相关论文: Applications of Zigzag Persistence to Topological …

Topological Data Analysis (TDA) is a rising field of computational topology in which the topological structure of a data set can be observed by persistent homology. By considering a sequence of sublevel sets, one obtains a filtration that…

We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window (or time-delay) embeddings, as seen through the lens of…

Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules,…

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates with data sets. Here we introduce a…

We characterize structures such as monotonicity, convexity, and modality in smooth regression curves using persistent homology. Persistent homology is a key tool in topological data analysis that detects higher-dimensional topological…

Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using persistence landscapes…

Persistent Homology (PH) is a fundamental tool in computational topology, designed to uncover the intrinsic geometric and topological features of data across multiple scales. Originating within the broader framework of Topological Data…

The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base…

Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines…

Topological data analysis (TDA) is a rising field in the intersection of mathematics, statistics, and computer science/data science. The cornerstone of TDA is persistent homology, which produces a summary of topological information called a…

Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…

In this note one shows that the four persistence diagrams and measures defined in [5] are particular cases of the maps and the measures discussed in [2] and [1]

Duality results connecting persistence modules for absolute and relative homology provides a fundamental understanding into persistence theory. In this paper, we study similar associations in the context of zigzag persistence. Our main…

Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques.…

The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of…

Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting, exploiting the theory of point…

Persistent homology and zigzag persistent homology are techniques which track the homology over a sequence of spaces, outputting a set of intervals corresponding to birth and death times of homological features in the sequence. This paper…

Topological data analysis is a relatively new branch of machine learning that excels in studying high dimensional data, and is theoretically known to be robust against noise. Meanwhile, data objects with mixed numeric and categorical…

Link prediction is an important learning task for graph-structured data. In this paper, we propose a novel topological approach to characterize interactions between two nodes. Our topological feature, based on the extended persistent…

Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. Additionally,…