Chuan-Shen Hu

Chuan-Shen Hu

Combinatorial and topological structures, such as graphs, simplicial complexes, and cell complexes, form the foundation of geometric and topological deep learning (GDL and TDL) architectures. These models aggregate signals over such domains, integrate local features, and generate representations for diverse real-world applications. However, the distribution and diffusion behavior of GDL and TDL features during training remains an open and underexplored problem. Motivated by this gap, we introduce a cellular sheaf theoretic framework for modeling and analyzing the local consistency and harmonicity of node features and edge weights in graph-based architectures. By tracking local feature alignments and agreements through sheaf structures, the framework offers a topological perspective on feature diffusion and aggregation. Furthermore, a multiscale extension inspired by topological data analysis (TDA) is proposed to capture hierarchical feature interactions in graph models. This approach enables a joint characterization of GDL and TDL architectures based on their underlying geometric and topological structures and the learned signals defined on them, providing insights for future studies on conventional tasks such as node classification, substructure detection, and community detection.

Yu-Min Chung, Sarah Day, Chuan-Shen Hu

The field of mathematical morphology offers well-studied techniques for image processing. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis. We demonstrate that morphological operations naturally form a multiparameter filtration and that persistent homology can then be used to extract information about both topology and geometry in the images as well as to automate methods for optimizing the study and rendering of structure in images. For illustration, we apply this framework to analyze noisy binary, grayscale, and color images.

Chuan-Shen Hu, Yu-Min Chung

This paper concerns a theoretical approach that combines topological data analysis (TDA) and sheaf theory. Topological data analysis, a rising field in mathematics and computer science, concerns the shape of the data and has been proven effective in many scientific disciplines. Sheaf theory, a mathematics subject in algebraic geometry, provides a framework for describing the local consistency in geometric objects. Persistent homology (PH) is one of the main driving forces in TDA, and the idea is to track changes of geometric objects at different scales. The persistence diagram (PD) summarizes the information of PH in the form of a multi-set. While PD provides useful information about the underlying objects, it lacks fine relations about the local consistency of specific pairs of generators in PD, such as the merging relation between two connected components in the PH. The sheaf structure provides a novel point of view for describing the merging relation of local objects in PH. It is the goal of this paper to establish a theoretic framework that utilizes the sheaf theory to uncover finer information from the PH. We also show that the proposed theory can be applied to identify the branch numbers of local objects in digital images.

Yu-Min Chung, Chuan-Shen Hu, Yu-Lun Lo et al.

Persistent homology (PH) is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general pipeline to apply PH to study time series; particularly the instantaneous heart rate time series for the heart rate variability (HRV) analysis. The first step is capturing the shapes of time series from two different aspects -- {the PH's and hence persistence diagrams of its} sub-level set and Taken's lag map. Second, we propose a systematic {and computationally efficient} approach to summarize persistence diagrams, which we coined {\em persistence statistics}. To demonstrate our proposed method, we apply these tools to the HRV analysis and the sleep-wake, REM-NREM (rapid eyeball movement and non rapid eyeball movement) and sleep-REM-NREM classification problems. The proposed algorithm is evaluated on three different datasets via the cross-database validation scheme. The performance of our approach is better than the state-of-the-art algorithms, and the result is consistent throughout different datasets.

Yu-Min Chung, Chuan-Shen Hu, Austin Lawson et al.

Skin cancer is one of the most common cancers in the United States. As technological advancements are made, algorithmic diagnosis of skin lesions is becoming more important. In this paper, we develop algorithms for segmenting the actual diseased area of skin in a given image of a skin lesion, and for classifying different types of skin lesions pictured in a given image. The cores of the algorithms used were based in persistent homology, an algebraic topology technique that is part of the rising field of Topological Data Analysis (TDA). The segmentation algorithm utilizes a similar concept to persistent homology that captures the robustness of segmented regions. For classification, we design two families of topological features from persistence diagrams---which we refer to as {\em persistence statistics} (PS) and {\em persistence curves} (PC), and use linear support vector machine as classifiers. We also combined those topological features, PS and PC, into ResNet-101 model, which we call {\em TopoResNet-101}, the results show that PS and PC are effective in two folds---improving classification performances and stabilizing the training process. Although convolutional features are the most important learning targets in CNN models, global information of images may be lost in the training process. Because topological features were extracted globally, our results show that the global property of topological features provide additional information to machine learning models.