Inductive Bias of Deep Convolutional Networks through Pooling Geometry

TL;DR

This paper investigates how the pooling geometry in deep convolutional networks influences their inductive bias, particularly their ability to model correlations in natural images, through theoretical analysis and empirical validation.

Contribution

It introduces the concept of separation rank to quantify correlations and shows how pooling geometry controls the network's bias towards certain input partitions, explaining the success of convolutional networks.

Findings

Deep networks support exponential separation ranks for certain partitions.

Pooling geometry determines which input correlations are favored.

Shallow networks are limited to linear separation ranks.

Abstract

Our formal understanding of the inductive bias that drives the success of convolutional networks on computer vision tasks is limited. In particular, it is unclear what makes hypotheses spaces born from convolution and pooling operations so suitable for natural images. In this paper we study the ability of convolutional networks to model correlations among regions of their input. We theoretically analyze convolutional arithmetic circuits, and empirically validate our findings on other types of convolutional networks as well. Correlations are formalized through the notion of separation rank, which for a given partition of the input, measures how far a function is from being separable. We show that a polynomially sized deep network supports exponentially high separation ranks for certain input partitions, while being limited to polynomial separation ranks for others. The network's pooling geometry effectively determines which input partitions are favored, thus serves as a means for controlling the inductive bias. Contiguous pooling windows as commonly employed in practice favor interleaved partitions over coarse ones, orienting the inductive bias towards the statistics of natural images. Other pooling schemes lead to different preferences, and this allows tailoring the network to data that departs from the usual domain of natural imagery. In addition to analyzing deep networks, we show that shallow ones support only linear separation ranks, and by this gain insight into the benefit of functions brought forth by depth - they are able to efficiently model strong correlation under favored partitions of the input.