The local-distance conjecture for autoencoder generalization

Let $\mathcal{D}_{train}$ and $\mathcal{D}_{test}$ be the training and test subsets of the dataset, and let $D_{model}\circ E_{model}$ be a trained autoencoder. Near each training point, compare this map with a projection map, equivalently measuring its local distance from the identity through its Jacobian. Local-distance conjecture. The further the map $D_{model}\circ E_{model}$ is from a projection near the training points, the worse the mean squared error will be for the autoencoder on the test points. This proposes a relationship between local geometric deviation from a projection on training data and test-set reconstruction error; the supplied text gives the relationship as a conjecture rather than an established result.