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This paper studies how neural representations form during training, not just where they end up, and reports that trajectory geometry (measured with Bures distance and quantum Fisher information) relates to the computation's interaction structure and to later stability. The findings are empirical and span synthetic vector inputs and CNNs on images, so their generality is still open. For ML researchers, the practical question is whether path-based metrics can serve as cheap diagnostics or training targets beyond endpoint evaluation.
The Information Geometry of Representation Formation
arXiv:2610.34015v1 Announce Type: new Abstract: Most analyses of neural representations focus on trained endpoints, leaving the process by which internal structure forms comparatively unmeasured. We instead treat representation formation itself as an object of study, asking whether the path a representation takes through training reveals information that its endpoint does not. We formalize this idea by viewing representation learning as a geometric trajectory, using Bures geometry to measure movement through representation space and quantum Fisher information to characterize local sensitivity to input factors. We find that learning concentrates local sensitivity on task-relevant factors, and that changing only the prediction target reorganizes this geometry around the corresponding visual factor. More strikingly, representation trajectories become systematically less direct as the learned computation requires higher-order interactions among input factors. This effect persists under matched-performance and training-duration controls and transfers from synthetic vector inputs to residual CNNs on rendered and natural images. Importantly, formation history contains predictive information about the stability of the learned representation: controlling for interaction order, path tortuosity predicts its subsequent drift under continued training and sensitivity to parameter perturbations, whereas endpoint displacement does not. Finally, representation-aware optimization lets us actively shape these trajectories, showing that formation geometry is not only measurable but actionable. Together, these results establish representation formation as a new source of information about learning: its geometry reveals properties of the computation being constructed, predicts subsequent representational stability, and provides a target for intervention during optimization.
Abstract:Most analyses of neural representations focus on trained endpoints, leaving the process by which internal structure forms comparatively unmeasured. We instead treat representation formation itself as an object of study, asking whether the path a representation takes through training reveals information that its endpoint does not. We formalize this idea by viewing representation learning as a geometric trajectory, using Bures geometry to measure movement through representation space and quantum Fisher information to characterize local sensitivity to input factors. We find that learning concentrates local sensitivity on task-relevant factors, and that changing only the prediction target reorganizes this geometry around the corresponding visual factor. More strikingly, representation trajectories become systematically less direct as the learned computation requires higher-order interactions among input factors. This effect persists under matched-performance and training-duration controls and transfers from synthetic vector inputs to residual CNNs on rendered and natural images. Importantly, formation history contains predictive information about the stability of the learned representation: controlling for interaction order, path tortuosity predicts its subsequent drift under continued training and sensitivity to parameter perturbations, whereas endpoint displacement does not. Finally, representation-aware optimization lets us actively shape these trajectories, showing that formation geometry is not only measurable but actionable. Together, these results establish representation formation as a new source of information about learning: its geometry reveals properties of the computation being constructed, predicts subsequent representational stability, and provides a target for intervention during optimization.
33 pages, 9 figures. Code available at this https URL
Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2610.34015 [cs.LG]
(or arXiv:2610.34015v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.34015 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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