Duality of Bures and Shape Distances with Implications for Comparing
  Neural Representations
Sarah E. Harvey Brett W. Larsen Alex H. Williams
A multitude of (dis)similarity measures between neural network
representations have been proposed, resulting in a fragmented research
landscape. Most of these measures fall into one of two categories.
  First, measures such as linear regression, canonical correlations analysis
(CCA), and shape distances, all learn explicit mappings between neural units to
quantify similarity while accounting for expected invariances. Second, measures
such as representational similarity analysis (RSA), centered kernel alignment
(CKA), and normalized Bures similarity (NBS) all quantify similarity in summary
statistics, such as stimulus-by-stimulus kernel matrices, which are already
invariant to expected symmetries. Here, we take steps towards unifying these
two broad categories of methods by observing that the cosine of the Riemannian
shape distance (from category 1) is equal to NBS (from category 2). We explore
how this connection leads to new interpretations of shape distances and NBS,
and draw contrasts of these measures with CKA, a popular similarity measure in
the deep learning literature.