Learning Operators for Symmetry Discovery: From Lie Groups to Local Symmetry Geometry

Pavan Karjol; Scindhiya Laxmi; Prathosh AP

cs.LGOct 12, 2026

Learning Operators for Symmetry Discovery: From Lie Groups to Local Symmetry Geometry

Authors: Pavan Karjol, Scindhiya Laxmi, Prathosh AP

Organizations: Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India · Department of Mathematics and Computing, IIT (ISM) Dhanbad, India

Abstract

Continuous symmetry discovery is typically formulated as a separate estimation problem for each function or dataset. We instead investigate whether the map from a function to its symmetry structure can be learned across a family of functions from finite observations. Using the infinitesimal differential characterization of continuous symmetries, we formulate this map as an inverse symmetry operator, analogous to an inverse PDE problem. For global Lie-group symmetries, we establish a representation-theoretic population characterization: the symmetry Lie algebra is the nullspace of a positive-semidefinite operator constructed from representation coefficients and the infinitesimal group action. We also show that the inverse symmetry operator is equivariant under coordinate changes through the adjoint action. This characterization separates the geometric symmetry map from finite-sample estimation, motivating a hybrid approach that amortizes statistically difficult components while retaining analytic structure. On SO(5) benchmarks, amortized equivariant estimation of harmonic coefficients improves symmetry recovery from sparse observations, whereas ordinary least squares suffices for SL(2,R). We extend this viewpoint to local symmetries, where symmetry directions form smoothly varying tangent distributions rather than a fixed Lie algebra. Under the Lie-bracket consistency condition, Frobenius' theorem guarantees local integral submanifolds along which observables remain unchanged. We recover this geometry using a hybrid estimator that learns observation weights while preserving analytic differential and geometric recovery. Across global, non-compact, local, and physically motivated synthetic benchmarks, our results support learning the statistically difficult finite-sample components while preserving the known geometric structure of symmetry recovery.

Figures & tables

Panel of Figure 1 — Figure 1: Finite-sample symmetry recovery. Top: global $\mathrm{SO}(5)$ symmetry discovery. Bottom: recovery of spatially varying local symmetry distributions. Amortization is most effective when applied to the statistically difficult finite-sample inference step while retaining the analytic symmetry construction.

	$N=32,\ \sigma=.05$		$N=1024$ , clean
Method	$\mathrm{SO}(4)$	$\mathrm{SO}(5)$	$\mathrm{SO}(4)$	$\mathrm{SO}(5)$
AEHE	.869/.598	.586/1.410	.961/.179	.892/.135
AHC	.948/.417	.546/1.452	1.000/.00241	.998/.0126
Analytic harmonic	.200/2.400	.167/3.333	1.000/ $4.9{\times}10^{-7}$	1.000/ $4.0{\times}10^{-17}$
Beyond-Affine	.288/1.886	.318/2.563	1.000/ $2.09{\times}10^{-5}$	1.000/ $3.71{\times}10^{-6}$

Table 1: Physically motivated synthetic applications. Entries report dimension accuracy / hard-projector error. We show a sparse noisy setting and a well-sampled clean setting; full sample-size and noise sweeps are reported in the appendix.

	$\mathrm{SO}(4)$ magnetic		$\mathrm{SO}(5)$ superspin
Method	Proj.	Angle	Proj.	Angle
AEHE	.1499	$12.53^{\circ}$	.0109	$4.03^{\circ}$
AHC	.00244	$1.98^{\circ}$	.0112	$4.08^{\circ}$
Beyond-Affine	$5.36{\times}10^{-5}$	$0.19^{\circ}$	$3.66{\times}10^{-4}$	$0.51^{\circ}$

Table 2: Oracle-dimension geometry at

N=1024

and

\sigma=.05

. Supplying the true symmetry dimension isolates geometric subspace recovery from dimension selection.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Notation	Description
$G$	Continuous transformation group acting on the domain
$\mathfrak{g}$	Lie algebra of $G$
$X$	Domain/manifold on which $G$ acts
$x$	Point in the domain $X$
$g$	Element of the transformation group $G$
$f$	Scalar-valued function whose symmetries are to be discovered

Appendix

Table 3: Summary of the notation used throughout the manuscript.

	AEHE			AHC		Analytic harmonic		Beyond-Affine
$N$	Dim.	Hard	Oracle	Dim.	Hard	Dim.	Hard	Dim.	Hard	Oracle
32	.668	1.265	1.107	.514	1.737	.111	7.440	.130	2.497	1.530
64	.834	.794	.705	.566	1.359	.123	7.146	.123	2.377	1.119
128	.931	.531	.500	.646	1.022	.201	4.299	.214	1.890	.300
256	.973	.306	.295	.746	.730	.226	2.210	.866	.159	.109
512	.989	.126	.117	.807	.530	.253	1.764	.879	.131	.082

Appendix

Table 4: Untouched

\mathrm{SO}(5)

final-test results. “Dim.” is symmetry- dimension accuracy, “hard” is

\|\widehat{P}-P_{\mathfrak{h}_{f}}\|_{F}^{2}

, and “oracle” supplies the true dimension only when converting the estimated stabilizer metric into a projector. Lower projector error is better.

Configuration	Mean dim. acc. (%)	Dim. acc. (%)	Hard acc. (%)	Hard projector MSE	Soft projector MSE
Supervised $L=32$	$70.251\pm.240$	$73.525\pm.874$	$80.934\pm.405$	$.291\pm.011$	$.282\pm.012$
Semi $L=32$	$85.519\pm.165$	$97.680\pm.072$	$99.215\pm.060$	$.024\pm.001$	$.018\pm.001$
Supervised $L=128$	$77.194\pm.753$	$86.163\pm 1.974$	$92.233\pm.755$	$.139\pm.020$	$.118\pm.025$
Semi $L=128$	$85.044\pm.426$	$97.261\pm.370$	$99.255\pm.201$	$.028\pm.003$	$.020\pm.002$
Supervised $L=512$	$82.445\pm.235$	$95.021\pm.430$	$97.907\pm.759$	$.051\pm.004$	$.036\pm.004$
Trace-only $L=512$	$82.629\pm.261$	$95.394\pm.133$	$98.068\pm.302$	$.047\pm.001$	$.035\pm.001$

Appendix

Table 5:

\mathrm{SO}(3)

untouched final-test results. Mean accuracy averages the five observation resolutions. The remaining columns use

N=512

. “Hard” denotes the no-

\ell=2

subset.

$N$	32	64	128	256	512	1024
Analytic $\ell=2$ (%)	23.12	36.25	53.75	74.38	90.62	93.12
Full learned (%)	79.73	89.23	96.39	99.16	99.89	99.99

Appendix

Table 6:

\mathrm{SO}(4)

symmetry-dimension accuracy for the analytic

\ell=2

estimator and the full learned-calibration model.

$N$	AEHE	AHC	Analytic LS	Beyond-Affine
32	.875/.5806	.942/.4119	.263/2.248	.672/.3638
64	.934/.3709	.991/.1378	1.000/ $4.194\!\times\!10^{-5}$	.797/.2036
128	.953/.2711	1.000/.03202	1.000/ $7.544\!\times\!10^{-6}$	.994/.007439
256	.956/.2193	1.000/.01308	1.000/ $2.594\!\times\!10^{-6}$	1.000/.0001293
512	.959/.1958	1.000/.004843	1.000/ $1.174\!\times\!10^{-6}$	1.000/ $4.472\!\times\!10^{-5}$
1024	.961/.1787	1.000/.002411	1.000/ $4.897\!\times\!10^{-7}$	1.000/ $2.087\!\times\!10^{-5}$

Appendix

Table 7:

\mathrm{SO}(4)

quaternion-energy sweep. Each method reports symmetry-dimension accuracy / hard projector error.

$N$	AEHE	AHC	Analytic LS	Beyond-Affine
32	.614/1.382	.537/1.403	.167/3.333	.586/.6798
64	.701/.8294	.714/.7667	.172/3.285	.956/.05366
128	.791/.4232	.875/.3199	1.000/ $9.181\!\times\!10^{-16}$	.997/.002937
256	.839/.2439	.951/.1023	1.000/ $2.237\!\times\!10^{-16}$	1.000/ $3.226\!\times\!10^{-5}$
512	.858/.1968	.991/.03344	1.000/ $9.118\!\times\!10^{-17}$	1.000/ $8.794\!\times\!10^{-6}$
1024	.892/.1348	.998/.01261	1.000/ $4.037\!\times\!10^{-17}$	1.000/ $3.710\!\times\!10^{-6}$

Appendix

Table 8:

\mathrm{SO}(5)

superspin-energy sweep. Each method reports symmetry-dimension accuracy / hard projector error.

	Mean projector error				Median largest angle ( ∘ )
$N$	ALW	AL	BA	Direct neural	ALW	AL	BA	Direct neural
64	.01608	.02055	.02132	.26016	3.65	4.12	4.21	17.77
128	.00856	.01017	.01717	.21298	2.63	2.93	3.73	15.84
256	.00490	.00617	.01588	.18710	2.02	2.30	3.53	14.77
512	.00293	.00382	.01529	.17501	1.57	1.85	3.45	14.32
1024	.00177	.00233	.01501	.17143	1.24	1.46	3.40	14.24

Appendix

Table 9: Local-symmetry final-test comparison. Projector error is the mean squared Frobenius error; angle is the median largest principal angle in degrees. Lower is better.

$N$	Method	Mean angle $\downarrow$	Proj. err. $\downarrow$
512	Harmonic calibration	$36.78^{\circ}$	.794
	LieGAN, general	$71.66^{\circ}$	1.307
	LieGAN, $\mathrm{SO}(5)$	$71.83^{\circ}$	1.307
1024	Harmonic calibration	$34.50^{\circ}$	.745
	LieGAN, general	$68.82^{\circ}$	1.269
	LieGAN, $\mathrm{SO}(5)$	$68.91^{\circ}$	1.266

Appendix

Table 10: Matched coupled-class pilot, averaged over the three subclasses and three optimization seeds or frozen checkpoints. Projector error is squared Frobenius error.

Class	Method	Angle $\downarrow$	Proj. $\downarrow$	Off-plane $\downarrow$	Ratio
$1{:}1$	Harmonic cal.	$36.96^{\circ}$	.769	.517	1.224
	LieGAN, gen.	$63.40^{\circ}$	1.189	.796	.725
	LieGAN, $\mathrm{SO}(5)$	$66.77^{\circ}$	1.228	.814	.833
$1{:}2$	Harmonic cal.	$14.25^{\circ}$	.346	.075	4.190
	LieGAN, gen.	$64.82^{\circ}$	1.220	.654	.836
	LieGAN, $\mathrm{SO}(5)$	$64.99^{\circ}$	1.224	.666	.909

Appendix

Table 11: Per-class matched results at

N=2048

. Angle, projector error, and off-plane leakage are means; rate ratio is the median. Ground-truth rate ratios are

1

2

, and

1.5

Class	Dim. acc.	Proj. err. $\downarrow$	Angle $\downarrow$	Median ratio
$1{:}1$	1.000	.00581	$2.46^{\circ}$	1.018
$1{:}2$	1.000	.00449	$2.40^{\circ}$	2.114
$2{:}3$	1.000	.00238	$1.87^{\circ}$	1.537

Appendix

Table 12: AEHE coupled-class recovery on the untouched 3E final test at

N=2048

. Projector error is squared Frobenius error.

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