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LIT-362: Learning the Irreducible Representations of Commutative Lie Groups

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Properties6
StatusActive — read in full 2026-09-29 (NOTE-307); worth reading as the earliest ML paper in this batch to state that the "elementary components" of a representation are the irreps of a symmetry group and to learn them from data; read it as learning the irreducible decomposition of an assumed abelian group class from transformation pairs on raw pixels, not as detecting an unknown symmetry in a trained network, and with an experiment that is a one-group proof of concept on noise patches.
Version2
Tagsrepresentation-learning, mathematics, probabilistic-modeling
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-307 — Learning the Irreducible Representations of Commutative Lie 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

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Cohen & Welling (2014), Proceedings of the 31st International Conference on Machine Learning (ICML 2014), Beijing, PMLR 32(2):1755–1763 (verified on the PMLR page, proceedings.mlr.press/v32/cohen14.html); arXiv v1 18 Feb 2014, v2 25 May 2014 (the version read); arXiv DOI 10.48550/arXiv.1402.4437 — arXiv:1402.4437

Standing in the record

Filed on 2026-09-29 as a supplemental reading. The close readings of the owner’s prior-art novelty map’s references found the map’s citation for this point wrong, or its source unreachable, and this work is the candidate replacement. published: is the first appearance (ADR-002).

It was filed Deferred, unread. NOTE-307 is the close reading of 2026-09-29, and it placed the work: Active — worth reading as the earliest ML paper in this batch to state that the “elementary components” of a representation are the irreps of a symmetry group and to learn them from data; read it as learning the irreducible decomposition of an assumed abelian group class from transformation pairs on raw pixels, not as detecting an unknown symmetry in a trained network, and with an experiment that is a one-group proof of concept on noise patches.

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