Cholesky-factorized sparse Kernel in support vector machines

Support Vector Machine (SVM) is one of the most powerful machine learning algorithms due to its convex optimization formulation and handling non-linear classification. However, one of its main drawbacks is the long time it takes to train large data sets. This limitation is often aroused when applying non-linear kernels (e.g. RBF Kernel) which are usually required to obtain better separation for linearly inseparable data sets. In this thesis, we study an approach that aims to speed-up the training time by combining both the better performance of RBF kernels and fast training by a linear solver, LIBLINEAR. The approach uses an RBF kernel with a sparse matrix which is factorized using Cholesky decomposition. The method is tested on large artificial and real data sets and compared to the standard RBF and linear kernels where both the accuracy and training time are reported. For most data sets, the result shows a huge training time reduction, over 90\%, whilst maintaining the accuracy

Convex Optimization for Binary Classifier Aggregation in Multiclass Problems

Multiclass problems are often decomposed into multiple binary problems that are solved by individual binary classifiers whose results are integrated into a final answer. Various methods, including all-pairs (APs), one-versus-all (OVA), and error correcting output code (ECOC), have been studied, to decompose multiclass problems into binary problems. However, little study has been made to optimally aggregate binary problems to determine a final answer to the multiclass problem. In this paper we present a convex optimization method for an optimal aggregation of binary classifiers to estimate class membership probabilities in multiclass problems. We model the class membership probability as a softmax function which takes a conic combination of discrepancies induced by individual binary classifiers, as an input. With this model, we formulate the regularized maximum likelihood estimation as a convex optimization problem, which is solved by the primal-dual interior point method. Connections of our method to large margin classifiers are presented, showing that the large margin formulation can be considered as a limiting case of our convex formulation. Numerical experiments on synthetic and real-world data sets demonstrate that our method outperforms existing aggregation methods as well as direct methods, in terms of the classification accuracy and the quality of class membership probability estimates.Comment: Appeared in Proceedings of the 2014 SIAM International Conference on Data Mining (SDM 2014

A Unifying View of Multiple Kernel Learning

Recent research on multiple kernel learning has lead to a number of approaches for combining kernels in regularized risk minimization. The proposed approaches include different formulations of objectives and varying regularization strategies. In this paper we present a unifying general optimization criterion for multiple kernel learning and show how existing formulations are subsumed as special cases. We also derive the criterion's dual representation, which is suitable for general smooth optimization algorithms. Finally, we evaluate multiple kernel learning in this framework analytically using a Rademacher complexity bound on the generalization error and empirically in a set of experiments