• Corpus ID: 162168840

Rank-based persistence

@article{Bergomi2019RankbasedP,
  title={Rank-based persistence},
  author={Mattia G. Bergomi and Pietro Vertechi},
  journal={ArXiv},
  year={2019},
  volume={abs/1905.09151}
}
Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category, so that functors to that category… 

Figures and Tables from this paper

Hochschild homology, and a persistent approach via connectivity digraphs
We introduce a persistent Hochschild homology framework for directed graphs. Hochschild homology groups of (path algebras of) directed graphs vanish in degree i ≥ 2. To extend them to higher degrees,
Topological graph persistence
Abstract Graphs are a basic tool in modern data representation. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional
Beyond Topological Persistence: Starting from Networks
Persistent homology enables fast and computable comparison of topological objects. We give some instances of a recent extension of the theory of persistence, guaranteeing robustness and computability
Steady and ranging sets in graph persistence
TLDR
In the category of graphs, a standard way of producing gp-functions is proposed: steady and ranging sets for a given feature; their meaning is investigated in three concrete networks.

References

SHOWING 1-10 OF 17 REFERENCES
Proximity of persistence modules and their diagrams
TLDR
This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.
A Mayer–Vietoris Formula for Persistent Homology with an Application to Shape Recognition in the Presence of Occlusions
TLDR
It is shown that persistence diagrams are able to recognize an occluded shape by showing a common subset of points and a Mayer–Vietoris formula involving the ranks of the persistent homology groups of X, A, B, and A∩B plus three extra terms is obtained.
Beyond Topological Persistence: Starting from Networks
Persistent homology enables fast and computable comparison of topological objects. We give some instances of a recent extension of the theory of persistence, guaranteeing robustness and computability
Persistence Theory - From Quiver Representations to Data Analysis
  • S. Oudot
  • Mathematics
    Mathematical surveys and monographs
  • 2015
TLDR
Theoretical foundations: Algebraic persistence, Stability Applications: Topological inference, and Perspectives: New trends in topological data analysis.
Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions
TLDR
The matching distance is shown to be resistant to perturbations, implying that it is always smaller than the natural pseudo-distance, and it is proved that the lower bound so obtained is sharp and cannot be improved by any other distance between size functions.
Persistent Homology — a Survey
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical
Topology and data
TLDR
This paper will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data, particularly high throughput data from microarray or other sources.
Unzerlegbare Darstellungen I
LetK be the structure got by forgetting the composition law of morphisms in a given category. A linear representation ofK is given by a map V associating with any morphism ϕ: a→e ofK a linear vector
Measuring shapes by size functions
  • P. Frosini
  • Mathematics, Computer Science
    Other Conferences
  • 1992
TLDR
The concept of deformation distance between manifolds is presented, a distance which measures the `difference in shape' of two manifolds and the link between deformation distances and size functions is pointed out.
Size Functions and Formal Series
  • P. Frosini, C. Landi
  • Mathematics
    Applicable Algebra in Engineering, Communication and Computing
  • 2001
TLDR
It is proved that every size function can be represented as a set of points and lines in the real plane, with multiplicities, which allows for an algebraic approach to size functions and the construction of new pseudo-distances between size functions for comparing shapes.
...
...