  We formulate a data independent latent space regularisation constraint for
general unsupervised autoencoders. The regularisation rests on sampling the
autoencoder Jacobian in Legendre nodes, being the centre of the Gauss-Legendre
quadrature. Revisiting this classic enables to prove that regularised
autoencoders ensure a one-to-one re-embedding of the initial data manifold to
its latent representation. Demonstrations show that prior proposed
regularisation strategies, such as contractive autoencoding, cause topological
defects already for simple examples, and so do convolutional based
(variational) autoencoders. In contrast, topological preservation is ensured
already by standard multilayer perceptron neural networks when being
regularised due to our contribution. This observation extends through the
classic FashionMNIST dataset up to real world encoding problems for MRI brain
scans, suggesting that, across disciplines, reliable low dimensional
representations of complex high-dimensional datasets can be delivered due to
this regularisation technique.
