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.

本文研究卷积网络对其输入区域之间相关性的建模能力，并通过理论分析和实证验证，证明深度网络的池化几何形状是控制感性偏差的一种手段，使其在输入不同类型的数据时可以适应不同的偏差方向。