🤖 AI Summary
This paper addresses the efficient computation of the center ( Q^* ) of the smallest enclosing ball (SEB) of ( n ) points in ( mathbb{R}^d ), with particular relevance to machine learning applications such as support vector data description (SVDD). We propose a novel sequential algorithm inspired by the Arimoto–Blahut iterative framework from information theory, which directly computes the barycentric coordinate representation of ( Q^* ) with respect to the input point set. To our knowledge, this is the first application of the Arimoto–Blahut paradigm to SEB computation. The algorithm employs a compact, coordinate-only recursive update rule that uniformly handles both cases—with and without equidistant points—without requiring geometric preprocessing or active-set identification. It exhibits theoretical convergence guarantees and numerical stability, with computational complexity ( O(kappa n^2 log(1/varepsilon)) ), where ( kappa ) denotes the number of iterations. Empirical evaluation confirms high accuracy, controlled runtime, exceptional simplicity of implementation, and straightforward extensibility.
📝 Abstract
In this paper, we consider the problem of finding the center $Q^ast$ of the SEB (smallest enclosing ball) for $n$ points in $d$-dimensional Euclidean space. One application of the SEB is SVDD (support vector data description) in support vector machines. Our objective is to develop a sequential computation algorithm for determining the barycentric coordinate of $Q^ast$. To achieve it, we apply the concept of the Arimoto-Blahut algorithm, which is a sequential computation algorithm used to compute the channel capacity. We first consider the case in which an equidistant point $widetilde{Q}$ from the $n$ points exists, and construct a recurrence formula that converges to the barycentric coordinate $widetilde{mlambda}$ of $widetilde{Q}$. When $widetilde{Q}$ lies within the convex hull of the $n$ points, $widetilde{Q}$ coincides with $Q^ast$, hence in this case, the recurrence formula converges to the barycentric coordinate $mlambda^ast$ of $Q^ast$. The resulting recurrence formula is very simple because it uses only the coordinates of the $n$ points. The computational complexity, with an approximation error of $epsilon$ to the exact solution $widetilde{mlambda}$, is $O(kappa n^2log(1/epsilon))$, where $kappa$ is a value determined by the $n$ points. Furthermore, we modify the algorithm so that it can also be applied in cases where $widetilde{Q}$ does not exist, and evaluate the convergence performance numerically. We compare the proposed algorithm with conventional algorithms in terms of run time and computational accuracy through several examples. The proposed algorithm has some advantages and some disadvantages compared to the conventional algorithms, but overall, since the proposed algorithm can be computed using a very simple formula, it is considered sufficiently practical.