Towards Scalable Topological Regularizers

2025-01-24

AI Summary

Conventional feature discrepancy measures for latent-space distribution matching are computationally expensive and neglect geometric-topological structure; meanwhile, persistent homology-based methods suffer from poor scalability and training instability. Method: This paper proposes a scalable topological regularization framework centered on a novel lightweight topological regularizer based on subsampled persistent homology. Contribution/Results: We theoretically prove that the regularizer’s gradient is continuous with respect to input density—ensuring stable backpropagation—and enable efficient GPU-accelerated computation, overcoming topological regularization bottlenecks for point clouds exceeding one thousand points. Extensive experiments on shape matching, image generation, and semi-supervised learning demonstrate significant improvements in training stability and scalability to large-scale settings.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

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.

Problem

• Feature Difference Measurement
• Computational Efficiency
• Persistent Homology

Innovation

• Persistent Homology
• GPU Parallel Computing
• Shape Feature Control