Smooth Support Vector Machines
نویسنده
چکیده
Smoothing methods, extensively used for solving important mathematical programming problems and applications, are proposed here to generate and solve an unconstrained smooth reformulation of support vector machines for pattern classification using completely arbitrary kernels. We term such reformulations smooth support vector machines (SSVMs). A fast Newton-Armijo algorithm for solving the SSVMs 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 five of these datasets. Using nonlinear kernels, SSVMs also obtained very distinct nonlinear separations for the checkerboard and the two-spiral datasets. A medical application is also proposed here. A linear support vector machine (SVM) is used to extract 6 features from a total of 31 features in a dataset of 253 breast cancer patients. Five features are nuclear features obtained during a non-invasive diagnostic procedure while one feature, tumor size, is obtained during surgery. The linear SVM selected the 6 features in the process of classifying the patients into node-positive (patients with some metastasized lymph nodes) and nodenegative (patients with no metastasized lymph nodes). The 6 features were then used in a Gaussian kernel nonlinear SVM to classify the patients into three prognostic groups: good (node-negative), intermediate (1 to 4 metastasized nodes) and poor (more than 4 metastasized nodes). Very well separated Kaplan-Meier survival curves were constructed for the three groups with pairwise p-value of less than 0.009 based on the logrank statistic. New patients can be assigned to one of these three prognostic groups with its associated survival curve, based only on 6 features obtained before and during surgery, but without the potentially risky procedure of removing lymph nodes to determine how many of them have metastasized.
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تاریخ انتشار 2000