Ensuring Topological Data-Structure Preservation under Autoencoder Compression Due to Latent Space Regularization in Gauss–Legendre Nodes

We formulate a data-independent latent space regularization constraint for general unsupervised autoencoders. The regularization relies on sampling the autoencoder Jacobian at Legendre nodes, which are the centers of the Gauss–Legendre quadrature. Revisiting this classic allows us to prove that regularized autoencoders ensure a one-to-one re-embedding of the initial data manifold into its latent representation. Demonstrations show that previously proposed regularization strategies, such as contractive autoencoding, cause topological defects even in simple examples, as do convolutional-based (variational) autoencoders. In contrast, topological preservation is ensured by standard multilayer perceptron neural networks when regularized using our approach. This observation extends from the classic FashionMNIST dataset to (low-resolution) MRI brain scans, suggesting that reliable low-dimensional representations of complex high-dimensional datasets can be achieved using this regularization technique.