CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizations
This addresses the problem of leveraging topological features in structured image analysis for domains like medical imaging and computer vision, offering a novel method for a known bottleneck.
The paper tackles the challenge of integrating Cubical Multiparameter Persistence into deep learning by introducing CuMPerLay, a differentiable vectorization layer that decomposes it into learnable single-parameter persistence, resulting in improved classification and segmentation performance on benchmark datasets, especially in limited-data scenarios.
We present CuMPerLay, a novel differentiable vectorization layer that enables the integration of Cubical Multiparameter Persistence (CMP) into deep learning pipelines. While CMP presents a natural and powerful way to topologically work with images, its use is hindered by the complexity of multifiltration structures as well as the vectorization of CMP. In face of these challenges, we introduce a new algorithm for vectorizing MP homologies of cubical complexes. Our CuMPerLay decomposes the CMP into a combination of individual, learnable single-parameter persistence, where the bifiltration functions are jointly learned. Thanks to the differentiability, its robust topological feature vectors can be seamlessly used within state-of-the-art architectures such as Swin Transformers. We establish theoretical guarantees for the stability of our vectorization under generalized Wasserstein metrics. Our experiments on benchmark medical imaging and computer vision datasets show the benefit CuMPerLay on classification and segmentation performance, particularly in limited-data scenarios. Overall, CuMPerLay offers a promising direction for integrating global structural information into deep networks for structured image analysis.