Towards Scalable Topological Regularizers
This work addresses a bottleneck in using topological features for machine learning tasks like adversarial attacks and domain adaptation, offering a scalable solution for researchers and practitioners.
The paper tackled the problem of high computational cost and unstable training in topological regularizers by proposing principal persistence measures, which compute persistent homology on many small subsamples, and provided a GPU implementation with continuous gradients. They demonstrated its efficacy on shape matching, image generation, and semi-supervised learning tasks, achieving scalable performance.
Latent space matching, which consists of matching distributions of features in latent space, is a crucial component for tasks such as adversarial attacks and defenses, domain adaptation, and generative modelling. Metrics for probability measures, such as Wasserstein and maximum mean discrepancy, are commonly used to quantify the differences between such distributions. However, these are often costly to compute, or do not appropriately take the geometric and topological features of the distributions into consideration. Persistent homology is a tool from topological data analysis which quantifies the multi-scale topological structure of point clouds, and has recently been used as a topological regularizer in learning tasks. However, computation costs preclude larger scale computations, and discontinuities in the gradient lead to unstable training behavior such as in adversarial tasks. We propose the use of principal persistence measures, based on computing the persistent homology of a large number of small subsamples, as a topological regularizer. We provide a parallelized GPU implementation of this regularizer, and prove that gradients are continuous for smooth densities. Furthermore, we demonstrate the efficacy of this regularizer on shape matching, image generation, and semi-supervised learning tasks, opening the door towards a scalable regularizer for topological features.