We consider the problem of computing the outer-radii of point sets. In this problem, we are given integers $n, d, k$ where $k \le d$, and a set $P$ of $n$ points in $R^d$. The goal is to compute the {\em outer $k$-radius} of $P$, denoted by $\kflatr(P)$, which is the minimum, over all $(d-k)$-dimensional … Read more
When the mean vectors and the covariance matrices of two classes are available in a binary classification problem, Lanckriet et al.\ \cite{mpm} propose a minimax approach for finding a linear classifier which minimizes the worst-case (maximum) misclassification probability. We extend the minimax approach to a multiple classification problem, where the number $m$ of classes could … Read more
Support Vector Machine has shown to have good performance in many practical classification settings. In this paper we propose, for multi-group classification, a biobjective optimization model in which we consider not only the generalization ability (modelled through the margin maximization), but also costs associated with the features. This cost is not limited to an economical … Read more
The widely used Support Vector Machine (SVM) method has shown to yield good results in Supervised Classification problems. The Binarized SVM (BSVM) is a variant which is able to automatically detect which variables are, by themselves, most relevant for the classifier. In this work, we extend the BSVM introduced by the authors to a method … Read more
The widely used Support Vector Machine (SVM) method has shown to yield good results in Supervised Classification problems. Other methods such as Classification Trees have become more popular among practitioners than SVM thanks to their interpretability, which is an important issue in Data Mining. In this work, we propose an SVM-based method that automatically detects … Read more
We present an optimization engine for large scale pattern recognition using Support Vector Machine (SVM). Our treatment is based on conversion of soft-margin SVM constrained optimization problem to an unconstrained form, and solving it using newly developed Sequential Subspace Optimization (SESOP) method. SESOP is a general tool for large-scale smooth unconstrained optimization. At each iteration … Read more
We propose minimum volume ellipsoids (MVE) clustering as an alternate clustering technique to k-means clustering for Gaussian data points and explore its value and practicality. MVE clustering allocates data points into clusters that minimizes the total volumes of each cluster’s covering ellipsoids. Motivations for this approach include its scale-invariance, its ability to handle asymmetric and … Read more
One of the fundamental clustering problems is to assign $n$ points into $k$ clusters based on the minimal sum-of-squares(MSSC), which is known to be NP-hard. In this paper, by using matrix arguments, we first model MSSC as a so-called 0-1 semidefinite programming (SDP). We show that our 0-1 SDP model provides an unified framework for … Read more
We present an original method for partitioning by automatic classi- fication, using the optimization technique of tabu search. The method uses a classical tabu search scheme based on transfers for the minimization of the within variance; it introduces in the tabu list the indicator of the object transfered. This method is compared with two stochastic … Read more
This paper presents a new data classification method based on mixed-integer programming. Traditional approaches that are based on partitioning the data sets into two groups perform poorly for multi-class data classification problems. The proposed approach is based on the use of hyper-boxes for defining boundaries of the classes that include all or some of the … Read more