LexiCan: Lexicographic Canonicalization of Point Clouds for Machine Learning Applications

Daniel Sanchez; Djordje Mihajlovic; Federica Valeau; Luka Benić; Raghuwansh Raj; Soutrik Sarangi; Snir Hordan

cs.LGOct 10, 2026

LexiCan: Lexicographic Canonicalization of Point Clouds for Machine Learning Applications

Authors: Daniel Sanchez, Djordje Mihajlovic, Federica Valeau, Luka Benić, Raghuwansh Raj, Soutrik Sarangi, Snir Hordan

Organizations: Technion - Israel Institute of Technology

Abstract

Canonicalization resolves the rotational and permutation ambiguity of 3D point clouds by selecting a unique orbit representative, maintaining their geometric information while allowing standard architectures, such as Transformers, to operate on them. Existing canonicalizations, however, exhibit discontinuity jumps on molecular configurations and non-trivial invariance errors in popular molecular datasets, which lead to ambiguity in their representation. To bridge this gap, we introduce LexiCan, a canonicalization that is provably invariant under either $O(3)$ or $SO(3)$ , empirically invariant to rotations to within $\sim 10^{-6}$ angstroms on the QM9 and QM-sym molecular datasets, and mitigates discontinuity jumps on QM-sym, while running orders of magnitude faster than an existing canonicalization. Finally, non-invariant models perform better on QM9 property prediction and ModelNet40 shape classification with LexiCan than with that canonicalization.

Figures & tables

Figure 1: Visualization of LexiCan algorithm. (1) Each point of the labeled and centered point cloud described by a string of values

w_{j}

. (2) A candidate set

\Pi(P)

for the axis of projection is chosen based on radius, labels (here, one point at distance

r_{\star}

). The projection axis is then oriented with

e_{3}

. (3) 2D canonicalization is performed on the projected point cloud, where the lexicographically minimal starting point (here

w_{2}

) is rotated around axis and aligned with

+y

. (4) This is performed across all choices of

p\in\Pi(P)

and the lexicographically minimal serialization yields the canonicalization.

ModelNet40
#Points	LexiCan	PCA	ASUN
$128$	$0.428{\pm}0.009$	$\mathbf{0.189{\pm}0.001}$	$56.8{\pm}0.2$
$256$	$0.467{\pm}0.008$	$\mathbf{0.204{\pm}0.001}$	$158.6{\pm}0.5$
$512$	$0.574{\pm}0.012$	$\mathbf{0.231{\pm}0.001}$	$562.7{\pm}1.4$
$1{,}024$	$0.782{\pm}0.023$	$\mathbf{0.279{\pm}0.001}$	$1{,}873.0{\pm}1.8$
$2{,}048$	$1.274{\pm}0.042$	$\mathbf{0.374{\pm}0.001}$	$6{,}883.9{\pm}12.4$

Table 1: Per-cloud canonicalization runtime in milliseconds (ms) on ModelNet40 and QM9, mean

\pm

standard error over clouds of the per-cloud median of

10

calls, one thread, for

\mathrm{SO}(3)

canonicalization. Best results are in bold.

Method	R1	R2	R3
PCA	$\mathbf{3.5{\times}10^{-16}}$	$5.4{\times}10^{-16}$	$8.9{\times}10^{-1}$
ASUN	$1.3{\times}10^{-15}$	$1.3{\times}10^{-15}$	$3.8{\times}10^{-2}$
LexiCan	$\mathbf{3.5{\times}10^{-16}}$	$\mathbf{4.2{\times}10^{-16}}$	$\mathbf{1.8{\times}10^{-6}}$

Table 2: Invariance error by rank: mean EMD (Å, two significant digits) between the canonical form of each QM9 molecule and that of a random rotation, translation and permutation of it. Best results are in bold.

		PCA		ASUN		LexiCan
Group	$N_{\mathrm{mol}}$	Mean	Fail%	Mean	Fail%	Mean	Fail%
$C_{2h}$	$81{,}564$	$0.564$	$99.9$	$0.616$	$50.3$	$\mathbf{0.000}$	$\mathbf{0.0}$
$C_{3h}$	$81{,}017$	$1.715$	$100.0$	$0.155$	$32.0$	$\mathbf{0.000}$	$\mathbf{0.0}$
$C_{4h}$	$25{,}668$	$1.447$	$100.0$	$0.267$	$60.0$	$\mathbf{0.000}$	$\mathbf{0.0}$
All	$188{,}249$	$1.180$	$99.9$	$0.370$	$43.8$	$\mathbf{0.000}$	$\mathbf{0.0}$

Table 3: Symmetry group canonicalization errors: mean EMD (in Å) over random transformations in QM-sym. Best results are in bold.

$p(\varepsilon)$	PCA	ASUN	LexiCan
$p(10^{-8})$	$0.171$	$0.031$	$\mathbf{0.000}$
$p(10^{-6})$	$0.515$	$0.032$	$\mathbf{0.001}$
$p(10^{-4})$	$0.519$	$0.094$	$\mathbf{0.093}$
$p(10^{-2})$	$0.543$	$0.786$	$\mathbf{0.522}$
$\alpha_{99}$	$2.7{\times}10^{7}$	$7.5{\times}10^{5}$	$\mathbf{3{\times}10^{3}}$
Jumps	$33/82$	$9/75$	$\mathbf{0/82}$

Table 4: The

\varepsilon\to 0

limit on QM-sym geometries.

\alpha_{99}

is the

99

-th percentile amplification, pooled over all

\varepsilon

, while Jumps specifies the number of geometries that show a jump on the pilot set. Best results are in bold.

Path	PCA	ASUN	LexiCan
$\mathrm{NH}_{3}$ stretch	$17$ ( $0.875$ )	$\mathbf{0}$ ( $0.001$ )	$\mathbf{0}$ ( $0.000$ )
$\mathrm{NH}_{3}$ bend	$17$ ( $0.860$ )	$\mathbf{0}$ ( $0.011$ )	$\mathbf{0}$ ( $0.001$ )
$\mathrm{CH}_{4}$ stretch	$5301$ ( $1.019$ )	$12$ ( $0.802$ )	$\mathbf{0}$ ( $0.001$ )
$\mathrm{C}_{6}\mathrm{H}_{6}$ stretch	$4$ ( $0.742$ )	$\mathbf{0}$ ( $0.018$ )	$\mathbf{0}$ ( $0.000$ )
$\mathrm{H}_{2}\mathrm{O}_{2}$ torsion	$\mathbf{3}$ ( $0.745$ )	$46$ ( $0.804$ )	$24$ ( $0.428$ )
Max move	$1.019$	$0.804$	$\mathbf{0.428}$

Table 5: Number of discontinuities (summed over

12

rotated copies) along controlled perturbation paths, with the largest single move (in Å) in parentheses. Best results are in bold.

	Flip rate	Median $\alpha$	p95 $\alpha$	Not invariant
PCA	$0.581$	$57.9$	$3.1{\times}10^{6}$	$1000/1000$
ASUN @ $10^{-2}$	$0.782$	$54.9$	$1.5{\times}10^{3}$	$456/1000$
LexiCan @ $10^{-3}$	$\mathbf{0.208}$	$\mathbf{1.52}$	$\mathbf{435}$	$\mathbf{0/1000}$

Table 6: Robustness in QM-sym (

1{,}000

molecules). Not invariant counts the molecules with one of

12

random views more than

0.1

Å (EMD) from the canonical form, out of those the algorithm canonicalizes. Best results are in bold.

Representation	MAE	RMSE
Regular	$0.81{\pm}0.07$	$1.22{\pm}0.09$
ASUN	$0.46{\pm}0.04$	$0.70{\pm}0.04$
LexiCan	$\mathbf{0.40{\pm}0.04}$	$\mathbf{0.60{\pm}0.06}$

Table 7: QM9 isotropic polarizability prediction (Transformer, Table 9 ). Test MAE and RMSE over three seeds (

\text{mean}\pm\text{STD}

), on randomly rotated, permuted and translated test molecules (see Appendix D.1 for more details).

ModelNet40 ( $n=512$ )
Model	$z/\mathrm{SO}(3)$	$\mathrm{SO}(3)/\mathrm{SO}(3)$	ASUN	LexiCan
SchNet	$27.8\pm 0.5$	$\bm{28.7\pm 0.3}$	$28.6\pm 0.7$	$28.1\pm 0.1$
EGNN	$\bm{34.9\pm 0.7}$	$31.4\pm 4.9$	$32.2\pm 0.8$	$33.4\pm 0.1$
GCN	$8.2\pm 0.6$	$15.2\pm 0.2$	$17.1\pm 2.5$	$\bm{28.6\pm 1.3}$
GCNII	$6.9\pm 0.8$	$19.1\pm 1.1$	$21.9\pm 1.3$	$\bm{29.1\pm 2.7}$
SchNet-Pos	$10.5\pm 1.1$	$29.2\pm 2.3$	$30.0\pm 0.8$	$\bm{37.1\pm 3.2}$

Table 8: Accuracy of classification (%, mean

\pm

STD over three seeds). Best results are in bold.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Transformer
Model dimension	$128$
Transformer layers	$6$
Attention heads	$8$
Feed-forward dimension	$512$
Encoder activation	GELU
Coordinate-MLP activation	SiLU

Appendix

Table 9: Final architecture and training hyperparameters selected for the Transformer model.

Hyperparameter	Value
Hidden dimension	$16$
Layers	$4$
Dropout	$0.5$
$k$ -NN graph	$k=20$
Readout	concatenated mean and max pooling, then a two-layer MLP
Trainable parameters	GCN $2{,}216$ ; GCNII $2{,}424$ ;

Appendix

Table 10: Architecture and training hyperparameters for the ModelNet40 models, shared by every model and representation.

	$n=200$				$n=1{,}024$
Model	$z/\mathrm{SO}(3)$	$\mathrm{SO}(3)/\mathrm{SO}(3)$	ASUN	LexiCan	$z/\mathrm{SO}(3)$	$\mathrm{SO}(3)/\mathrm{SO}(3)$	ASUN	LexiCan
SchNet	$27.6\pm 0.8$	$\bm{28.2\pm 0.8}$	$28.1\pm 0.4$	$27.8\pm 1.2$	$27.6\pm 0.2$	$\bm{28.2\pm 1.1}$	$27.9\pm 0.1$	$27.5\pm 1.6$
EGNN	$33.9\pm 1.7$	$33.3\pm 0.8$	$\bm{36.3\pm 0.1}$	$32.1\pm 1.1$	$25.7\pm 5.3$	$\bm{30.8\pm 2.9}$	$27.0\pm 2.5$	$28.9\pm 2.1$
GCN	$7.4\pm 0.3$	$14.7\pm 0.5$	$16.6\pm 1.5$	$\bm{25.3\pm 1.3}$	$8.9\pm 0.6$	$18.9\pm 0.6$	$20.7\pm 1.4$	$\bm{27.1\pm 1.4}$
GCNII	$7.7\pm 0.9$	$21.1\pm 1.1$	$24.4\pm 0.4$	$\bm{29.5\pm 2.1}$	$7.2\pm 1.8$	$20.8\pm 1.1$	$25.2\pm 0.3$	$\bm{27.3\pm 2.8}$
SchNet-Pos	$10.7\pm 1.0$	$28.6\pm 0.4$	$29.9\pm 0.6$	$\bm{33.1\pm 1.8}$	$10.2\pm 0.7$	$27.7\pm 1.6$	$31.3\pm 1.7$	$\bm{33.5\pm 3.1}$

Appendix

Table 11: Accuracy of classification on ModelNet40 for

n=200

and

n=1{,}024

(%, mean

\pm

STD over three seeds). Best results are in bold.

	PCA			ASUN			LexiCan
tol	Time	Flip	Non-inv.	Time	Flip	Non-inv.	Time	Flip	Non-inv.
$10^{-9}$	$\mathbf{0.184}$	$\mathbf{0.762}$	$\mathbf{1.000}$	$\mathbf{5.838}$	$0.418^{*}$	$\mathbf{0.353}$	$\mathbf{2.051}$	$0.623$	$\mathbf{0.000}$
$10^{-8}$	$\mathbf{0.184}$	$\mathbf{0.762}$	$\mathbf{1.000}$	$\underline{5.929}$	$0.418^{*}$	$\mathbf{0.353}$	$\mathbf{2.096}$	$0.623$	$\mathbf{0.000}$
$10^{-7}$	$\mathbf{0.184}$	$\mathbf{0.762}$	$\mathbf{1.000}$	$5.982$	$0.418^{*}$	$\mathbf{0.353}$	$\mathbf{2.082}$	$0.600$	$\mathbf{0.000}$
$10^{-6}$	$\mathbf{0.184}$	$\mathbf{0.762}$	$\mathbf{1.000}$	$6.001$	$0.408^{*}$	$\mathbf{0.353}$	$\mathbf{2.084}$	$0.503$	$\mathbf{0.000}$
$10^{-5}$	$\mathbf{0.184}$	$\mathbf{0.762}$	$\mathbf{1.000}$	$5.987$	$0.393^{*}$	$\mathbf{0.353}$	$\mathbf{2.094}$	$0.404$	$\mathbf{0.000}$

Appendix

Table 12: Tolerance sweep across the

18

labeled point clouds. Time is the mean per-cloud canonicalization time in ms (median over 10 calls per cloud). Flip is the fraction of perturbations whose canonical form moved more than

10\times

further than the input did, pooled over six noise magnitudes (

\sigma=10^{-7},\ldots,10^{-2}

of the cloud’s RMS radius) and twelve draws per cloud. Non-inv. is the fraction of clouds whose random rotations give different canonical forms. Per method and column, bold marks the best tol and underline the second best (times compared to two significant figures);

*

marks a tol at which the method fails to return a canonicalization for some inputs. LexiCan ranks labels rarest first. PCA has no tolerance (one setting, repeated in every row). Best results are in bold.

	PCA			ASUN			LexiCan
tol	Time	Flip	Non-inv.	Time	Flip	Non-inv.	Time	Flip	Non-inv.
$10^{-9}$	$\mathbf{0.206}$	$\mathbf{0.581}$	$\mathbf{1.000}$	$38.6$	$0.897$	$\mathbf{0.276}$	$\mathbf{1.349}$	$0.534$	$\mathbf{0.000}$
$10^{-8}$	$\mathbf{0.206}$	$\mathbf{0.581}$	$\mathbf{1.000}$	$\underline{38.1}$	$0.897$	$\mathbf{0.276}$	$\underline{1.388}$	$0.534$	$\mathbf{0.000}$
$10^{-7}$	$\mathbf{0.206}$	$\mathbf{0.581}$	$\mathbf{1.000}$	$\underline{37.9}$	$0.896$	$\underline{0.297}$	$1.690$	$0.519$	$\mathbf{0.000}$
$10^{-6}$	$\mathbf{0.206}$	$\mathbf{0.581}$	$\mathbf{1.000}$	$\underline{37.8}$	$0.875$	$0.468$	$1.795$	$0.457$	$\mathbf{0.000}$
$10^{-5}$	$\mathbf{0.206}$	$\mathbf{0.581}$	$\mathbf{1.000}$	$\underline{37.6}$	$0.861$	$0.413$	$1.802$	$0.373$	$\mathbf{0.000}$

Appendix

Table 13: Tolerance sweep across

1{,}000

QM-sym molecules Time is the mean per-cloud canonicalization time in ms (median over 10 calls per cloud). Flip is the fraction of perturbations whose canonical form moved more than

10\times

further than the input did, pooled over six noise magnitudes (

\sigma=10^{-7},\ldots,10^{-2}

of the cloud’s RMS radius) and twelve draws per cloud. Non-inv. is the fraction of molecules with one of

12

random views more than

0.1

Å (EMD) from the canonical form. Per method and column, bold marks the best tol and underline the second best (times compared to two significant figures);

*

Figure 2: Each point represents the fraction of $500$ random perturbation directions, at $\varepsilon=10^{-8}$ , that move the canonical form by more than $0.1$ of the cloud’s RMS radius, for each geometry. The geometries are sorted descending on the x axis independently per method. ASUN is drawn over the $75$ geometries it can process; it raises an assertion on the octahedron and the six bipyramids at every tolerance.

	Flip rate	Median $\alpha$	p95 $\alpha$	Not invariant
PCA	$0.762$	$1.4{\times}10^{3}$	$7.5{\times}10^{6}$	$18/18$
ASUN @ $10^{-2}$	$0.356$	$2.14$	$1.7{\times}10^{6}$	$6/17$
LexiCan @ $10^{-3}$	$\mathbf{0.196}$	$\mathbf{1.25}$	$\mathbf{516}$	$\mathbf{0/18}$
LexiCan @ $10^{-5}$	$0.404$	$1.66$	$5.4{\times}10^{4}$	$\mathbf{0/18}$

Appendix

Table 14: Algorithms applied to the

18

labeled point clouds spanning polyhedra, molecules, and adversarial cases (see Figure 3 ). Flip rate is the fraction of perturbations whose canonical form varied

10\times

further than the input moved,

\alpha

is the output movement divided by the input movement (1 means no amplification), and not invariant counts the clouds whose random rotations give different canonical forms, out of those the algorithm canonicalizes. Best results are in bold.

Figure 3: The

18

labeled point clouds used in the perturbation study. The set represents a mix of four molecules, symmetric polyhedra, unstructured clouds and three adversarial cases. Color indicates species, while edges are drawn only to make the shape legible and carry no information beyond the points themselves. Each cloud is scaled independently, so relative size across panels is not meaningful.

Representation	MAE	RMSE
Regular	$0.81{\pm}0.07$	$1.22{\pm}0.09$
ASUN	$0.46{\pm}0.04$	$0.70{\pm}0.04$
LexiCan ${}_{@10^{-3}}$	$0.40{\pm}0.04$	$0.60{\pm}0.06$
LexiCan ${}_{@10^{-5}}$	$\mathbf{0.38{\pm}0.02}$	$\mathbf{0.59{\pm}0.03}$

Appendix

Table 15: QM9 isotropic polarizability prediction (Transformer). Test MAE and RMSE over three seeds (

\text{mean}\pm\text{STD}

), on randomly rotated, permuted and translated test molecules, as in Table 7 . Regular is trained on the stored QM9 poses.

ModelNet40 ( $n=512$ )
Model	$z/\mathrm{SO}(3)$	$\mathrm{SO}(3)/\mathrm{SO}(3)$	ASUN	LexiCan ${}_{@10^{-3}}$	LexiCan ${}_{@10^{-5}}$
SchNet	$27.8\pm 0.5$	$\bm{28.7\pm 0.3}$	$28.6\pm 0.7$	$28.1\pm 0.1$	$28.7\pm 1.1$
EGNN	$\bm{34.9\pm 0.7}$	$31.4\pm 4.9$	$32.2\pm 0.8$	$33.4\pm 0.1$	$33.7\pm 4.0$
GCN	$8.2\pm 0.6$	$15.2\pm 0.2$	$17.1\pm 2.5$	$\bm{28.6\pm 1.3}$	$27.9\pm 1.7$
GCNII	$6.9\pm 0.8$	$19.1\pm 1.1$	$21.9\pm 1.3$	$\bm{29.1\pm 2.7}$	$27.8\pm 0.4$
SchNet-Pos	$10.5\pm 1.1$	$29.2\pm 2.3$	$30.0\pm 0.8$	$\bm{37.1\pm 3.2}$	$31.8\pm 0.4$

Appendix

Table 16: Accuracy of classification (%, mean

\pm

STD over three seeds). Best results are in bold.

Invariant Representation Learning

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When and How to Canonize: A Generalization Perspective

Yonatan Sverdlov, Benjamin Friedman, Snir Hordan, Nadav Dym

Technion – Israel Institute of Technology

Invariant Representation Learning Generalization Bounds

Open paper AskPDF arXiv

While invariant architectures are standard for processing symmetric data, there is growing interest in achieving invariance by applying group averaging or canonization to non-invariant backbones. However, the theoretical generalization properties of these alternative strategies remain poorly understood. We introduce a theoretical framework to analyze the generalization error of these methods by bounding their covering numbers. We establish a rigorous generalization hierarchy: the error bounds of canonized models are at best equal to the error bounds of structurally invariant and group-averaged models, and at worst equal to the bounds of non-invariant baselines. Furthermore, we show that there exist optimal canonizations which attain the optimal error bounds, and poor canonizations which attain the non-invariant error bounds, and that this depends on the regularity of the canonization. Finally, applying this framework to permutation groups in point cloud processing, we rigorously prove that the covering number of lexicographical sorting grows exponentially with point cloud dimension, whereas Hilbert curve canonization guarantees polynomial growth. This provides the first formal theoretical justification for the empirical success of Hilbert curve serialization in state-of-the-art point cloud architectures. We conclude with experiments that support our theoretical claims. Code is available at https://github.com/yonatansverdlov/Canonization

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