Multi-Parameter Topological VAEs for Conditioning Image Generation

Topological Data Analysis (TDA) is the field of data science whose aim is to detect and encode topological patterns in data sets. As such, it has become common in machine learning to complement traditional models with TDA-based features and losses to enforce geometrical constraints into the models behaviour. However, while several methods can ensure that topology can be preserved at a global level (eg, total number of components or cycles), constraining topology locally (eg, imposing the presence of topological features at specific areas in space) has been relatively unexplored, despite its usefulness in several application domains. In this article, we propose to fill this gap with multi-parameter persistent homology. More precisely, we leverage a recently proposed descriptor, called the Multi-parameter Module Approximation (MMA), to design new losses for generative models. Indeed, MMA descriptors are easily interpretable and are thus more amenable to the construction of losses than other standard descriptors. On the theoretical side, we prove that MMA descriptors are locally Lipschitz and definable (under suitable assumptions), so that stochastic subgradient descent converges, which allows us to use them for steering neural net behaviours at training time. On the practical side, we showcase the efficiency of our localized topological constraints for mask and image generation with variational autoencoders, by providing superior results over both single-parameter and non-topological baselines.