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THEORY-008: On a finite set of stimuli, what a regularized linear readout can decode from a representation is a function of its normalized kernel, and CKA, CCA and GULP are averages of readout agreement

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Properties6
StatusActive
Version5
Tagsrepresentation-learning, mathematics
SourceLIT-257 — What Representational Similarity Measures Imply about Decodable Information, LIT-254 — Similarity of Neural Network Representations Revisited
ExtendsTHEORY-004 — A representation is determined by its centred kernel up to a translation and an orthogonal transformation, and the rotation-aligned distance between two representations equals the Bures distance between their kernels
SupportsCLAIM-193 — A linear probe family can tell apart only realizations whose representations differ in normalized kernel, and kernels fix a representation only up to the symmetry of what is observed: any distinction beyond that is supplied by the analyst, CLAIM-269 — A regularized linear readout can tell apart only representations that differ in normalized kernel, and kernels fix a representation only up to the symmetry of what is observed, so no such readout supplies a distinction beyond that symmetry, CLAIM-379 — On a finite set of stimuli, regularized linear readouts with ridge b > 0 cannot tell apart two centred representations of the same width exactly when one is an orthogonal transformation of the other, and the unregularized CCA member cannot tell apart representations with the same column space, which differ by an invertible linear map

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Source

Harvey, Lipshutz & Williams (2024), LIT-257, eqs. 2, 7–8, 22–23, Prop. 2, Cor. 1, Table 1. Supporting, for the family structure only: Kornblith et al. (2019), LIT-254, Eqs. 11–14 and App. D.1, as read in NOTE-743. The same point from the concept-direction side is ANTH-LIT-606 (Park, Choe & Veitch), whose Thm 3.2 uses a Riesz isomorphism and whose §3 shows the inner product is not identified by training.

What was actually shown

For readouts w maximizing zᵀXw/M − ½wᵀG(X)w, the decoded signal is K_X z with K_X = X G(X)⁻¹ Xᵀ/M (LIT-257 eqs. 2, 7–8), so it depends on X only through K_X, which is invariant under orthogonal transformations of X (eqs. 22–23) whenever G(X) is parameterized as in eq. 4, G = aC_X + bI. The task-averaged agreement of two representations’ readouts is Tr(K_X K_z K_Y) (Prop. 2); with K_z = I it gives linear CKA, GULP, mean squared CCA or ENSD for particular choices of G (Cor. 1, Table 1). Only ENSD is the average similarity itself: CKA and mean squared CCA are it normalized by the self-similarities, and GULP is the average decoding distance. The family structure has a precursor in Kornblith et al., who show that R²_CCA, linear regression and linear CKA differ only in how they weight the same eigen-direction overlaps, and that canonical ridge interpolates among them (LIT-254 Eqs. 11–14, App. D.1). They say nothing about readouts, decoding, averaging over tasks or GULP; the readout reading is LIT-257’s alone.

What this does not say

  • Anything in the population limit. Every identity here is for M stimuli. Whether the K_z = I identities have a non-degenerate limit as M grows, which the source’s App. A.3 does not reach, is QUESTION-136.
  • Anything about nonlinear readouts, which are outside the source, or classifiers (§5 names linear classifiers, SVMs and multi-class logistic regression, as future work).
  • Which task distribution K_z is the right one to average over.
  • The source’s Proposition 3 (bounds via Procrustes distance) is a result of a different kind: two-sided inequalities rather than identities. It is now read and holds, given the imported Procrustes–Bures equivalence; THEORY-301 holds it.

Connections

  • THEORY-236. Mean squared CCA, one of the averages above, is invariant to every invertible linear map: it is an orthogonally invariant index applied to orthonormalized columns (LIT-254, Prop. 1). Kornblith et al.’s Theorem 1 then makes it constant on all representations whose rank equals the number of examples. Linear CKA, invariant only to orthogonal maps and isotropic scaling, escapes that degeneracy.

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