CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizations

2025-10-14

AI Summary

Cubical multiparameter persistence (CMP) is difficult to vectorize and integrate into deep learning pipelines due to its inherent complexity and lack of differentiability. Method: This paper proposes CuMPerLay—a differentiable, end-to-end trainable CMP vectorization layer. It decomposes CMP into learnable combinations of single-parameter persistent homology, jointly optimizes dual filtration functions, and provides theoretical stability guarantees under the generalized Wasserstein metric. CuMPerLay is architecture-agnostic and seamlessly integrates with mainstream models (e.g., Swin Transformer) without modifying downstream network structures. Contribution/Results: Experiments on medical imaging and standard CV benchmarks demonstrate significant performance gains in image classification and segmentation—particularly under few-shot settings. To our knowledge, this is the first work to realize differentiable vectorization of CMP, establishing a novel paradigm for tightly integrating topological features into deep learning frameworks.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Mining of Visual, Multimedia & Multimodal Data

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Integrating cubical multiparameter persistence into deep learning pipelines
• Overcoming complexity in vectorizing multifiltration topological structures
• Enabling differentiable topological features for image analysis tasks

Innovation

• Integrates cubical multiparameter persistence into deep learning
• Decomposes CMP into learnable single-parameter persistence components
• Provides differentiable topological features compatible with modern architectures