Spectral Re-Basin for Linear Mode Connectivity

Linear mode connectivity is often analysed through exact parameter symmetries, especially hidden-unit permutations that leave the network function unchanged. Yet, many relationships between trained networks are not captured by bijective coordinate relabelings: widths may differ, neurons may split or clone, and explicit parameter symmetries may be broken. In this paper, we introduce *spectral re-basin*, taking an operator perspective on hidden-unit geometry and studying mode connectivity through the functional geometry of hidden units. Specifically, we represent each hidden layer as a graph of *neurons* built from layerwise descriptors (weights or activations), associate it with a graph shift operator, and study two coupled objects: layerwise graph functional symmetries, which preserve the relational geometry, and intertwining correspondences across layers from two models, which align their graph structures. We show that permutations arise as a special case of this viewpoint and derive results for non-bijective constructions. Our analysis connects correspondence quality to preserved spectral modes, functional exchangeability of neurons, and bounds on endpoint distortion and linear path barriers. These findings suggest that linear mode connectivity can be governed beyond explicit parameter symmetry by broader spectral operator compatibility in hidden-unit geometry.