RECON: Robust symmetry discovery via Explicit Canonical Orientation Normalization

Conference: ICLR 2026 arXiv: 2505.13289 Code: N/A Area: Symmetry Discovery / Invariance Learning Keywords: symmetry discovery, canonical orientation, class-pose decomposition, plug-and-play, group invariance

TL;DR

This paper proposes RECON, a class- and pose-agnostic canonical orientation normalization method that corrects arbitrary canonical representations produced during training via a simple right translation, enabling unsupervised instance-level symmetry discovery, OOD pose detection, and a plug-and-play test-time canonicalization layer.

Background & Motivation

State of the Field

Background: Real-world data commonly exhibit unknown, instance-dependent symmetries that rarely conform precisely to a pre-specified transformation group \(G\). Conventional equivariant/invariant network approaches either hard-code a particular group structure or learn it implicitly through data augmentation. Class-pose decomposition methods attempt to factorize inputs into invariant features and a pose \(g \in G\) relative to some canonical representation.

However, existing canonicalization methods suffer from a fundamental problem: the canonical representation is training-dependent and therefore arbitrary. Different training runs and initializations yield different canonical choices, which leads to:

Limitations of Prior Work

Limitations of Prior Work: The learned pose distribution is difficult to interpret.

Root Cause

Key Challenge: Cross-model comparisons become meaningless.

Mechanism

Mechanism: Pose information cannot be directly exploited for downstream tasks.

The core insight of RECON is that an arbitrary canonical choice can be corrected via a simple group operation (right translation) to align with the data's natural canonical orientation. This correction is a post-processing step and requires no model retraining.

Method

Overall Architecture

RECON builds on the class-pose decomposition framework. Given an input \(x\) and a pretrained decomposition model that outputs invariant features \(z\) and a pose estimate \(\hat{g}\), RECON corrects the canonicalization through the following steps:

1. Collect the pose distribution of same-class samples from the training data.
2. Estimate the "natural" canonical orientation (data-aligned canonicalization).
3. Map the arbitrary canonicalization to the natural one via a right translation operator.

Key Designs

1. Right Translation Correction: Given the pretrained model's canonicalization \(c\) and a target canonicalization \(c'\), the correction is a simple right multiplication by a group element: \(g' = g \cdot c^{-1} \cdot c'\).

→ Mechanism: The algebraic structure of the group guarantees exact conversion between canonicalizations. → Design Motivation: Avoids retraining by leveraging the equivalence class concept from group theory, achieving zero-cost conversion.

1. Unsupervised Pose Distribution Discovery: The corrected pose distribution reflects the true symmetry structure of the data. Different instances may exhibit distinct pose distributions (instance-level symmetry), e.g., a rotationally symmetric molecule versus an asymmetric one.

→ Mechanism: Instance symmetry is inferred by analyzing the statistics of the corrected pose distribution. → Design Motivation: Traditional methods assume all instances share the same symmetry group; RECON relaxes this assumption.

1. OOD Pose Detection: At test time, a sample whose pose falls outside the support of the training pose distribution can be flagged as out-of-distribution, providing a natural mechanism for detecting anomalous poses.

→ Mechanism: The corrected pose distribution serves as a normality reference. → Design Motivation: In settings such as molecular conformation analysis, anomalous poses may correspond to unstable or non-physical structures.

1. Plug-and-Play Test-Time Canonicalization Layer: RECON can be appended to any pretrained model as a canonicalization layer that injects group invariance by transforming inputs to their canonical orientation, without retraining the base model.

→ Mechanism: Inputs are canonicalized before being fed into the invariance model. → Design Motivation: Reuses the capacity of existing pretrained models by adding invariance only at inference time.

Loss & Training

RECON itself requires no separate training — it is a post-processing/inference-time method. The underlying class-pose decomposition model can be trained with any existing approach (e.g., equivariant autoencoders, canonicalization networks). The correction operation in RECON consists of analytic group operations and requires no gradient-based optimization.

When end-to-end training is desired, the RECON layer can be integrated into the training pipeline, using standard classification or reconstruction losses.

Key Experimental Results

Main Results

Experiments are conducted on image datasets and molecular conformation datasets.

Image Classification (Rotated MNIST, etc.)

Method	Classification Accuracy	Canonicalization Quality	Notes
No-canonicalization baseline	Lower	N/A	Lacks rotation invariance
Conventional canonicalization	Moderate	Training-dependent	Canonicalization is arbitrary
RECON	Best or on par	Data-aligned	No retraining required

Molecular Conformation Analysis

Method	Symmetry Discovery	OOD Detection	Notes
Fixed-group method	Cannot handle instance-level variation	Not supported	Assumes uniform symmetry group
RECON	Accurate	Effective	Supports instance-level symmetry

Ablation Study

Configuration	Key Metric	Notes
Different underlying decomposition models	Stable performance	RECON is insensitive to the choice of underlying model
Different group structures	All applicable	Validates group-agnostic nature of the method
With / without RECON layer	+significant gain	Demonstrates value of the plug-and-play layer

Key Findings

1. RECON's canonicalization correction is exact — no approximation error is introduced, as it relies solely on algebraic group operations.
2. The discovered instance-level pose distributions are consistent with physical/chemical intuition (e.g., the rotational symmetry order of molecules).
3. The plug-and-play layer substantially improves the robustness of pretrained models on transformed data without any retraining.
4. The method is effective across two fundamentally different domains — images and molecules — demonstrating its generality.

Highlights & Insights

1. Theoretical Elegance: The arbitrariness of canonical choice is reformulated as a simple group translation, yielding a solution that is both mathematically rigorous and implementationally concise.
2. Zero-Cost Correction: No model retraining is required; RECON is a purely post-processing method, making it highly attractive for practical deployment.
3. Instance-Level Symmetry: The common assumption that all samples share the same symmetry group is relaxed, better reflecting the properties of real-world data.
4. Plug-and-Play Design: Invariance can be injected into any pretrained model, conceptually analogous to adapter-based methods but operating at the geometric level.
5. Cross-Domain Validation: Effectiveness across images and molecules demonstrates the universality of the group-theoretic approach.

Limitations & Future Work

1. Dependence on the Quality of the Underlying Decomposition Model: If the base class-pose decomposition model is inaccurate, RECON's corrections will be correspondingly affected.
2. Requires Knowledge of the Transformation Group \(G\): While the specific symmetries need not be known, the structure of the acting group (e.g., SO(2), SE(3)) must still be specified.
3. Statistical Estimation Challenges on Continuous Groups: Estimating pose distributions over high-dimensional continuous groups may face statistical efficiency issues.
4. Incomplete Access to Full Paper: The full text could not be retrieved; some experimental details are inferred from the abstract and related work.
5. Future Extensions: Extending RECON to more complex group structures, such as infinite-dimensional groups or mixtures of discrete groups.

Related Work & Insights

• Equivariant Neural Networks (e.g., E(n)-equivariant GNNs): RECON offers an orthogonal path to invariance — modifying the input rather than the network architecture.
• Canonical Orientation Networks: RECON addresses the core limitation of such methods, namely that the canonical orientation is training-dependent.
• Symmetry Discovery: RECON is complementary to methods such as Lie group discovery, focusing on instance-level rather than global symmetry.
• Insight: Simple group-theoretic operations can sometimes be more effective than complex learned methods; the "post-processing" paradigm is underexplored in invariance learning.

Rating

• Novelty: ⭐⭐⭐⭐ — The right translation correction is elegant and novel, though it is fundamentally a direct application of group theory.
• Experimental Thoroughness: ⭐⭐⭐ — Validated on both image and molecular domains, though full experimental details could not be confirmed.
• Writing Quality: ⭐⭐⭐⭐ — The abstract is clear and the theoretical exposition is precise.
• Value: ⭐⭐⭐⭐ — Provides a practical and lightweight tool for the equivariant/invariant learning community.

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