  Learning informative representations of data is one of the primary goals of
deep learning, but there is still little understanding as to what
representations a neural network actually learns. To better understand this,
subspace match was recently proposed as a method for assessing the similarity
of the representations learned by neural networks. It has been shown that two
networks with the same architecture trained from different initializations
learn representations that at hidden layers show low similarity when assessed
with subspace match, even when the output layers show high similarity and the
networks largely exhibit similar performance on classification tasks. In this
note, we present a simple example motivated by standard results in commutative
algebra to illustrate how this can happen, and show that although the subspace
match at a hidden layer may be 0, the representations learned may be isomorphic
as vector spaces. This leads us to conclude that a subspace match comparison of
learned representations may well be uninformative, and it points to the need
for better methods of understanding learned representations.
