  Understanding what makes high-dimensional data learnable is a fundamental
question in machine learning. On the one hand, it is believed that the success
of deep learning lies in its ability to build a hierarchy of representations
that become increasingly more abstract with depth, going from simple features
like edges to more complex concepts. On the other hand, learning to be
insensitive to invariances of the task, such as smooth transformations for
image datasets, has been argued to be important for deep networks and it
strongly correlates with their performance. In this work, we aim to explain
this correlation and unify these two viewpoints. We show that by introducing
sparsity to generative hierarchical models of data, the task acquires
insensitivity to spatial transformations that are discrete versions of smooth
transformations. In particular, we introduce the Sparse Random Hierarchy Model
(SRHM), where we observe and rationalize that a hierarchical representation
mirroring the hierarchical model is learnt precisely when such insensitivity is
learnt, thereby explaining the strong correlation between the latter and
performance. Moreover, we quantify how the sample complexity of CNNs learning
the SRHM depends on both the sparsity and hierarchical structure of the task.
