Deep Learning with Set-Valued Inputs

In this chapter, standard inputs in deep learning networks are extended to set-valued inputs. In this framework, a set-valued activation map is given that extends the known softsign function. Optimal weights and biases can be computed by minimizing the well-known Hausdorff distance between true and predicted sets. Two additional distances for special classes of sets are introduced, leading to simpler optimization problems. Finally, the theory is specialized to the simple case of inputs in the form of closed bounded intervals.