Related papers: 2-Categorical Foundations for Multiparameter Persi…
Given a finite dimensional, bigraded module over the polynomial ring in two variables, we define its two-parameter count, a natural number, and its end-curves, a set of plane curves. These are two-dimensional analogues of the notions of…
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…
Persistent homology studies the birth and death of cycles in a parameterized family of spaces. In this paper, we study the birth and death of cycles in a multifiltration of a chain complex with the goal of producing a persistence diagram…
We study the recently introduced notion of output-input stability, which is a robust variant of the minimum-phase property for general smooth nonlinear control systems. The subject of this paper is developing the theory of output-input…
Motivation: Radiomics refers to the high-throughput mining of quantitative features from radiographic images. It is a promising field in that it may provide a non-invasive solution for screening and classification. Standard machine learning…
Membrane particles such as proteins and lipids organize into zones that perform unique functions. Here, I introduce a topological and category-theoretic framework to represent particle and zone intra-scale interactions and inter-scale…
Expansion of the categorical point of view on many areas of the mathematics and mathematical physics will cause to deeper understanding of genuine features of these problems. New applications of categorical methods are connected with new…
Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to…
The analysis of stability of community structure is an important problem for scientists from many fields. Here, we propose a new framework to reveal hidden properties of community structure by quantitatively analyzing the dynamics of Potts…
A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed.…
We introduce a construction and an algorithm, both based on Topological Data Analysis (TDA), to tackle the problem of the isomorphism check of Orthogonal Arrays (OAs). Specifically, we associate to any binary OA a persistence diagram, one…
Deep neural network algorithms are difficult to analyze because they lack structure allowing to understand the properties of underlying transforms and invariants. Multiscale hierarchical convolutional networks are structured deep…
The multiple-biomarker classifier problem and its assessment are reviewed against the background of some fundamental principles from the field of statistical pattern recognition, machine learning, or the recently so-called "data science". A…
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 following article is an application of commutative algebra to the study of multiparameter persistent homology in topological data analysis. In particular, the theory of finite free resolutions of modules over polynomial rings is applied…
Secondary homological stability is a recently discovered stability pattern for the homology of a sequence of spaces exhibiting homological stability in a range where homological stability does not hold. We prove secondary homological…
Novelty detection methods aim at partitioning the test units into already observed and previously unseen patterns. However, two significant issues arise: there may be considerable interest in identifying specific structures within the…
We treat the problem of characterizing in a systematic way the qualitative features of two-dimensional dynamical systems. To that end, we construct a representation of the topological features of phase portraits by means of diagrams that…
Extended persistence is a technique from topological data analysis to obtain global multiscale topological information from a graph. This includes information about connected components and cycles that are captured by the so-called…
The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…