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LIT-305: On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups

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Properties6
StatusActive — read in full 2026-09-29 (NOTE-282); worth reading as the standard ML statement of "equivariance ⇔ convolution" for compact groups and as a compact introduction to isotypic decomposition and noncommutative Fourier analysis for neural networks. Read the "only if" proof as an outline. It is written out for finite or countable groups and scalar channels only, and one step (removing the pointwise nonlinearity) does not hold as stated for ReLU.
Version2
Tagsrepresentation-learning, mathematics, anthology-candidate
SupportsCLAIM-385 — If a symmetric operator on ℝⁿ commutes with an orthogonal representation of a compact group, each eigenspace is a sum of real irreducibles and each multiplicity a sum of multiples of their dimensions, so SO(2) and ℤ/n for n ≥ 3 force paired eigenvalues; conversely the spectrum fixes neither the group nor a basis inside an eigenspace
Cited as “paper” byNOTE-282 — On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups
Cited as “source” byTHEORY-019 — Symmetry forces spectral degeneracy but degeneracy does not identify a symmetry: an operator commuting with a group has eigenspaces built from its real irreducibles, so abelian rotation groups force pairs over ℝ, and the spectrum fixes neither the group nor a basis inside a multiplet, THEORY-022 — The reverse-engineered grokking network computes modular addition by multiplying characters of ℤ/p at a few frequencies, the same real two-dimensional irreducibles later found as circles in language models; in both the group comes from the task, not the network

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Risi Kondor, Shubhendu Trivedi (2018), ICML 2018 (PMLR 80) — ARXIV-1802.03690

Standing in the record

Registered on 2026-09-29 at the owner’s request, because the owner’s prior-art novelty map (a private working document on concepts as projectors, the linear representation hypothesis and their neighbours) cites it: row 7 must-cite (“Kondor”): equivariance via group representation theory / irreps — the closest owner of characters-and-irreps machinery in ML. It is seeded from metadata and published descriptions and has not been read here.

The map names this reference only loosely; this entry is one reasonable reading of it, and the choice is the owner’s to revise.

Identification notes: Map says bare “Kondor”. Proposed Kondor & Trivedi 2018 because row 7 is about characters/irreps/equivariance and this paper ties equivariance to noncommutative Fourier analysis over compact groups. Alternatives the owner may prefer: Kondor 2008 PhD thesis “Group theoretical methods in machine learning” (Columbia); Kondor, Lin & Trivedi 2018 “Clebsch–Gordan Nets” (NeurIPS, arXiv 1806.09231). Owner to confirm.

It was filed Deferred, unread. NOTE-282 is the close reading of 2026-09-29, and it placed the work: Active — worth reading as the standard ML statement of “equivariance ⇔ convolution” for compact groups and as a compact introduction to isotypic decomposition and noncommutative Fourier analysis for neural networks. Read the “only if” proof as an outline. It is written out for finite or countable groups and scalar channels only, and one step (removing the pointwise nonlinearity) does not hold as stated for ReLU.

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