Manifold approximation
Scope of local manifold studies
This page examines how a Taylor-series expansion describes a local class transition, and what that description does and does not establish about prediction.
The perturbation expansion
The expansion writes the label y(Horse given x) as a function of the latent representation z, obtained through the chain of primitive features edges, corners, and formal parts. The base class β is shifted by a difference term Δβ.
Treating the shift as a smooth perturbation gives a second-order approximation in Δβ. This is a statement about a local neighbourhood of the base class, not a global model of unseen classes.
Theory versus practical prediction
Local zero-shot adaptation
Within a small neighbourhood of the base class, the expansion is a coherent local description. It is a theoretical claim about smoothness, and it is the strongest part of the argument.
Out-of-distribution failure modes
Smoothness does not guarantee accuracy across distant shifts. Where the true transition departs from the second-order approximation, predictions may degrade, and no evidence in this material measures how often that happens.
Direct channel
Send a technical inquiry
Direct communication remains open for questions about the formulated representations and the expansion described above.