  Recent work has sought to understand the behavior of neural networks by
comparing representations between layers and between different trained models.
We examine methods for comparing neural network representations based on
canonical correlation analysis (CCA). We show that CCA belongs to a family of
statistics for measuring multivariate similarity, but that neither CCA nor any
other statistic that is invariant to invertible linear transformation can
measure meaningful similarities between representations of higher dimension
than the number of data points. We introduce a similarity index that measures
the relationship between representational similarity matrices and does not
suffer from this limitation. This similarity index is equivalent to centered
kernel alignment (CKA) and is also closely connected to CCA. Unlike CCA, CKA
can reliably identify correspondences between representations in networks
trained from different initializations.
