Models, networks and algorithmic complexity

TL;DR

This paper explores the deep connections between models, networks, and algorithmic complexity, demonstrating their equivalence and hierarchical relationships, with implications for cognitive neuroscience and neural network analysis.

Contribution

It establishes the algorithmic equivalence of models, networks, and datasets, and introduces a framework linking neural network types to recursive function theory.

Findings

Neural networks implement models in various forms.

Input and node perturbations propagate strongly in trained networks.

Different network architectures fall into a hierarchy aligned with recursive function theory.

Abstract

I aim to show that models, classification or generating functions, invariances and datasets are algorithmically equivalent concepts once properly defined, and provide some concrete examples of them. I then show that a) neural networks (NNs) of different kinds can be seen to implement models, b) that perturbations of inputs and nodes in NNs trained to optimally implement simple models propagate strongly, c) that there is a framework in which recurrent, deep and shallow networks can be seen to fall into a descriptive power hierarchy in agreement with notions from the theory of recursive functions. The motivation for these definitions and following analysis lies in the context of cognitive neuroscience, and in particular in Ruffini (2016), where the concept of model is used extensively, as is the concept of algorithmic complexity.