Related papers: On the Structural Theorem of Persistent Homology
A central challenge in topological data analysis is the interpretation of barcodes. The classical algebraic-topological approach to interpreting homology classes is to build maps to spaces whose homology carries semantics we understand and…
This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…
Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of…
A topological shape analysis is proposed and utilized to learn concepts that reflect shape commonalities. Our approach is two-fold: i) a spatial topology analysis of point cloud segment constellations within objects. Therein constellations…
Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page.…
Frame theory provides a robust method for recovering vectors in a Hilbert space from inner product data, though the associated decomposition formula can be computationally demanding. We relax the frame condition by studying sequences that…
We generalize a recent result by J.F. Carlson to finite tensor categories having finitely generated cohomology. Specifically, we show that if the Krull dimension of the cohomology ring is sufficiently large, then there exist infinitely many…
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…
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…
Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that utilize persistence diagrams in statistical inference, a full Bayesian treatment remains…
The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of data, the persistence barcode tracks the evolution of its homology groups. In this paper, we…
We study the interaction of structural subtyping with parametric polymorphism and recursively defined type constructors. Although structural subtyping is undecidable in this setting, we describe a notion of parametricity for type…
We introduce a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology. From a zigzag filtration of complexes $(K_i)$, we introduce a zigzag Morse…
This work introduces a new class of symmetric matrix structures, called harmonic structures, which enable the generation of all possible directed transitions $(x_i, x_{i+1})$ over a set of $n$ symbols, without internal repetitions. Unlike…
While topological data analysis has emerged as a powerful paradigm for structural inference, its foundational tools, notably persistent homology and the persistent Laplacian, are frequently insensitive to localized structural fluctuations…
In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across a filtration. In this work, we extend this setting, and propose the use of bipath persistent homology,…
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional…
In data clustering, it is often desirable to find not just a single partition into clusters but a sequence of partitions that describes the data at different scales (or levels of coarseness). A natural problem then is to analyse and compare…
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and…
In a recent paper, an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices -- the so called `matrix properties'. In this paper we extend this algorithm to…