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Self-Supervised Detection of Perfect and Partial Input-Dependent Symmetries

Alonso Urbano, David W. Romero

TL;DR

This work tackles the problem that fixed group symmetries can misalign with real data, causing overly constrained models. It introduces a self-supervised framework that learns input-dependent symmetry distributions by constraining an Invariant-Equivariant Autoencoder (IE-AE) and training a symmetry-boundary predictor, with the symmetry boundary denoted by $\theta_x$ and the boundary predictor $\\Theta = \\omega \\circ \\phi$. Key contributions include (i) per-input symmetry detection without labels, (ii) a constraint that collapses canonical representations to the center of symmetry $c_{[x]}$ via $\\psi(c_{[x]}) = e$, (iii) the ability to detect out-of-distribution symmetries and to standardize data by removing symmetries, and (iv) generality to distributions such as Gaussian and cyclic. The framework supports both continuous and discrete symmetry families and enables symmetry standardization, which improves downstream performance for non-equivariant models and enables effective OOD symmetry detection with practical applicability.

Abstract

Group equivariance can overly constrain models if the symmetries in the group differ from those observed in data. While common methods address this by determining the appropriate level of symmetry at the dataset level, they are limited to supervised settings and ignore scenarios in which multiple levels of symmetry co-exist in the same dataset. In this paper, we propose a method able to detect the level of symmetry of each input without the need for labels. Our framework is general enough to accommodate different families of both continuous and discrete symmetry distributions, such as arbitrary unimodal, symmetric distributions and discrete groups. We validate the effectiveness of our approach on synthetic datasets with different per-class levels of symmetries, and demonstrate practical applications such as the detection of out-of-distribution symmetries. Our code is publicly available at https://github.com/aurban0/ssl-sym.

Self-Supervised Detection of Perfect and Partial Input-Dependent Symmetries

TL;DR

This work tackles the problem that fixed group symmetries can misalign with real data, causing overly constrained models. It introduces a self-supervised framework that learns input-dependent symmetry distributions by constraining an Invariant-Equivariant Autoencoder (IE-AE) and training a symmetry-boundary predictor, with the symmetry boundary denoted by and the boundary predictor . Key contributions include (i) per-input symmetry detection without labels, (ii) a constraint that collapses canonical representations to the center of symmetry via , (iii) the ability to detect out-of-distribution symmetries and to standardize data by removing symmetries, and (iv) generality to distributions such as Gaussian and cyclic. The framework supports both continuous and discrete symmetry families and enables symmetry standardization, which improves downstream performance for non-equivariant models and enables effective OOD symmetry detection with practical applicability.

Abstract

Group equivariance can overly constrain models if the symmetries in the group differ from those observed in data. While common methods address this by determining the appropriate level of symmetry at the dataset level, they are limited to supervised settings and ignore scenarios in which multiple levels of symmetry co-exist in the same dataset. In this paper, we propose a method able to detect the level of symmetry of each input without the need for labels. Our framework is general enough to accommodate different families of both continuous and discrete symmetry distributions, such as arbitrary unimodal, symmetric distributions and discrete groups. We validate the effectiveness of our approach on synthetic datasets with different per-class levels of symmetries, and demonstrate practical applications such as the detection of out-of-distribution symmetries. Our code is publicly available at https://github.com/aurban0/ssl-sym.
Paper Structure (21 sections, 6 theorems, 26 equations, 8 figures, 6 tables)

This paper contains 21 sections, 6 theorems, 26 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariant-Equivariant Autoencoder
  4. Partial Group Equivariant Convolutional Neural Networks
  5. Self-Supervised Detection of Perfect and Partial Input-Dependent Symmetries
  6. Learning input-depending symmetries from data
  7. Learning constrained group action estimators
  8. Estimation of the symmetry boundary
  9. Self-supervised learning of the symmetry boundaries
  10. Symmetry standardization
  11. Considering other groups and symmetry distributions
  12. Related work
  13. Experiments
  14. Conclusions and limitations
  15. Background
  16. ...and 6 more sections

Key Result

Proposition 3.1

Consider a ${\mathcal{G}}$-invariant autoencoder $\delta \circ \eta$ and a group action estimator $\psi$. Under the assumption of uniformity of symmetries in ${\mathcal{X}}$, the following statements are equivalent:

Figures (8)

  • Figure 1: Self-supervised detection of input-dependent symmetries. In real world scenarios, different classes of objects present different levels of symmetries (left). Nevertheless, existing methods assume the same distribution of symmetries for all elements of the dataset. Our method can identify and determine the distribution of symmetries inherent to each input (right).
  • Figure 2: Overview of our proposed method.
  • Figure 3: Canonical orientations obtained during inference by the IE-AE winter2022unsupervised(left) and our method (right). Both models are trained on MNISTRot60-90, a dataset exhibiting uniform rotational symmetries within $[-60^\circ, 60^\circ]$ for digits $0$ to $4$ and $[-90^\circ, 90^\circ]$ for digits $5$ to $9$. Our method is able to consistently choose the center of each input's symmetry distribution as the canonical representation (Prop. \ref{['theorem:uniform']}(ii)).
  • Figure 4: Distribution of the transformations predicted by $\psi$ with the IE-AE and our method in class 8 from MNISTRot60-90 (top). Our group action estimator $\bar{\psi}$ correctly captures the different input-dependent distributions in the dataset by means of the constraint in Proposition \ref{['theorem:uniform']}(i)(bottom).
  • Figure 5: Prediction of input-dependent levels of symmetry on rotation augmented variants of MNIST and FashionMNIST.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • ...and 3 more