Alex Williams
Researcher with public evidence across AI research.
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The difference between Σ s_i² and Σ s_i seems minor, but intuitively the former (CKA/RSA) places greater emphasis on the top PCs. Procrustes is often a more stringent notion of similarity, and this paper by Nathan Cloos shows that the difference can matter. proceedings.iclr.cc…
Your observation that "if s_i are the singular values of the centered cross-covariance, CKA weights shared modes through Σ s_i², while Procrustes uses Σ s_i" is exactly what we discuss here, alongside equivalence of Procrustes to Bures distance on Gram matrices proceedings.mlr…
§A.13.3 notes RSA and CKA differ over the diagonal vs row/column means. Indeed, adding a mean-centering step to RSA makes the two equivalent, for linear and nonlinear variants as shown here: proceedings.mlr.press/v285/william...
Enjoying reading this! Your Lemma 22 bounds Procrustes error by CKA in one direction, with a rank factor. Proposition 3 from our paper derives a two-sided version with a sharp constant that depends on participation ratio: proceedings.mlr.press/v285/harvey2...
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