The use of grossone in elastic net regularization and sparse support vector machines
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METHODOLOGIES AND APPLICATION
The use of grossone in elastic net regularization and sparse support vector machines Renato De Leone1
· Nadaniela Egidi1
· Lorella Fatone1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract New algorithms for the numerical solution of optimization problems involving the l0 pseudo-norm are proposed. They are designed to use a recently proposed computational methodology that is able to deal numerically with finite, infinite and 1 (grossone) infinitesimal numbers. This new methodology introduces an infinite unit of measure expressed by the numeral 1 and indicating the number of elements of the set IN, of natural numbers. We show how the numerical system built upon 1 can be successfully used in the solution of elastic net and the proposed approximation of the l0 pseudo-norm in terms of regularization problems and sparse support vector machines classification problems. Keywords Elastic net regularization · Grossone · Sparse support vector machines
1 Introduction Given a vector x of n components, the l0 pseudo-norm x0 := number of nonzero components in x, has often been used in optimization problems arising in various fields. However, the introduction of x0 makes these problems extremely complicated to solve, so that approximations and iterative schemes have been proposed in the scientific literature to efficiently solve them. The use of this pseudo-norm arises in many different fields, such as machine learning, signal processing, pattern recognition, portfolio optimization, subset selection problem in regression and elastic-net regularization. Cardinality constrained optimization problems are difficult to solve, and a common approach is to apply global discrete optimization techniques. Communicated by V. Loia.
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Renato De Leone [email protected] Nadaniela Egidi [email protected] Lorella Fatone [email protected]
1
School of Science and Technology, University of Camerino, Camerino, Italy
Quite recently Sergeyev, in a book and in a series of papers, proposed a novel approach to infinite and infinitesimal numbers. By introducing the new numeral grossone (indicated 1 defined as the number of elements of the set of the by ), natural numbers, Sergeyev demonstrated how it is possible to operate with finite, infinite and infinitesimal quantities using the same arithmetics. This new numerical system allows to treat infinite and infinitesimal numbers as particular cases of a single structure and offers a new view and an alternative approach for many fundamental aspects of mathematics such as limits, derivatives, sums of series and so on. The aim of this paper is to show how this new numeral sys1 can be used in different optimization tem and in particular problems, by replacing the l0 pseudo-norm x0 with x0, 1 −1 :=
n
xi2
i=1
1 −1 xi2 +
.
Indeed in literature, there are many contributes for approximating the l0 pseudo-norm. For example in Rinaldi et al. (2010), two new smooth approximations of the l0 pseudonorm are presente
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