth support vector machine for /spl epsi/-insensitive regression

A new smoothing strategy for solving /spl epsi/-support vector regression (/spl epsi/-SVR), tolerating a small error in fitting a given data set linearly or nonlinearly, is proposed in this paper. Conventionally, /spl epsi/-SVR is formulated as a constrained minimization problem, namely, a convex quadratic programming problem. We apply the smoothing techniques that have been used for solving the support vector machine for classification, to replace the /spl epsi/-insensitive loss function by an accurate smooth approximation. This will allow us to solve /spl epsi/-SVR as an unconstrained minimization problem directly. We term this reformulated problem as /spl epsi/-smooth support vector regression (/spl epsi/-SSVR). We also prescribe a Newton-Armijo algorithm that has been shown to be convergent globally and quadratically to solve our /spl epsi/-SSVR. In order to handle the case of nonlinear regression with a massive data set, we also introduce the reduced kernel technique in this paper to avoid the computational difficulties in dealing with a huge and fully dense kernel matrix. Numerical results and comparisons are given to demonstrate the effectiveness and speed of the algorithm.

Locality-constrained group sparse representation for robust face recognition

This paper presents a novel sparse representation for robust face recognition. We advance both group sparsity and data locality and formulate a unified optimization framework, which produces a locality and group sensitive sparse representation (LGSR) for improved recognition. Empirical results confirm that our LGSR not only outperforms state-of-the-art sparse coding based image classification methods, our approach is robust to variations such as lighting, pose, and facial details (glasses or not), which are typically seen in real-world face recognition problems.

Survival-Time Classification of Breast Cancer Patients

The identification of breast cancer patients for whom chemotherapy could prolong survival time is treated here as a data mining problem. This identification is achieved by clustering 253 breast cancer patients into three prognostic groups: Good, Poor and Intermediate. Each of the three groups has a significantly distinct Kaplan-Meier survival curve. Of particular significance is the Intermediate group, because patients with chemotherapy in this group do better than those without chemotherapy in the same group. This is the reverse case to that of the overall population of 253 patients for which patients undergoing chemotherapy have worse survival than those who do not. We also prescribe a procedure that utilizes three nonlinear smooth support vector machines (SSVMs) for classifying breast cancer patients into the three above prognostic groups. These results suggest that the patients in the Good group should not receive chemotherapy while those in the Intermediate group should receive chemotherapy based on our survival curve analysis. To our knowledge this is the first instance of a classifiable group of breast cancer patients for which chemotherapy can possibly enhance survival.

Nonlinear Dimension Reduction with Kernel Sliced Inverse Regression

Sliced inverse regression (SIR) is a renowned dimension reduction method for finding an effective low-dimensional linear subspace. Like many other linear methods, SIR can be extended to nonlinear setting via the ldquokernel trick.rdquo The main purpose of this paper is two-fold. We build kernel SIR in a reproducing kernel Hilbert space rigorously for a more intuitive model explanation and theoretical development. The second focus is on the implementation algorithm of kernel SIR for fast computation and numerical stability. We adopt a low-rank approximation to approximate the huge and dense full kernel covariance matrix and a reduced singular value decomposition technique for extracting kernel SIR directions. We also explore kernel SIRs ability to combine with other linear learning algorithms for classification and regression including multiresponse regression. Numerical experiments show that kernel SIR is an effective kernel tool for nonlinear dimension reduction and it can easily combine with other linear algorithms to form a powerful toolkit for nonlinear data analysis.

Variant Methods of Reduced Set Selection for Reduced Support Vector Machines

In dealing with large datasets the reduced support vector machine (RSVM) was proposed for the practical objective to overcome the computational difficulties as well as to reduce the model complexity. In this paper, we propose two new approaches to generate representative reduced set for RSVM. First, we introduce Clustering Reduced Support Vector Machine (CRSVM) that builds the model of RSVM via RBF (Gaussian kernel) construction. Applying clustering algorithm to each class, we can generate cluster centroids of each class and use them to form the reduced set which is used in RSVM. We also estimate the approximate density for each cluster to get the parameter used in Gaussian kernel which will save a lot of tuning time. Secondly, we present Systematic Sampling RSVM (SSRSVM) that incrementally selects the informative data points to form the reduced set while the RSVM used random selection scheme. SSRSVM starts with an extremely small initial reduced set and adds a portion of misclassified points into the reduced set iteratively based on the current classifier until the validation set correctness is large enough. We also show our methods, CRSVM and SSRSVM with smaller size of reduced set, have superior performance than the original random selection scheme.