Consistency constraints for overlapping data clustering

Culbertson, Jared; Guralnik, Dan P.; Hansen, Jakob; Stiller, Peter F.

doi:10.48550/arxiv.1608.04331

Cited by 4 publications

(13 citation statements)

References 14 publications

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“…Now recall that the set of connected components of the δ-Vietoris-Rips complex of (X, d X ) is the partioning of X into subsets with maximum pairwise distance no greater than δ. Given a ∈ (0, 1], an example of a flat clustering functor on Met is the a-single linkage functor SL(a), which maps a metric space to the connected components of its −log(a)-Vietoris-Rips complex [12,4,1]. Given a 1 , a 2 ∈ (0, 1], an example of a flat clustering functor on Met bij is the robust single linkage functor SL R (a 1 , a 2 ) which maps a metric space (X, d X ) to the connected components of the −log(a 2 )-Vietoris-Rips complex of (X, d a1 X ) where:…”

Section: Multiparameter Hierarchical Clusteringmentioning

confidence: 99%

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Flattening Multiparameter Hierarchical Clustering Functors

Shiebler¹

2021

Preprint

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We bring together topological data analysis, applied category theory, and machine learning to study multiparameter hierarchical clustering. We begin by introducing a procedure for flattening multiparameter hierarchical clusterings. We demonstrate that this procedure is a functor from a category of multiparameter hierarchical partitions to a category of binary integer programs. We also include empirical results demonstrating its effectiveness. Next, we introduce a Bayesian update algorithm for learning clustering parameters from data. We demonstrate that the composition of this algorithm with our flattening procedure satisfies a consistency property.

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Section: Multiparameter Hierarchical Clusteringmentioning

confidence: 99%

“…In this paper we will characterize and study clustering algorithms as functors, similarly to [1,4,12]. We will particularly focus on multiparameter hierarchical clustering algorithms with partially ordered hyperparameter spaces.…”

Section: Introductionmentioning

confidence: 99%

Flattening Multiparameter Hierarchical Clustering Functors

Shiebler¹

2021

Preprint

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“…If M u (δ) is the collection of maximal simplices of K u (δ), then M u (δ) forms a non-nested flag cover of X. Define R : Weight → Sieve by R(X, u) t = (X, M u (t)), which we will call the Rips sieving functor (elsewhere, including in [12], this functor is denoted ML and called maximal-linkage). On morphisms, R sends a nonexpansive map of weight spaces to the same underlying set function.…”

Section: Clustering With Overlaps: Sievesmentioning

confidence: 99%

“…Many other examples can be derived from the clustering functors introduced in our work with Hansen in [12].…”

Section: Clustering With Overlaps: Sievesmentioning

confidence: 99%

“…Thus, our notion of stationarity is analogous to these conditions in the sense of the following theorem. In our first look at sieving functors, we defined the Čech sieve of Example 16, which is not a stationary sieving functor: for a fixed weight space, after sufficiently many applications (interleaving applications of the functor J ) it gives the result of the single-linkage sieve (see the discussion of L k clustering in our previous paper with Hansen [12], where we describe Čech sieving in terms of set relations). In that paper, we exhibited many different sieving functors that fail to be stationary.…”

Section: Clustering Domainsmentioning

confidence: 99%

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Functorial hierarchical clustering with overlaps

Culbertson

Guralnik

Stiller

2018

Discrete Applied Mathematics

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This work draws inspiration from three important sources of research on dissimilarity-based clustering and intertwines those three threads into a consistent principled functorial theory of clustering. Those three are the overlapping clustering of Jardine and Sibson, the functorial approach of Carlsson and Mémoli to partition-based clustering, and the Isbell/Dress school's study of injective envelopes. 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. 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. This obviates some of the unpleasant effects of chaining that occur, for example with single-linkage clustering. We prove an equivalence between these general overlapping clustering functors and projections of weight spaces to what we term clustering domains, by focusing on the order structure determined by the morphisms. As a specific application of this machinery, we are able to prove that there are no functorial projections to cut metrics, or even to tree metrics. Finally, although we focus less on the construction of clustering methods (clustering domains) derived from injective envelopes, we lay out some preliminary results, that hopefully will give a feel for how the third leg of the stool comes into play.

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Functorial Manifold Learning

Shiebler

2022

Electron. Proc. Theor. Comput. Sci.

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We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and that factor through hierarchical clustering functors. We then use this characterization to prove refinement bounds on manifold learning loss functions and construct a hierarchy of manifold learning algorithms based on their equivariants. We express several popular manifold learning algorithms as functors at different levels of this hierarchy, including Metric Multidimensional Scaling, IsoMap, and UMAP. Next, we use interleaving distance to study the stability of a broad class of manifold learning algorithms. We present bounds on how closely the embeddings these algorithms produce from noisy data approximate the embeddings they would learn from noiseless data. Finally, we use our framework to derive a set of novel manifold learning algorithms, which we experimentally demonstrate are competitive with the state of the art.

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Consistency constraints for overlapping data clustering

Cited by 4 publications

References 14 publications

Flattening Multiparameter Hierarchical Clustering Functors

Flattening Multiparameter Hierarchical Clustering Functors

Functorial hierarchical clustering with overlaps

Functorial Manifold Learning

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