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A multitude of (dis)similarity measures between neural networkrepresentations have been proposed, resulting in a fragmented researchlandscape. 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 toquantify similarity while accounting for expected invariances. Second, measuressuch as representational similarity analysis (RSA), centered kernel alignment(CKA), and normalized Bures similarity (NBS) all quantify similarity in summarystatistics, such as stimulus-by-stimulus kernel matrices, which are alreadyinvariant to expected symmetries. Here, we take steps towards unifying thesetwo broad categories of methods by observing that the cosine of the Riemannianshape distance (from category 1) is equal to NBS (from category 2). We explorehow this connection leads to new interpretations of shape distances and NBS,and draw contrasts of these measures with CKA, a popular similarity measure inthe deep learning literature.

Duality of Bures and Shape Distances with Implications for Comparing Neural Representations