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.

我们提出了一种基于数据无关的潜在空间正则化约束，用于一对一重新嵌入初始数据流形到其潜在表示。通过神经网络的正则化，我们实现了拓扑保持，从 FashionMNIST 数据集到 MRI 脑部扫描等真实世界的编码问题，这种正则化技术可以提供可靠的复杂高维数据的低维表示。