Spherical separation with infinitely far center
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Spherical separation with infinitely far center Annabella Astorino1
· Antonio Fuduli2
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We tackle the problem of separating two finite sets of samples by means of a spherical surface, focusing on the case where the center of the sphere is fixed. Such approach reduces to the minimization of a convex and nonsmooth function of just one variable (the radius), revealing very effective in terms of computational time. In particular, we analyze the case where the center of the sphere is selected far from both the two sets, embedding the grossone idea and obtaining a kind of linear separation. Some numerical results are presented on classical binary data sets drawn from the literature. Keywords Spherical separation · Classification · Grossone
1 Introduction Classification problems in mathematical programming concern separation of sample sets by means of an appropriate surface. This field, entered by many researchers in optimization community in the last years, is a part of the more general machine learning area, aimed at providing automated systems able to learn from human experiences. In machine learning, classification can be addressed on the basis of different paradigms (Astorino et al. 2008). The most common one is the supervised approach, where the samples in each set are equipped with the class label heavily exploited in the learning phase. A well-established supervised technique is the support vector machine (SVM) (Cristianini and Shawe-Taylor 2000; Vapnik 1995), which revealed a powerful classification tool in many applicative areas. A widely adopted alternative is called unsupervised, since no class label is known in advance; the aim is to cluster the data on the basis of their similarities (Celebi 2015). In the middle, we find the semisupervised techniques (Chapelle et al. 2006), which are a compromise between the supervised and
the unsupervised approaches; in such case, the learning task is characterized by the exploitation of the overall information coming from both labeled and unlabeled samples. Some useful references are Chapelle and Zien (2005) and Astorino and Fuduli (2007), the latter being a semisupervised version of the SVM technique. A more recent classification framework is constituted by the multiple instance learning (MIL) (Herrera et al. 2016), which can be interpreted as a kind of weak supervised approach; it consists in categorizing bags of samples, being available only the class label of the bags instead of the class label of each sample inside them. A seminal SVM-type MIL paper is Andrews et al. (2003), while some recent articles are Astorino et al. (2018, 2019a, b, 2020), Avolio and Fuduli (2020), Gaudioso et al. (2020), and Plastria et al. (2014). In this work, we present an extension of the supervised binary classification approach reported in Astorino and Gaudioso (2009) and based on the spherical separation of two finite sets of samples (points in IRn ), say A = {a1 , . . . , am }, with ai ∈ IRn , i = 1, . . . ,
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