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

Sarah E. Harvey, Brett W. Larsen, Alex H. Williams

Research output: Contribution to journalConference articlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)60-75
Number of pages16
JournalProceedings of Machine Learning Research
StatePublished - 2023
Event1st Workshop on Unifying Representations in Neural Models, UniReps 2023 - New Orleans, United States
Duration: Dec 15 2023 → …

ASJC Scopus subject areas

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability


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