The geometry of meaning in neural manifolds

Just as different languages express the same concept through different words, neural networks express similar functions through incompatible coordinate systems. Different architectures, random initializations, and optimization trajectories produce incompatible parameterizations in extrinsic coordinates, obscuring functional relationships between independently trained models. The question is: when models independently solve similar tasks on related data, do their learned representations share exploitable structure despite incompatible coordinates? This thesis shows that semantic correspondence is a fundamental primitive revealing shared geometric structure beneath incompatible parameterizations. By identifying which elements refer to the same semantic content, correspondence breaks the symmetry of arbitrary implementation choices, anchoring incompatible systems to a shared semantic frame and revealing the underlying intrinsic geometry that ultimately enables zero-shot transformation between independently trained models. Building on this principle, we walk through fourteen scientific contributions exploring correspondence-based alignment across diverse settings: zero-shot model stitching, cross-modal translation, policy transfer, activation steering, and neural decoding. These methods span domains (vision, language, reinforcement learning, neuroscience), architectures (CNNs, Transformers, graph networks), and scales (from small supervised models to billion-parameter generative systems). Each exploits different geometric properties—angular, metric, spectral, sparse, or distributional—yet all instantiate the same underlying principle. Remarkably, correspondence-based alignment extends beyond artificial systems: fMRI data show that different human brains processing identical stimuli exhibit alignable geometric structure, suggesting a general principle of learning systems. From a practical standpoint, correspondence-based alignment enables model reuse without expensive joint training, knowledge transfer between independently developed systems, efficient transformation by exploiting low-dimensional semantic structure, and direct evaluation of representations in latent space without requiring downstream task deployment. The methods we developed have achieved top-tier recognition (oral and spotlight presentations) in ML conferences and widespread adoption, contributing to weakly supervised pretraining, out-of-distribution detection, federated learning, and policy transfer in RL, and formed the conceptual foundation for the UniReps workshops at NeurIPS (2023–2025). By revealing intrinsic geometric structure beneath extrinsic parameterization, semantic correspondence transforms model isolation into interoperability.