Group-Theoretic Structure Governing Identifiability in Inverse Problems
Isshin Arai, Tomoaki Itano
TL;DR
The paper addresses identifiability in inverse causal inference for symmetric physical systems by casting reconstruction as finding an $SO(3)$-equivariant group homomorphism $F \in \mathrm{Hom}_{SO(3)}(V_{\mathcal{C}_N^*}, V_{\mathcal{D}})$ between representation spaces. It leverages the Clebsch–Gordan decomposition $V_1 \otimes V_1 = V_0 \oplus V_1 \oplus V_2$ to derive theoretical identifiability limits as a function of the number of observations $N$, predicting when each component of the velocity-gradient tensor is reconstructible. The authors verify these limits numerically with an $SO(3)$-equivariant neural network (VGN) implemented in the e3nn framework, showing that learning respects the symmetry and closely follows the predicted identifiability boundaries, with substantially reduced covariance error compared to non-equivariant baselines. The work provides a universal, symmetry-driven principle for reconstructability in causal inverse problems and offers concrete design guidance for equivariant estimators, including when to expect full recovery of causal factors and how to structure representations.
Abstract
In physical systems possessing symmetry, reconstructing the underlying causal structure from observational data constitutes an inverse problem of fundamental importance. In this work, we formulate the inverse problem of causal inference within the framework of group-representation theory, clarifying the structure of the representation spaces to which the {\it causality} and estimation maps belong. This formulation leads to both theoretical and practical limits of reconstructability (identifiability). We show that the local velocity-gradient tensor, regarded as a {\it causal factor}, can be reconstructed from the orientational motion of suspended particles. In this setting, the estimation map must act as a group homomorphism between the observation and causal spaces, and the reconstructable subspace is constrained by the decomposition structure of the SO(3) representation. Based on this principle, we construct an SO(3)-equivariant neural network (implemented with the e3nn framework) and verify that the identifiability determined by the group-representation structure is reproduced in the actual learning process. These results demonstrate a fundamental principle that the group-representation structure determines the reconstructability (identifiability limit) in inverse problems of causal inference.