Group-Theoretic Upper Bounds on Reconstructability in Inverse Problems
Reconstructing the causal structure of physical systems from observational data constitutes a fundamental inverse problem. Here we show that the reconstruction dimension---defined as an upper bound on the number of recoverable components---is determined by the group-representation structure of the observation spaces and reconstruction maps. This formulation provides an explicit and operational characterization of reconstructability and reconstruction dimension, extending ideas that are often understood only intuitively in equivariant representation theory. As a concrete example, we demonstrate the reconstruction of the local velocity-gradient tensor from orientational measurements of particles suspended in flows, where the observation and velocity-gradient tensor spaces form SO(3) representations with constrained equivariant maps between them. Using an SO(3)-equivariant neural network (implemented with e3nn), we show that the reconstructable subspaces predicted by the representation decomposition are qualitatively consistent with those found in practice. Our formulation shows that the representation structure constrains reconstructability by determining an upper bound sector by sector, while our numerical results suggest that the actual saturation of the bound depends on the physics and data geometry. Beyond providing a useful theoretical framework, this work also connects the abstract representation-theoretic structure to concrete inverse reconstruction problems in fluid physics.