Fixed Points of Cone Mapping with the Application to Neural Networks

TL;DR

This paper establishes new conditions for the existence of fixed points in cone mappings without requiring scalability, with applications to neural networks that challenge traditional assumptions about weights and monotonicity.

Contribution

It introduces a novel analysis of fixed points for neural networks considering non-negative data and weights, relaxing previous monotonicity and scalability constraints.

Findings

Fixed points can exist without scalability assumptions.

Neural networks with non-negative data may have weights with negative values.

Weakening of monotonicity assumptions broadens applicability.

Abstract

We derive conditions for the existence of fixed points of cone mappings without assuming scalability of functions. Monotonicity and scalability are often inseparable in the literature in the context of searching for fixed points of interference mappings. In applications, such mappings are approximated by non-negative neural networks. It turns out, however, that the process of training non-negative networks requires imposing an artificial constraint on the weights of the model. However, in the case of specific non-negative data, it cannot be said that if the mapping is non-negative, it has only non-negative weights. Therefore, we considered the problem of the existence of fixed points for general neural networks, assuming the conditions of tangency conditions with respect to specific cones. This does not relax the physical assumptions, because even assuming that the input and output are to be non-negative, the weights can have (small, but) less than zero values. Such properties (often found in papers on the interpretability of weights of neural networks) lead to the weakening of the assumptions about the monotonicity or scalability of the mapping associated with the neural network. To the best of our knowledge, this paper is the first to study this phenomenon.