Duality of Bures and Shape Distances with Implications for Comparing Neural Representations
This work provides a theoretical unification for researchers comparing neural representations, though it appears incremental as it connects existing methods rather than introducing new ones.
The paper tackled the fragmented landscape of similarity measures for neural network representations by showing that the cosine of the Riemannian shape distance equals normalized Bures similarity, unifying two broad categories of methods. It explored new interpretations and contrasts with popular measures like CKA.
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.