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Visualization in the emerging field of topological data analysis has progressed from persistence barcodes and persistence diagrams to display of two-parameter persistent homology. Although persistence barcodes and diagrams have permitted…
A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent…
Topological features based on persistent homology capture high-order structural information so as to augment graph neural network methods. However, computing extended persistent homology summaries remains slow for large and dense graphs and…
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine…
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…
Time-delay embedding is a fundamental technique in Topological Data Analysis (TDA) for reconstructing the phase space dynamics of time-series data. Persistent homology effectively identifies global topological features, such as loops…
Persistent homology is a tool that can be employed to summarize the shape of data by quantifying homological features. When the data is an object in $\mathbb{R}^d$, the (augmented) persistent homology transform ((A)PHT) is a family of…
Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data. Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not…
We improve and extend persistence spheres, introduced in~\cite{pegoraro2025persistence}. Persistence spheres map an integrable measure $\mu$ on the upper half-plane, including persistence diagrams (PDs) as counting measures, to a function…
Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially…
We explore extended B-splines as a stable basis for isogeometric analysis with trimmed parameter spaces. The stabilization is accomplished by an appropriate substitution of B-splines that may lead to ill-conditioned system matrices. The…
The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the $\infty$-norm of…
\v{C}ech Persistence diagrams (PDs) are topological descriptors routinely used to capture the geometry of complex datasets. They are commonly compared using the Wasserstein distances $OT_{p}$; however, the extent to which PDs are stable…
This article studies the robust version of persistent homology based on trimming methodology to capture the geometric feature through support of the data in presence of outliers. Precisely speaking, the proposed methodology works when the…
Persistent homology is an important methodology in topological data analysis which adapts theory from algebraic topology to data settings. Computing persistent homology produces persistence diagrams, which have been successfully used in…
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…
We propose a novel method for topological analysis of unweighted graphs which is based on \textit{persistent homology}. The proposed method maps the input graph to a complete weighted graph where the weighting function maps each edge to a…
Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of…
Distance measures play an important role in shape classification and data analysis problems. Topological distances based on Reeb graphs and persistence diagrams have been employed to obtain effective algorithms in shape matching and scalar…
Topological Data Analysis (TDA), a relatively new field of data analysis, has proved very useful in a variety of applications. The main persistence tool from TDA is persistent homology in which data structure is examined at many scales.…