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.

Comparison of data representations is a complex multi-aspect problem. We propose a method for comparing two data representations. We introduce the Representation Topology Divergence (RTD) score measuring the dissimilarity in multi-scale topology between two point clouds of equal size with a one-to-one correspondence between points. The two data point clouds can lie in different ambient spaces. The RTD score is one of the few topological data analysis based practical methods applicable to real machine learning datasets. Experiments show the agreement of RTD with the intuitive assessment of data representation similarity. The proposed RTD score is sensitive to the data representation’s fine topological structure. We use the RTD score to gain insights on neural networks representations in computer vision and NLP domains for various problems: training dynamics analysis, data distribution shift, transfer learning, ensemble learning, disentanglement assessment.

We propose a framework for comparing data manifolds, aimed, in particular, towards the evaluation of deep generative models. We describe a novel tool, Cross-Barcode(P,Q), that, given a pair of distributions in a high-dimensional space, tracks multiscale topology spacial discrepancies between manifolds on which the distributions are concentrated. Based on the Cross-Barcode, we introduce the Manifold Topology Divergence score (MTop-Divergence) and apply it to assess the performance of deep generative models in various domains: images, 3D-shapes, time-series, and on different datasets: MNIST, Fashion MNIST, SVHN, CIFAR10, FFHQ, market stock data, ShapeNet. We demonstrate that the MTop-Divergence accurately detects various degrees of mode-dropping, intra-mode collapse, mode invention, and image disturbance. Our algorithm scales well (essentially linearly) with the increase of the dimension of the ambient high-dimensional space. It is one of the first TDA-based methodologies that can be applied universally to datasets of different sizes and dimensions, including the ones on which the most recent GANs in the visual domain are trained. The proposed method is domain agnostic and does not rely on pre-trained networks.

Click logs provide a unique and highly valuable source of human judgments on ads' relevance. However, clicks are heavily biased by lots of factors. Two main factors that are widely acknowledged to be the most influential ones are neighboring ads and presentation order. The latter is referred to as positional effect. A popular practice to recover the ads quality cleaned from positional bias is to adopt click models based on examination or cascade hypothesis originally developed for organic search. In this paper we show the strong evidence that this practice is far from perfection when considering the top ads block on a search engine result page (SERP). We show that cascade hypothesis is the most questionable one because of important differences between organic and sponsored search results that may encourage users to analyze the whole ads-block before clicking. Additionally, we design a testing setup for an unbiased evaluation of click model prediction accuracy.