The Ring of Algebraic Functions on Persistence Bar Codes
@article{Adcock2013TheRO, title={The Ring of Algebraic Functions on Persistence Bar Codes}, author={Aaron B. Adcock and Erik Carlsson and Gunnar E. Carlsson}, journal={arXiv: Rings and Algebras}, year={2013} }
We study the ring of algebraic functions on the space of persistence barcodes, with applications to pattern recognition.
Figures and Tables from this paper
105 Citations
Persistent homology and applied homotopy theory
- Mathematics
- 2020
This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence…
Computational Tools in Weighted Persistent Homology
- MathematicsChinese Annals of Mathematics, Series B
- 2017
In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted…
Tropical Coordinates on the Space of Persistence Barcodes
- Computer ScienceFound. Comput. Math.
- 2019
The purpose of this paper is to identify tropical coordinates on the space of barcodes and prove that they are stable with respect to the bottleneck distance and Wasserstein distances.
Tropical Coordinates on the Space of Persistence Barcodes
- Computer Science
- 2016
The purpose of this paper is to identify tropical coordinates on the space of barcodes and prove that they are stable with respect to the bottleneck distance and Wasserstein distances.
Further Properties and Applications of Weighted Persistent Homology
- Mathematics
- 2017
In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for…
Persistence Paths and Signature Features in Topological Data Analysis
- Computer ScienceIEEE Transactions on Pattern Analysis and Machine Intelligence
- 2020
A new feature map for barcodes as they arise in persistent homology computation that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.
Amplitudes on abelian categories
- Mathematics
- 2021
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…
Stratifying the space of barcodes using Coxeter complexes
- Mathematics
- 2021
We use tools from geometric group theory to produce a stratification of the space Bn of barcodes with n bars. The top-dimensional strata are indexed by permutations associated to barcodes as defined…
Curvature Sets Over Persistence Diagrams
- Mathematics
- 2021
We study an invariant of compact metric spaces which combines the notion of curvature sets introduced by Gromov in the 1980s together with the notion of VietorisRips persistent homology. For given…
Vectorization of persistence barcode with applications in pattern classification of porous structures
- Computer ScienceComput. Graph.
- 2020
References
SHOWING 1-10 OF 28 REFERENCES
Partially Ordered Rings and Semi-Algebraic Geometry
- Mathematics
- 1980
Introduction 1. Partially Ordered Rings 2. Homomorphisms and Convex Ideals 3. Localization 4. Some Categorical Notions 5. The Prime Convex Ideal Spectrum 6. Polynomials 7. Ordered Fields 8. Affine…
Gromov‐Hausdorff Stable Signatures for Shapes using Persistence
- MathematicsComput. Graph. Forum
- 2009
We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on the persistence diagrams of suitable filtrations built on top of these spaces. We…
Persistence barcodes for shapes
- Mathematics, Computer ScienceSGP '04
- 2004
This paper initiates a study of shape description and classification via the application of persistent homology to two tangential constructions on geometric objects, obtaining a shape descriptor, called a barcode, that is a finite union of intervals.
Computing persistent homology
- Mathematics, Computer ScienceSCG '04
- 2004
The analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields and derives an algorithm for computing individual persistent homological groups over an arbitrary principal ideal domain in any dimension.
Topological pattern recognition for point cloud data*
- MathematicsActa Numerica
- 2014
The definition and computation of homology in the standard setting of simplicial complexes and topological spaces are discussed, then it is shown how one can obtain useful signatures, called barcodes, from finite metric spaces, thought of as sampled from a continuous object.
On the Local Behavior of Spaces of Natural Images
- MathematicsInternational Journal of Computer Vision
- 2007
A theoretical model for the high-density 2-dimensional submanifold of ℳ showing that it has the topology of the Klein bottle and a polynomial representation is used to give coordinatization to various subspaces ofℳ.
GEOMETRIC INVARIANT THEORY
- Economics
- 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and…
Classification of hepatic lesions using the matching metric
- Computer ScienceComput. Vis. Image Underst.
- 2014
Topology of viral evolution
- BiologyProceedings of the National Academy of Sciences
- 2013
This method effectively characterizes clonal evolution, reassortment, and recombination in RNA viruses and provides an evolutionary perspective that not only captures reticulate events precluding phylogeny, but also indicates the evolutionary scales where phylogenetic inference could be accurate.