Open AccessPosted Content
Classifying Clustering Schemes
Reads0
Chats0
TLDR
A framework is constructed for studying what happens when one imposes various structural conditions on the clustering schemes, under the general heading of functoriality, and it is shown that, within this framework, one can prove a theorem analogous to one of Kleinberg (Becker et al).Abstract:
Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result.
We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.read more
Citations
More filters
Posted Content
UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction
Leland McInnes,John Healy +1 more
TL;DR: The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance.
Journal Article
Characterization, Stability and Convergence of Hierarchical Clustering Methods
TL;DR: It is shown that within this framework, one can prove a theorem analogous to one of Kleinberg (2002), in which one obtains an existence and uniqueness theorem instead of a non-existence result.
Proceedings ArticleDOI
Axiomatic construction of hierarchical clustering in asymmetric networks
TL;DR: It is shown that any clustering method that satisfies the axioms of value and transformation lies between reciprocal and nonreciprocal clustering in a well defined sense.
Journal ArticleDOI
A Population Background for Nonparametric Density-Based Clustering
TL;DR: In this article, the authors provide an explicit formulation for the ideal population goal of the modal clustering methodology, which understand clusters as regions of high density, and present two new loss functions, applicable in fact to any clustering methodologies, to evaluate the performance of a data-based clustering algorithm with respect to the desired population goal, where mild conditions on a sequence of density estimators are needed to ensure that the sequence of modal clusterings that they induce is consistent.
Proceedings Article
Persistent path homology of directed networks
Samir Chowdhury,Facundo Mémoli +1 more
TL;DR: The persistent path homology (PPH) as mentioned in this paper is an extension of persistent homology to the directed setting, and it has been shown that PPH agrees with Cech persistence on symmetric spaces, satisfying a local condition called square-freeness.
References
More filters
Proceedings Article
A density-based algorithm for discovering clusters a density-based algorithm for discovering clusters in large spatial databases with noise
TL;DR: In this paper, a density-based notion of clusters is proposed to discover clusters of arbitrary shape, which can be used for class identification in large spatial databases and is shown to be more efficient than the well-known algorithm CLAR-ANS.
Proceedings Article
A density-based algorithm for discovering clusters in large spatial Databases with Noise
TL;DR: DBSCAN, a new clustering algorithm relying on a density-based notion of clusters which is designed to discover clusters of arbitrary shape, is presented which requires only one input parameter and supports the user in determining an appropriate value for it.
Book
Categories for the Working Mathematician
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Journal ArticleDOI
Uncovering the overlapping community structure of complex networks in nature and society
TL;DR: After defining a set of new characteristic quantities for the statistics of communities, this work applies an efficient technique for exploring overlapping communities on a large scale and finds that overlaps are significant, and the distributions introduced reveal universal features of networks.