Expectation of topological invariants

In this paper, we study the expectation values of topological invariants of the Vietoris-Rips complex and \v{C}ech complex for a finite set of sample points on a Riemannian manifold. We show that the Betti number and Euler characteristic of the complexes are Lipschitz functions of the scale parameter and that there is an interval such that the Betti curve converges to the Betti number of the underlying manifold.Comment: 9 pages; 1 figur

An Arrhythmia Classification-Guided Segmentation Model for Electrocardiogram Delineation

Accurate delineation of key waveforms in an ECG is a critical initial step in extracting relevant features to support the diagnosis and treatment of heart conditions. Although deep learning based methods using a segmentation model to locate P, QRS and T waves have shown promising results, their ability to handle signals exhibiting arrhythmia remains unclear. In this study, we propose a novel approach that leverages a deep learning model to accurately delineate signals with a wide range of arrhythmia. Our approach involves training a segmentation model using a hybrid loss function that combines segmentation with the task of arrhythmia classification. In addition, we use a diverse training set containing various arrhythmia types, enabling our model to handle a wide range of challenging cases. Experimental results show that our model accurately delineates signals with a broad range of abnormal rhythm types, and the combined training with classification guidance can effectively reduce false positive P wave predictions, particularly during atrial fibrillation and atrial flutter. Furthermore, our proposed method shows competitive performance with previous delineation algorithms on the Lobachevsky University Database (LUDB)