Absolute abstraction: a renormalisation group approach

TL;DR

This paper proposes a renormalisation group approach to understanding abstraction in neural networks, emphasizing the importance of training set breadth for developing truly abstract representations, supported by theoretical and numerical evidence.

Contribution

It introduces the Hierarchical Feature Model as a candidate for absolute abstraction and demonstrates its relevance through experiments with deep belief networks and auto-encoders.

Findings

Representations approach the Hierarchical Feature Model with broader data and increased depth.

Broader training sets lead to more abstract neural network representations.

Theoretical predictions align with numerical experiments.

Abstract

Abstraction is the process of extracting the essential features from raw data while ignoring irrelevant details. It is well known that abstraction emerges with depth in neural networks, where deep layers capture abstract characteristics of data by combining lower level features encoded in shallow layers (e.g. edges). Yet we argue that depth alone is not enough to develop truly abstract representations. We advocate that the level of abstraction crucially depends on how broad the training set is. We address the issue within a renormalisation group approach where a representation is expanded to encompass a broader set of data. We take the unique fixed point of this transformation -- the Hierarchical Feature Model -- as a candidate for a representation which is absolutely abstract. This theoretical picture is tested in numerical experiments based on Deep Belief Networks and auto-encoders trained on data of different breadth. These show that representations in neural networks approach the Hierarchical Feature Model as the data get broader and as depth increases, in agreement with theoretical predictions.