Transformation Laws in Neural Representations: Structure, Realisability, and Construction

Abstract:How neural representations preserve the structure of input changes connects representation analysis with internal intervention. We study operable representational content through compatible actions of reference transformations on neural features. We characterise when a transformation descends through an encoder, and give a linear setting in which the defect is governed by the transformation's demand for discarded information, measured in the metric the representation induces. On a rectifier the failure to realise a transformation has two distinguishable sources --- what the source region has already made unrecoverable, and what it costs to satisfy every region the transformation visits with one operator --- and for a \textit{measured} harmonic carrier the same question has a closed answer: a linear realisation exists exactly when the retained harmonic blocks are invariant under the action. Using colour as the in-depth instance, we find that hue orbits in frozen visual features concentrate 84--88\% of their energy in the first two harmonics with rotation planes shared across shapes, that this organisation is substantially inherited from input and architecture and is reshaped by training and depth, and that the measured structure supports prediction, transport from new starting states, and composition --- with global and local realisations differing sharply in which they achieve. Guided by the measurements, we construct a compact interface whose rotation action is fixed by the structure and never fitted: it reads hue zero-shot at 3.4$^\circ$ median error on unseen shapes. Theory, structural measurement, and construction together establish transformation laws as a concrete object connecting the understanding of neural representations to their design.

Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE)

I.2.6; I.4.10; I.5.1

arXiv:2609.18190 [cs.LG]

(or arXiv:2609.18190v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.18190 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.