What question did this study set out to answer?

The aim is to develop a post hoc method that provides stable and accurate explanations in deep learning models under group transformations.

April 10, 2026Open Access

Equivariant Transition Matrices for Explainable Deep Learning: A Lie Group Linearization Approach

Key Points

The aim is to develop a post hoc method that provides stable and accurate explanations in deep learning models under group transformations.
Proposed Equivariant Transition Matrices that incorporate Lie-group structural constraints.
Estimated infinitesimal generators in both formal and mental feature spaces.
Implemented diagnostics for symmetry validation and introduced an unsupervised strategy for selecting regularization weights.
Solved a convex Least-Squares problem using singular value decomposition for small networks.
Reduced symmetry defect from 13,100 to 0.0425 on a synthetic benchmark.
Mean squared error increased from 0.00367 to 0.00524 slightly.
On the MNIST dataset, symmetry defect decreased by 72.6 percent (from 141.19 to 38.65).
Changed structural similarity and achieved peak signal-to-noise ratios below 0.03 percent and 0.06 percent.

Abstract

Deep learning systems deployed in regulated settings require explanations that are accurate and stable under nuisance transformations, yet classical post hoc transition matrices rely on fidelity-only fitting that fails to guarantee consistent explanations under spatial rotations or other group actions. In this work, we propose Equivariant Transition Matrices, a post hoc approach that augments transition matrices with Lie-group-aware structural constraints to bridge this research gap. Our method estimates infinitesimal generators in the formal and mental feature spaces, enforces an approximate intertwining relation at the Lie algebra level, and solves the resulting convex Least-Squares problem via singular value decomposition for small networks or implicit operators for large systems. We introduce diagnostics for symmetry validation and an unsupervised strategy for regularization weight selection. On a controlled synthetic benchmark, our approach reduces the symmetry defect from 13,100 to 0.0425 while increasing the mean squared error marginally from 0.00367 to 0.00524. On the MNIST dataset, the symmetry defect decreases by 72.6 percent (141.19 to 38.65) with changes in structural similarity and peak signal-to-noise ratio below 0.03 percent and 0.06 percent, respectively. These results demonstrate that explanation-level equivariance can be reliably imposed post-training, providing geometrically consistent interpretations for fixed deep models.

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