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Functorial hierarchical clustering with overlaps

Jared Culbertson, Dan P. Guralnik, Peter F. Stiller
2018 Discrete Applied Mathematics  
Our first series of results extends their theory of functorial clustering schemes to methods that allow overlapping clusters in the spirit of Jardine and Sibson.  ...  Carlsson and Mémoli introduce the idea of viewing clustering methods as functors from a category of metric spaces to a category of clusters, with functoriality subsuming many desirable properties.  ...  Any fibered functor with values in Sieve then provides a hierarchical method of clustering that respects the constraints imposed by the morphisms in Weight and produces potentially overlapping clusters  ... 
doi:10.1016/j.dam.2017.10.015 fatcat:pqgvvod7zzfvvcrb3cdt2e5mtm

Consistency constraints for overlapping data clustering [article]

Jared Culbertson, Dan P. Guralnik, Jakob Hansen, Peter F. Stiller
2016 arXiv   pre-print
We examine overlapping clustering schemes with functorial constraints, in the spirit of Carlsson--Memoli. This avoids issues arising from the chaining required by partition-based methods.  ...  Our principal result shows that any clustering functor is naturally constrained to refine single-linkage clusters and be refined by maximal-linkage clusters.  ...  The two previous theorems then give us a family of hierarchical clustering schemes, {ML, . . . , L k , . . . , SL}. Note, however, that there are many more functorial hierarchical clustering schemes.  ... 
arXiv:1608.04331v1 fatcat:odmly6jcvjch3ngljwlnuq7nsa

Functorial Manifold Learning [article]

Dan Shiebler
2021 arXiv   pre-print
We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning.  ...  We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and factor through hierachical clustering functors.  ...  Preliminaries on Functorial Hierarchical Overlapping Clustering We briefly review some definitions from the functorial perspective on hierarchical overlapping clustering.  ... 
arXiv:2011.07435v5 fatcat:a2v4efqq2vgrdgvowlgh2tqjnu

Reoccurring Patterns in Hierarchical Protein Materials and Music: The Power of Analogies

Tristan Giesa, David I. Spivak, Markus J. Buehler
2011 BioNanoScience  
Specifically, natural hierarchical materials like spider silk exhibit properties comparable to classical music in terms of their hierarchical structure and function.  ...  Complex hierarchical structures composed of simple nanoscale building blocks form the basis of most biological materials.  ...  Therefore, the structure and function dimension in Table 1 overlap.  ... 
doi:10.1007/s12668-011-0022-5 fatcat:hijl2rctf5allobxuulqh7y4iy

Persistent Clustering and a Theorem of J. Kleinberg [article]

Gunnar Carlsson, Facundo Memoli
2008 arXiv   pre-print
We construct a framework for studying clustering algorithms, which includes two key ideas: persistence and functoriality.  ...  The first encodes the idea that the output of a clustering scheme should carry a multiresolution structure, the second the idea that one should be able to compare the results of clustering algorithms as  ...  Agglomerative hierarchical clustering, in standard form, as described for example in [JD88] , begins with point cloud data and constructs a binary tree (or dendrogram) which describes the merging of clusters  ... 
arXiv:0808.2241v1 fatcat:ssw7v2veofewlgcwfkuhivntjy

Flattening Multiparameter Hierarchical Clustering Functors [article]

Dan Shiebler
2021 arXiv   pre-print
We begin by introducing a procedure for flattening multiparameter hierarchical clusterings.  ...  We bring together topological data analysis, applied category theory, and machine learning to study multiparameter hierarchical clustering.  ...  Our contributions are: -We describe an algorithm for flattening multiparameter hierarchical clusterings, which we demonstrate is a functorial map from a category of multiparameter hierarchical partitions  ... 
arXiv:2104.14734v1 fatcat:e3jps6cktjfe7ehuux3v2jcdna

Detecting poisoning attacks on hierarchical malware classification systems

Dan P. Guralnik, Bill Moran, Ali Pezeshki, Omur Arslan, Igor V. Ternovskiy, Peter Chin
2017 Cyber Sensing 2017  
Anti-virus software based on unsupervised hierarchical clustering (HC) of malware samples has been shown to be vulnerable to poisoning attacks.  ...  ABSTRACT Anti-virus software based on unsupervised hierarchical clustering (HC) of malware samples has been shown to be vulnerable to poisoning attacks.  ...  ) hierarchical clustering.  ... 
doi:10.1117/12.2266556 fatcat:tbzntmwaufgxlmbejykibvel3u

Beyond Hartigan Consistency: Merge Distortion Metric for Hierarchical Clustering [article]

Justin Eldridge, Mikhail Belkin, Yusu Wang
2015 arXiv   pre-print
Hierarchical clustering is a popular method for analyzing data which associates a tree to a dataset.  ...  We proceed to introduce a merge distortion metric between hierarchical clusterings and show that convergence in our distance implies both separation and minimality.  ...  ) is the hierarchical clustering tree T r n equipped with the height function f .  ... 
arXiv:1506.06422v2 fatcat:rfgs2uwlznh6znoeligxmqdctq

STRUCTURAL EQUIVALENCE OF INDIVIDUALS IN SOCIAL NETWORKS [chapter]

FRANÇOIS LORRAIN, HARRISON C. WHITE
1977 Social Networks  
The notion of functorial mapping is of central importance in the 'theory of categories,' a branch of modern algebra with numerous applications to algebra, topology, logic.  ...  The global patterns are derived algebraically through a 'functorial' mapping of the original pattern.  ...  Naturally this in no way excludes that the given domain be also part of a wider system and interact or even overlap with other systems.  ... 
doi:10.1016/b978-0-12-442450-0.50012-2 fatcat:cdysht2givb2poshkoorseiauu

Structural equivalence of individuals in social networks

François Lorrain, Harrison C. White
1971 The Journal of mathematical sociology  
The notion of functorial mapping is of central importance in the 'theory of categories,' a branch of modern algebra with numerous applications to algebra, topology, logic.  ...  The global patterns are derived algebraically through a 'functorial' mapping of the original pattern.  ...  Naturally this in no way excludes that the given domain be also part of a wider system and interact or even overlap with other systems.  ... 
doi:10.1080/0022250x.1971.9989788 fatcat:2eq2m72nwrfkpbtvm34gisxbjy

Foreword to the Special Focus on Mathematics, Data and Knowledge

Xiaoyu Chen, Dongming Wang, Xiangliang Zhang
2013 Mathematics in Computer Science  
The present foreword provides a short review of some key ideas and techniques on how mathematics interacts with data and knowledge, together with a few selected research directions and problems and a brief  ...  With increasingly solid mathematical foundations, various methods and techniques have been studied and developed for data mining, modeling, and processing, and knowledge representation, organization, and  ...  In the process of reasoning about the nature of clustering methods, the notion of functoriality plays a key role [10] .  ... 
doi:10.1007/s11786-013-0169-2 fatcat:tjzthjeaerdrfjjwifghsuhgbm

A functorial Dowker theorem and persistent homology of asymmetric networks [article]

Samir Chowdhury, Facundo Mémoli
2018 arXiv   pre-print
We study two methods for computing network features with topological underpinnings: the Rips and Dowker persistent homology diagrams.  ...  As a stepping stone for our constructions, we prove a functorial generalization of a theorem of Dowker, after whom our constructions are named.  ...  Note that with some exceptions, networks corresponding to the same arena are clustered together.  ... 
arXiv:1608.05432v3 fatcat:gatqwgbyajcxnisykavpm7qduy

Topological Analysis of Syntactic Structures [article]

Alexander Port, Taelin Karidi, Matilde Marcolli
2019 arXiv   pre-print
We analyze relations between syntactic parameters in terms of dimensionality, of hierarchical clustering structures, and of non-trivial loops.  ...  the trees describing the merging structure of persistent connected components for languages in different language families and we show that they partly correlate to historical phylogenetic trees but with  ...  phylogenetic trees but as hierarchical clustering structures of syntactic features.  ... 
arXiv:1903.05181v1 fatcat:i2rqz3kzcjed3kh7ygd6e3t25u

Complex Non-linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks

I. C. Baianu, R. Brown, G. Georgescu, J. F. Glazebrook
2006 Axiomathes  
An algebraic formulation of variable 'next-state functions' is extended to a Łukasiewicz-Moisil Topos with an nvalued Łukasiewicz-Moisil Algebraic Logic subobject classifier description that represents  ...  Łukasiewicz-Moisil Algebraic-Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components  ...  The notion of colimit in a category generalises the notion of forming the union X [ Y of two overlapping sets, with intersection X\Y.  ... 
doi:10.1007/s10516-005-3973-8 fatcat:oo5f4pv62jhglpqrns6bvov6d4

Category-Theoretic Formulation of the Model-Based Systems Architecting Cognitive-Computational Cycle

Yaniv Mordecai, James Fairbanks, Edward F. Crawley
2021 Applied Sciences  
We demonstrate the CMGVC to transform a conceptual system architecture model built with the Object Process Modeling Language (OPM) into dual graphs and a stakeholder-informing matrix that stimulates system  ...  Finally, we thank the anonymous referees who helped improve this paper with constructive suggestions.  ...  Therefore, the matrix kernel must be integrated with hierarchical representations of row and column entities.  ... 
doi:10.3390/app11041945 fatcat:oghumyj2xvf3xafd7mrjldiu5q
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