Graphcodes handle datasets that are filtered along two real-valued scale parameters.

Datasets often possess an intrinsic multiscale structure with meaningful descriptions at different levels of coarseness. Such datasets are naturally described as multi-resolution clusterings, i.e., not necessarily hierarchical sequences of partitions across scales. To analyse and compare such sequences, we use tools from topological data analysis and define the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter filtration of abstract simplicial complexes that encodes cluster intersection patterns across scales. The MCbiF can be interpreted as a higher-order extension of Sankey diagrams and reduces to a dendrogram for hierarchical sequences. We show that the multiparameter persistent homology (MPH) of the MCbiF yields a finitely presented and block decomposable module, and its stable Hilbert functions characterise the topological autocorrelation of the sequence of partitions. In particular, at dimension zero, the MPH captures violations of the refinement order of partitions, whereas at dimension one, the MPH captures higher-order inconsistencies between clusters across scales. We demonstrate through experiments the use of MCbiF Hilbert functions as topological feature maps for downstream machine learning tasks. MCbiF feature maps outperform information-based baseline features on both regression and classification tasks on synthetic sets of non-hierarchical sequences of partitions. We also show an application of MCbiF to real-world data to measure non-hierarchies in wild mice social grouping patterns across time.

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.

Graphcodes handle datasets that are filtered along two real-valued scale parameters.

We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale parameters. Such multi-parameter topological summaries are usually based on complicated theoretical foundations and difficult to compute; in contrast, graphcodes yield an informative and interpretable summary and can be computed as efficient as one-parameter summaries. Moreover, a graphcode is simply an embedded graph and can therefore be readily integrated in machine learning pipelines using graph neural networks. We describe such a pipeline and demonstrate that graphcodes achieve better classification accuracy than state-of-the-art approaches on various datasets.