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A Kernel for Multi-Parameter Persistent Homology [article]

René Corbet, Ulderico Fugacci, Michael Kerber, Claudia Landi, Bei Wang
2019 arXiv   pre-print
We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines.  ...  Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques.  ...  We thank all members of the breakout session for the multi-dimensional kernel for their valuable suggestions.  ... 
arXiv:1809.10231v2 fatcat:3x6iu4oahzcp3jlonrq7xtiivm

A kernel for multi-parameter persistent homology

René Corbet, Ulderico Fugacci, Michael Kerber, Claudia Landi, Bei Wang
2019 Computers & Graphics: X  
We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines.  ...  Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques with applicability on shape analysis, recognition and classification.  ...  A feature map for multi-parameter persistent homology Let φ be a feature map (such as the scale-space kernel) that assigns to a mono-filtration a function in L 2 ( (1) ).  ... 
doi:10.1016/j.cagx.2019.100005 pmid:33367228 pmcid:PMC7755142 fatcat:mbx7ib4a4zdmbpe54ea6xo4ywu

Multidimensional Persistence Module Classification via Lattice-Theoretic Convolutions [article]

Hans Riess, Jakob Hansen
2020 arXiv   pre-print
We consider the use of lattice-based convolutional neural network layers as a tool for the analysis of features arising from multiparameter persistence modules.  ...  Multiparameter persistent homology has been largely neglected as an input to machine learning algorithms.  ...  Acknowledgments The authors would like to thank Chris Wendler for replicating our original experiment and pointing out a coding error in our pytorch implementation.  ... 
arXiv:2011.14057v1 fatcat:3ho3jgwfyjda3mbcgadgajtoi4

Persistence weighted Gaussian kernel for topological data analysis [article]

Genki Kusano, Kenji Fukumizu, Yasuaki Hiraoka
2016 arXiv   pre-print
This paper proposes a kernel method on persistence diagrams to develop a statistical framework in TDA.  ...  The proposed kernel satisfies the stability property and provides explicit control on the effect of persistence. Furthermore, the method allows a fast approximation technique.  ...  Kernel methods for persistence diagrams We propose a kernel for persistence diagrams, called the Persistence Weighted Gaussian Kernel (PWGK), to embed the diagrams into an RKHS.  ... 
arXiv:1601.01741v2 fatcat:hnbsw7fyyzg6rgr6cxo63cn4fu

Kernel method for persistence diagrams via kernel embedding and weight factor [article]

Genki Kusano, Kenji Fukumizu, Yasuaki Hiraoka
2017 arXiv   pre-print
Topological data analysis is an emerging mathematical concept for characterizing shapes in multi-scale data.  ...  Nowadays, it is highly desired to develop a statistical framework on persistence diagrams to deal with practical data. This paper proposes a kernel method on persistence diagrams.  ...  Acknowledgement We thank Ulrich Bauer for giving us useful comments in Section 4.  ... 
arXiv:1706.03472v1 fatcat:yovma4dcjvcaxpnfjxogsklhdm

Smart Vectorizations for Single and Multiparameter Persistence [article]

Baris Coskunuzer and CUneyt Gurcan Akcora and Ignacio Segovia Dominguez and Zhiwei Zhen and Murat Kantarcioglu and Yulia R. Gel
2021 arXiv   pre-print
In this work, we introduce two new and easily interpretable topological summaries for single and multi-parameter persistence, namely, saw functions and multi-persistence grid functions, respectively.  ...  The extracted patterns, or homological features, along with information on how long such features persist throughout the considered filtration of a scale parameter, convey a critical insight into salient  ...  In this work, we provide new and easily interpretable topological summaries for single and multi-parameter persistence, namely, saw functions and multi-persistence grid functions (MPGFs).  ... 
arXiv:2104.04787v1 fatcat:epaesy5zlbetjngo455oc4mbji

Learning metrics for persistence-based summaries and applications for graph classification [article]

Qi Zhao, Yusu Wang
2019 arXiv   pre-print
We study this problem and develop a new weighted kernel, called WKPI, for persistence summaries, as well as an optimization framework to learn a good metric for persistence summaries.  ...  Recently a new feature representation and data analysis methodology based on a topological tool called persistent homology (and its corresponding persistence diagram summary) has started to attract momentum  ...  We would also like to thank Giorgio Ascoli for helping provide the neuron dataset.  ... 
arXiv:1904.12189v2 fatcat:e5l5nh5pbzhodnwwwtvnekwqwq

Stabilizing the unstable output of persistent homology computations [article]

Paul Bendich and Peter Bubenik and Alexander Wagner
2018 arXiv   pre-print
We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done.  ...  The persistent homology algorithm is usually viewed as a procedure which starts with a filtered complex and ends with a persistence diagram.  ...  In this figure we see that there is a competitor for the location of the generator of the 28th longest bar.  ... 
arXiv:1512.01700v4 fatcat:bxnxo4vihrgbnpj4x42mdumyu4

Persistence Codebooks for Topological Data Analysis [article]

Bartosz Zielinski, Michal Lipinski, Mateusz Juda, Matthias Zeppelzauer, Pawel Dlotko
2019 arXiv   pre-print
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points.  ...  Persistence codebooks represent PDs in a convenient way for machine learning and statistical analysis and have a number of favorable practical and theoretical properties including 1-Wasserstein stability  ...  REPRESENTATION PARAMETERS (IF ANY) SCORE 2-WASSERSTEIN DISTANCE 0.79 ± 0.00 MULTI-SCALE KERNEL σ = {0.5, 1, 1.5} 0.58 ± 0.00 SLICED WASSERSTEIN KERNEL n = {50, 100, 150, 200, 250} 0.95 ± 0.00 PERSISTENCE  ... 
arXiv:1802.04852v4 fatcat:x4auvgtqojfsrmh4dd66emvyb4

Persistence paths and signature features in topological data analysis [article]

Ilya Chevyrev, Vidit Nanda, Harald Oberhauser
2018 IEEE Transactions on Software Engineering   accepted
We introduce a new feature map for barcodes that arise in persistent homology computation.  ...  The composition of these two operations - barcode to path, path to tensor series - results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness  ...  For the purposes of this introduction, it suffices to think of a barcode as a (multi)set of intervals [b • , d • ), each identifying those values of a scale parameter ≥ 0 at which some topological feature  ... 
doi:10.1109/tpami.2018.2885516 pmid:30530312 arXiv:1806.00381v2 fatcat:5pnkug4mong2bdix5an7lhzdcm

Compression for 2-Parameter Persistent Homology [article]

Ulderico Fugacci, Michael Kerber, Alexander Rolle
2021 arXiv   pre-print
For multi-parameter persistent homology, compression is a necessary step in any computational pipeline, since standard constructions lead to large inputs, and computational tasks in this area tend to be  ...  The first method extends the multi-chunk algorithm for one-parameter persistent homology, returning the smallest chain complex among all the ones quasi-isomorphic to the input.  ...  This naturally calls for the development of a multi-parameter extension of persistent homology.  ... 
arXiv:2107.10924v1 fatcat:7osctj7orzcmdm5tik4tee3dyy

A stable multi-scale kernel for topological machine learning

Jan Reininghaus, Stefan Huber, Ulrich Bauer, Roland Kwitt
2015 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)  
In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data.  ...  Yet, so far we lack a theoretically sound connection to popular kernelbased learning techniques, such as kernel SVMs or kernel PCA.  ...  The persistence scale-space kernel We propose a stable multi-scale kernel k σ for the set of persistence diagrams D.  ... 
doi:10.1109/cvpr.2015.7299106 dblp:conf/cvpr/ReininghausHBK15 fatcat:eqdrumfcg5horkcew65h3rqzha

Consistent manifold representation for topological data analysis

Tyrus Berry, Timothy Sauer
2019 Foundations of Data Science  
Thus CkNN produces a single graph that captures all topological features simultaneously, in contrast to persistent homology, which represents each homology generator at a separate scale.  ...  It is proved for compact (and conjectured for noncompact) manifolds that CkNN is the unique unweighted construction that yields a geometry consistent with the connected components of the underlying manifold  ...  In Fig. 9 we show the persistence of the correct homology in terms of the percentage of edges as a function of the parameter β that defines the multi-scale graph construction.  ... 
doi:10.3934/fods.2019001 fatcat:4guwusi2zra2nmrlgjmoglszs4

Consistent Manifold Representation for Topological Data Analysis [article]

Tyrus Berry, Timothy Sauer
2019 arXiv   pre-print
Thus CkNN produces a single graph that captures all topological features simultaneously, in contrast to persistent homology, which represents each homology generator at a separate scale.  ...  It is proved for compact (and conjectured for noncompact) manifolds that CkNN is the unique unweighted construction that yields a geometry consistent with the connected components of the underlying manifold  ...  In Fig. 9 we show the persistence of the correct homology in terms of the percentage of edges as a function of the parameter β that defines the multi-scale graph construction.  ... 
arXiv:1606.02353v3 fatcat:fvxrjfu7nfdxfo6a2j2jhbpk6q

Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology [article]

Michael Lesnick, Matthew Wright
2022 arXiv   pre-print
Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology.  ...  Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded K[x,y]-module M, where K is a field.  ...  3.8; Ulrich Bauer, for many enlightening discussions about 1-parameter persistence computation, which influenced this work; Roy Zhao, for helpful conversations about the clearing optimization and persistence  ... 
arXiv:1902.05708v6 fatcat:4mjpzly3hvdezhlsrfeb3qnxxe
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