Ensuring Topological Data-Structure Preservation under Autoencoder Compression due to Latent Space Regularization in Gauss--Legendre nodes
This work addresses the challenge of maintaining data topology in unsupervised learning for reliable compression across domains like medical imaging and fashion datasets, though it appears incremental as it builds on existing regularization techniques.
The authors tackled the problem of preserving topological data structures in autoencoder compression by introducing a data-independent latent space regularization based on Gauss-Legendre nodes, proving that it ensures a one-to-one re-embedding of the data manifold. Demonstrations on datasets like FashionMNIST and MRI brain scans showed that this method prevents topological defects where prior strategies fail, delivering reliable low-dimensional representations.
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.