Approximating Smallest Enclosing Balls With Applications to Machine Learning

Abstract

In this paper, we first survey prior work for computing exactly or approximately the smallest enclosing balls of point or ball sets in Euclidean spaces. We classify previous work into three categories: (1) purely combinatorial, (2) purely numerical, and (3) recent mixed hybrid algorithms based on coresets. We then describe two novel tailored algorithms for computing arbitrary close approximations of the smallest enclosing Euclidean ball of balls. These deterministic heuristics are based on solving relaxed decision problems using a primal-dual method. The primal-dual method is interpreted geometrically as solving for a minimum covering set, or dually as seeking for a minimum piercing set. Finally, we present some applications in machine learning of the exact and approximate smallest enclosing ball procedure, and discuss about its extension to non-Euclidean informationtheoretic spaces.

Secure Long-Distance Quantum Communication over Optical Fiber Quantum Channels

“…From this viewpoint, the DPS protocol has been analyzed by Takesue et al (Takesue et al, 2005). In this section we show that the complexity of the DPS QKD protocol's security analysis can be decreased dramatically, using fast computational information geometric methods , (Gyongyosi & Imre, 2010a), (Nielsen et al, 2007), , (Nielsen & Nock, 2008a), (Nielsen & Nock, 2009). …”

Numerosity Reduction for Resource Constrained Learning

“…The ranking style of the LO ranking scheme resembles Minimum Enclosing Ball (MEB) [2], [19], [23], [30]. However the core concepts and computational problems which each addresses, are significantly different.…”

“…However the core concepts and computational problems which each addresses, are significantly different. Most parametric MEB algorithms and their implementations compute the smallest enclosing balls of point sets in Euclidean spaces [2], [23]. As highlighted in Ref.…”

“…As highlighted in Ref. [23], application of MEB in other spaces is still an open issue. Although, theoretically, MEB can be used to dynamically reduce numerosity of data for a generic use, it is bound to be costful and we have not come across an implementation which does so.…”

A faster dual algorithm for the Euclidean minimum covering ball problem

“…Hubbard (1996), and Larsson et al (2016); machine learning, see e.g. Kumar et al (2003), Nielsen and Nock (2009), and the references therein; etc.…”