Topological methods for comparing weighted graphs are valuable in various learning tasks but often suffer from computational inefficiency on large datasets. We introduce RTD-Lite, a scalable algorithm that efficiently compares topological features, specifically connectivity or cluster structures at arbitrary scales, of two weighted graphs with one-to-one correspondence between vertices. By leveraging minimal spanning trees in auxiliary graphs, RTD-Lite captures topological discrepancies with $O(n^2)$ time and memory complexity. This efficiency enables its application in tasks like dimensionality reduction and neural network training. Experiments on synthetic and real-world datasets demonstrate that RTD-Lite effectively identifies topological differences while significantly reducing computation time compared to existing methods. Moreover, integrating RTD-Lite into neural network training as a loss function component enhances the preservation of topological structures in learned representations. Our code is publicly available at \url{https://github.com/ArGintum/RTD-Lite.}

International audience

We propose TopDis (Topological Disentanglement), a method for learning disentangled representations via adding a multi-scale topological loss term. Disentanglement is a crucial property of data representations substantial for the explainability and robustness of deep learning models and a step towards high-level cognition. The state-of-the-art methods are based on VAE and encourage the joint distribution of latent variables to be factorized. We take a different perspective on disentanglement by analyzing topological properties of data manifolds. In particular, we optimize the topological similarity for data manifolds traversals. To the best of our knowledge, our paper is the first one to propose a differentiable topological loss for disentanglement learning. Our experiments have shown that the proposed TopDis loss improves disentanglement scores such as MIG, FactorVAE score, SAP score, and DCI disentanglement score with respect to state-of-the-art results while preserving the reconstruction quality. Our method works in an unsupervised manner, permitting us to apply it to problems without labeled factors of variation. The TopDis loss works even when factors of variation are correlated. Additionally, we show how to use the proposed topological loss to find disentangled directions in a trained GAN.