This paper presents a hybrid morphological neural network for regression tasks called linear dilation-erosion regression ($\ell$-DER). In few words, an $\ell$-DER model is given by a convex combination of the composition of linear and elementary morphological operators. As a result, they yield continuous piecewise linear functions and, thus, are universal approximators. Apart from introducing the $\ell$-DER models, we present three approaches for training these models: one based on stochastic descent gradient and two based on the difference of convex programming problems. Finally, we evaluate the performance of the $\ell$-DER model using 14 regression tasks. Although the approach based on SDG revealed faster than the other two, the $\ell$-DER trained using a disciplined convex-concave programming problem outperformed the others in terms of the least mean absolute error score.

Dilation and erosion are two elementary operations from mathematical morphology, a non-linear lattice computing methodology widely used for image processing and analysis. The dilation-erosion perceptron (DEP) is a morphological neural network obtained by a convex combination of a dilation and an erosion followed by the application of a hard-limiter function for binary classification tasks. A DEP classifier can be trained using a convex-concave procedure along with the minimization of the hinge loss function. As a lattice computing model, the DEP classifier assumes the feature and class spaces are partially ordered sets. In many practical situations, however, there is no natural ordering for the feature patterns. Using concepts from multi-valued mathematical morphology, this paper introduces the reduced dilation-erosion (r-DEP) classifier. An r-DEP classifier is obtained by endowing the feature space with an appropriate reduced ordering. Such reduced ordering can be determined using two approaches: One based on an ensemble of support vector classifiers (SVCs) with different kernels and the other based on a bagging of similar SVCs trained using different samples of the training set. Using several binary classification datasets from the OpenML repository, the ensemble and bagging r-DEP classifiers yielded in mean higher balanced accuracy scores than the linear, polynomial, and radial basis function (RBF) SVCs as well as their ensemble and a bagging of RBF SVCs.