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In order to choose a neural network architecture that will be effective for aparticular modeling problem, one must understand the limitations imposed byeach of the potential options. These limitations are typically described interms of information theoretic bounds, or by comparing the relative complexityneeded to approximate example functions between different architectures. Inthis paper, we examine the topological constraints that the architecture of aneural network imposes on the level sets of all the functions that it is ableto approximate. This approach is novel for both the nature of the limitationsand the fact that they are independent of network depth for a broad family ofactivation functions.

Deep, Skinny Neural Networks are not Universal Approximators