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Revisiting strategies for fitting logistic regression for positive and unlabeled data

Wawrzeńczyk Adam, Mielniczuk Jan

International Journal of Applied Mathematics and Computer Science

2022

Vol. 32, no. 2

299--309

Positive unlabeled (PU) learning is an important problem motivated by the occurrence of this type of partial observability in many applications. The present paper reconsiders recent advances in parametric modeling of PU data based on empirical likelihood maximization and argues that they can be significantly improved. The proposed approach is based on the fact that the likelihood for the logistic fit and an unknown labeling frequency can be expressed as the sum of a convex and a concave function, which is explicitly given. This allows methods such as the concave-convex procedure (CCCP) or its variant, the disciplined convex-concave procedure (DCCP), to be applied. We show by analyzing real data sets that, by using the DCCP to solve the optimization problem, we obtain significant improvements in the posterior probability and the label frequency estimation over the best available competitors.

Modern regularization techniques for inverse modelling: a comparative study

Krawczyk-Stańdo D., Rudnicki M., Stańdo J.

Journal of Applied Computer Science

2011

Vol. 19, nr 1

51-63

Regularization techniques are used for computing stable solutions to ill-posed problems. The well-known form of regularization is that of Tikhonov in which the regularized solution is searched as a minimiser of the weighted combination of the residual norm and a side constraint-controlled by the regularization parameter. For the practical choice of regularization parameter we can use the L-curve approach, U-curve criterion introduced by us [1] and empirical risk method [2]. We present a comparative study of different strategies for the regularization parameter choice on examples of function approximation by radial basis neural networks. Such networks are universal approximators and can learn any nonlinear mapping. e.g. representing an magnetic inverse problem. Some integral equations of the first kind are considered as well.