Subhomogeneous operator theory for deep learning and optimization

Subhomogeneous operators are a broad class of function that generalize the notion of homogeneous operator. Exploiting their properties, we prove a theorem of existence and uniqueness of fixed points, where the subhomogeneous operators are the key ingredient. We apply this result in two different contexts. The first one is implicit layer models, or, more precisely deep equilibrium networks, a new Deep Learning model-type that solves a fixed point equation in order to calculate the output. Although they require a lower amount of memory, we need to ensure the well-posedness of the architecture ensuring the existence and uniqueness of the fixed point. We use the aforementioned result from subhomogenous operator theory in order to achieve this. The second application of our fixed point theorem is in the field of mathematical optimization. It is possible to ensure the existence and uniqueness of the solution of two optimization problems. Moreover, we derive a method that attempts to solve the maximum clique problem.