Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.2000 Mathematics Subject Classification. 55B55, 68U05, 68Q17, 13P25 (primary) .
The pairwise interaction paradigm of graph machine learning has predominantly governed the modelling of relational systems. However, graphs alone cannot capture the multi-level interactions present in many complex systems and the expressive power of such schemes was proven to be limited. To overcome these limitations, we propose Message Passing Simplicial Networks (MP-SNs), a class of models that perform message passing on simplicial complexes (SCs) -topological objects generalising graphs to higher dimensions. To theoretically analyse the expressivity of our model we introduce a Simplicial Weisfeiler-Lehman (SWL) colouring procedure for distinguishing non-isomorphic SCs. We relate the power of SWL to the problem of distinguishing non-isomorphic graphs and show that SWL and MPSNs are strictly more powerful than the WL test and not less powerful than the 3-WL test. We deepen the analysis by comparing our model with traditional graph neural networks with ReLU activations in terms of the number of linear regions of the functions they can represent. We empirically support our theoretical claims by showing that MPSNs can distinguish challenging strongly regular graphs for which GNNs fail and, when equipped with orientation equivariant layers, they can improve classification accuracy in oriented SCs compared to a GNN baseline. Additionally, we implement a library for message passing on simplicial complexes that we envision to release in due course.
Gastric cancer is one of the deadliest cancers worldwide. An accurate prognosis is essential for effective clinical assessment and treatment. Spatial patterns in the tumor microenvironment (TME) are conceptually indicative of the staging and progression of gastric cancer patients. Using spatial patterns of the TME by integrating and transforming the multiplexed immunohistochemistry (mIHC) images as Cell-Graphs, we propose a graph neural network-based approach, termed Cell−GraphSignatureorCGSignature, powered by artificial intelligence, for the digital staging of TME and precise prediction of patient survival in gastric cancer. In this study, patient survival prediction is formulated as either a binary (short-term and long-term) or ternary (short-term, medium-term, and long-term) classification task. Extensive benchmarking experiments demonstrate that the CGSignature achieves outstanding model performance, with Area Under the Receiver Operating Characteristic curve of 0.960 ± 0.01, and 0.771 ± 0.024 to 0.904 ± 0.012 for the binary- and ternary-classification, respectively. Moreover, Kaplan–Meier survival analysis indicates that the “digital grade” cancer staging produced by CGSignature provides a remarkable capability in discriminating both binary and ternary classes with statistical significance (P value < 0.0001), significantly outperforming the AJCC 8th edition Tumor Node Metastasis staging system. Using Cell-Graphs extracted from mIHC images, CGSignature improves the assessment of the link between the TME spatial patterns and patient prognosis. Our study suggests the feasibility and benefits of such an artificial intelligence-powered digital staging system in diagnostic pathology and precision oncology.
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