Abstraction requires breadth: a renormalisation group approach

Abstract: Abstraction is the process of extracting the essential features from raw data while ignoring irrelevant details. This is similar to the process of focusing on large-scale properties, systematically removing irrelevant small-scale details, implemented in the renormalisation group of statistical physics. This analogy is suggestive because the fixed points of the renormalisation group offer an ideal candidate of a truly abstract -- i.e. data independent -- representation. It has been observed that abstraction emerges with depth in neural networks. Deep layers of neural network capture abstract characteristics of data, such as "cat-ness" or "dog-ness" in images, by combining the 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 an abstract representation. This theoretical picture is tested in numerical experiments based on Deep Belief Networks trained on data of different breadth. These show that representations in deep layers of neural networks approach the Hierarchical Feature Model as the data gets broader, in agreement with theoretical predictions.