CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizations
Caner Korkmaz, Brighton Nuwagira, Barış Coşkunuzer, Tolga Birdal
TL;DR
CuMPerLay tackles the challenge of integrating Cubical Multiparameter Persistence into deep learning by slicing the multiparameter grid into single-parameter filtrations and applying learnable PersLay-style vectorizations per row, enabling end-to-end differentiable training with CNNs and Swin transformers. The authors establish stability under generalized $W_p$ metrics and provide a CUDA GPU implementation, combining a learnable multifiltration decoder with a four-part topology-aware learning block. Empirically, the approach yields improvements over single-parameter and non-learnable MP baselines on medical imaging and semantic segmentation, with pronounced benefits in limited-data scenarios. Overall, the work offers a practical pathway to infuse global topological structure into modern deep learning architectures for structured image analysis.
Abstract
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.