Efficient Learning and Symmetry Discovery under Exact Invariances
Ashkan Soleymani, Behrooz Tahmasebi, Patrick Jaillet, Stefanie Jegelka
cs.LG · stat.ML
Quick verdict
Proves poly-time exact-invariant regression for infinite groups and provable symmetry discovery; rigorous but purely theoretical, no experiments.
Summary
This theory paper resolves two open problems in learning with group invariances. First, building on the spectral-averaging framework of Soleymani et al. (2025b) — which achieved polynomial-time exact invariance only for finite groups, via generating sets of size log|G| — the authors observe that generating the entire group is unnecessary: only the fixed-point subspace of the group representation matters. Proposition 4.1 proves that randomly sampling T = O(r² + r log(1/δ)) group elements, processed by a deterministic intersection-update procedure, yields a constraint set that characterizes the invariant subspace of an r-dimensional representation with probability 1−δ, independently of |G|. Theorem 4.2 consequently delivers the first polynomial-time algorithm for exactly invariant regression over arbitrary compact groups, finite or infinite, with excess risk O(n^{−s/(s+d/2)}) in Sobolev settings and minimax-optimal O(dim(H_G)/n) in finite-dimensional feature spaces. Second, for symmetry discovery where the invariance group H is an unknown bounded-index subgroup (|H| ≥ κ|G|) of a known finite group G, Theorem 4.4 shows that rejection sampling over random group elements, combined with held-out error comparisons, provably recovers a generating set for H and returns an exactly H-invariant estimator in poly(n, d, 1/κ, log|G|, log(1/δ)) time, matching known-symmetry sample complexity in finite dimensions. The analysis relies on random Cayley graphs and expanders (Alon–Roichman). Limitations: purely theoretical, with no experiments or code; oracle assumptions of Haar sampling from G and representation query access; exact invariance holds only with probability 1−δ; the Sobolev rate is not the sharp invariant minimax rate; and the discovery threshold η depends on unknown population quantities.
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Assessment at a glance
4
risk flags identified
● high 0
● medium 2
● low 2
5 strengths · 5 weaknesses · 5 author questions
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Key claims under review
- There exists a randomized poly(n, d)-time algorithm producing exactly G-invariant estimators for arbitrary compact groups, finite or infinite, with excess risk O(n^{−s/(s+d/2)}) in Sobolev settings or minimax-optimal O(dim(H_G)/n) in finite-dimensional feature spaces (Theorem 4.2).
- Random subsets of at most r group elements characterize the invariant subspace of an r-dimensional unitary representation with probability 1−δ using T = O(r² + r log(1/δ)) samples, independent of |G| (Proposition 4.1).
- For an unknown invariance group H ≤ G with |H| ≥ κ|G|, Algorithm 3 recovers a generating set for H and returns an exactly H-invariant estimator in poly(n, d, 1/κ, log|G|, log(1/δ)) time, matching known-symmetry sample complexity in finite dimensions (Theorem 4.4).
- The Sobolev rate achieved is the no-invariance minimax rate; attaining the sharp invariant rate in polynomial time remains open (Section 4.2).
Strengths
Genuine removal of the group-size dependence
Proposition 4.1 shows O(r² + r log(1/δ)) random group elements (with |S| ≤ r) capture the exact invariant subspace of an r-dimensional representation; Theorem 4.2 thereby gives the first poly(n, d) exact-invariance algorithm uniform over finite and infinite compact groups, resolving the log|G| barrier left by Soleymani et al. (2025b).
Elegant, reusable core insight
Generating G as a group is unnecessary — only the fixed-point subspace ∩_g ker(I − D(g)) matters — so redundant random constraints are as informative as generators; the A(S) ≥ 2/r gap argument in Appendix B.1 is simple and likely applicable wherever random elements can replace generating sets.
Symmetry discovery with end-to-end guarantees
Honest scoping throughout
Complete proofs
Weaknesses
No empirical component whatsoever
Existential threshold in Algorithm 3
Probabilistic exactness
Sobolev guarantee is the non-invariant rate
Oracle model excludes real pipeline costs
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Scope & Claims
Pure learning-theory paper (arXiv:2609.07031, cs.LG/stat.ML) on the computational foundations of exact group invariance in supervised regression. Two claims. (1) *Known group*: where Soleymani et al. (2025b) enforced exact invariance in poly(n, d, log|G|) time via generating sets — inherently finite-group — this paper proves (Proposition 4.1) that T = O(r² + r log(1/δ)) Haar-random group elements, processed by a deterministic basis-intersection procedure, characterize the invariant subspace of any r-dimensional unitary representation with probability 1−δ, using |S| ≤ r elements and with no dependence on |G|. Theorem 4.2 upgrades this into the first polynomial-time exactly invariant regression algorithm uniform over finite and infinite compact groups, with excess risk O(n^{−s/(s+d/2)}) for Sobolev targets (s > d/2; truncation r = n^{1/(1+α)}, α = 2s/d) and minimax-optimal O(dim(H_G)/n) in finite-dimensional feature spaces. (2) *Unknown group* (symmetry discovery): for a finite reference G and unknown H ≤ G with bounded index (|H| ≥ κ|G|), Theorem 4.4 states that rejection sampling over group elements plus held-out error comparisons recovers a generating set of H and an exactly H-invariant estimator in poly(n, d, 1/κ, log|G|, log(1/δ)) time, matching known-symmetry sample complexity in finite dimensions. Claims are precisely scoped; the abstract's strongest wording (it “matches the minimax-optimal sample complexity”) is correctly restricted to finite-dimensional feature spaces.
The remaining sections cover Methodology Assessment, Statistical Evidence, Results & Reproducibility, Limitations & Risks, and Verdict & Recommendations.
- All 5 strengths & 5 weaknesses, analyzed in detail
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Frequently asked questions
What is the main result of this paper in simple terms?
It proves that you can learn regression functions that are exactly invariant under a possibly infinite continuous group (like all rotations) in polynomial time — previous methods were efficient only for finite groups — and that if you do not know which symmetries your data obeys (but know they come from a finite group where the true symmetry contains at least a fraction κ of the elements), you can discover them from labeled data and still learn efficiently with optimal statistical accuracy.
Does the paper include code or experiments?
No. This is a purely theoretical contribution with full proofs in appendices (A–B); there are no experiments, datasets, or code releases mentioned anywhere in the text.
What are the key assumptions behind the results?
Data lies on a smooth compact manifold with uniform density; the symmetry group is compact and acts smoothly and isometrically (finite groups and compact Lie groups qualify); the learner can sample uniformly (Haar) from the group and evaluate its matrix representation. Symmetry discovery additionally assumes the unknown group is a subgroup of a known finite group with index at most 1/κ, and that the target function is isotropic (generic) within its invariant subspace so the symmetry is identifiable.
How does this compare to previous spectral averaging work?
Prior work (Soleymani et al., ICML 2025) needed a generating set of size log|G|, which is fine for finite groups but meaningless for infinite ones. This paper shows a random set of O(r) elements — independent of group size — imposes exactly the right invariance constraints, extending guarantees to infinite compact groups and enabling provable symmetry discovery under a bounded-index assumption.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- PDF extraction mangles mathematical notation (lost sub/superscripts, fraction bars, and big-O parentheses — e.g., the trial-count bounds of Proposition 4.1 and the separation parameter τ in Appendix B.3); exact constants were reconstructed from context, not verified character-for-character.
- Both the metadata abstract and the abstract inside the extracted text cut off mid-sentence at the known-symmetry setting clause; the missing content is recoverable from Theorem 4.4 (minimax-optimal finite-dimensional rate, poly(n, d, 1/κ, log|G|, log(1/δ)) time), so the review treats it as known.
- The extracted text ends at the close of the Appendix B.3 proof (p. 29), which matches the table of contents; no section appears truncated.
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