Tropical Polynomial Division and Neural Networks

Minimax algebra [1] and tropical geometry [2] are fields of ma thematics with applications in a variety of domains, such as the analysis of dynamic systems [3], [4], [5], optimization [6], [7], idempotent functional analysis [8] and morphological meth ods for computer vision [9], [10]. Recent works [11], [12] have expanded the links of this branc h of mathematics in the domain of neural networks with piecewise linear activations, demo nstrating a profound connection between the two. It is apparent that further study is needed, given these new i nsights, in the role that this particular type of algebra plays, in the context of neural ne tworks, in order to better identify the inner workings of the latter. Such an accomplishment wou ld potentially have applications in the problem of network minimization, given that, as demon strated by the results of [13], [14], pruning a network can lead to considerable improvemen ts in both network size and runtime, without significant loss of accuracy. In this work, we seek to expand the link between these fields, b y examining the process of Tropical Polynomial Division.