Spectral concentration and greedy k-clustering

Dey, Tamal K.; Pan, Peng; Rossi, Alfred; Sidiropoulos, Anastasios

doi:10.1016/j.comgeo.2018.09.001

Cited by 6 publications

(6 citation statements)

References 21 publications

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“…Beyond these good properties of Laplacian eigenmaps classically studied in the literature, one important remaining question is whether Laplacian eigenmaps preserve the clusters through the transportation of the high-dimensional (possibly manifold supported) distribution on the low-dimensional space. In the case where the clusters are well separated, it is a well known fact that clusters can be recovered from the sign patterns in the components of the eigenvectors of the Laplacian matrix [13], [14]; see also [15]. Understanding spectral clustering through the lens of statistical phase transition phenomena under a very different model has been the focus of extensive research activity in recent years, via the analysis of the stochastic block model [16,17] and its generalisations to e.g.…”

Section: Challenges and Achieved Resultsmentioning

confidence: 99%

Clustering on Laplacian-embedded latent manifolds when clusters overlap

Chrétien

Jagan

Barton

2020

Meas. Sci. Technol.

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The purpose of clustering is to identify groups in a dataset in the hope of revealing some unforeseen latent discrete variable. Clustering however, is known to be one of the most difficult tasks in practice for two main reasons: (i) high dimensionality of data which requires appropriate feature extraction; and (ii) computational complexity of the associated optimisation problems.Spectral clustering was designed as a joint embedding and clustering technique that first embeds the data into a low dimensional space and then delineates between the clusters by considering the sign of the components of (a linearly transformed version of) the second eigenvector of a similarity matrix. Hence, spectral clustering seemingly solves the two main challenges associated with clustering problems, at least when the clusters are well separated.In this paper, we address the question of clustering when clusters overlap. In this regime, one relevant approach to clustering is to consider the modes of the point cloud distribution and, in particular, how the modes of the distribution of the raw data are mapped to the modes of the embedded data via the Laplacian eigenmap.The main contribution of the present paper is to provide a simulation study of the (approximate) mode-preserving property of Laplacian eigenmaps and how relevant the mode chasing approach is to high dimensional clustering. As a consequence of the mode-preserving property of Laplacian eigenmaps, this method is as good as finding modes in the original high dimensional space but with much better computational efficiency. The method is illustrated on simulated data and on satellite data relating to ground movement.

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Section: Challenges and Achieved Resultsmentioning

confidence: 99%

Clustering on Laplacian-embedded latent manifolds when clusters overlap

Chrétien

Jagan

Barton

2020

Meas. Sci. Technol.

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“…These approaches are less accurate, however [2,25]. The approximability of temporal versions of k-means clustering and its variants was studied by Dey et al [10].…”

Section: Return Truementioning

confidence: 99%

“…The parameterized complexity of the local approach of aggregating clusterings into one has been studied for multi-layer [70,71] and temporal graphs [16,17]. The approximability of temporal versions of k-means clustering and its variants was studied by Dey et al [72].…”

Section: Related Workmentioning

confidence: 99%

Cluster Editing for Multi-Layer and Temporal Graphs

Molter

Sorge

Suchý

2023

Preprint

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Motivated by the recent rapid growth of research for algorithms to cluster multi-layer and temporal graphs, we study extensions of the classical Cluster Editing problem. In Multi-Layer Cluster Editing we receive a set of graphs on the same vertex set, called layers and aim to transform all layers into cluster graphs (disjoint unions of cliques) that differ only slightly. More specifically, we want to mark at most d vertices and to transform each layer into a cluster graph using at most k edge additions or deletions per layer so that, if we remove the marked vertices, we obtain the same cluster graph in all layers. In Temporal Cluster Editing we receive a sequence of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most k edge additions or deletions and to mark a distinct set of d vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters d and k, among others. Despite the similar definition, the two problems behave quite differently: In particular, Multi-Layer Cluster Editing is fixed-parameter tractable with running time kO(k+d) sO(1) for inputs of size s, whereas Temporal Cluster Editing is W[1]-hard with respect to k even if d = 3.

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“…They also showed that, if a graph satisfies 𝜆 k+1 > 0 , there is a polynomial time algorithm that outputs a -way partition of the graph that is a (Ω( 2 k+1 ∕k 4 ), O(k 6 √ k ))-clustering where 1 ≤ ≤ k . Dey et al (2019) developed a greedy algorithm that partitions the node set of a graph into clusters with a large inside conductance and small outside conductance. They showed that, if there is a large gap between k and k+1 , the output of the algorithm is close to a (Ω( k+1 ∕k), O(k 3 √ k ))-clustering.…”

Section: Related Workmentioning

confidence: 99%

Convex programming based spectral clustering

Mizutani

2021

Mach Learn

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Clustering is a fundamental task in data analysis, and spectral clustering has been recognized as a promising approach to it. Given a graph describing the relationship between data, spectral clustering explores the underlying cluster structure in two stages. The first stage embeds the nodes of the graph in real space, and the second stage groups the embedded nodes into several clusters. The use of the k-means method in the grouping stage is currently standard practice. We present a spectral clustering algorithm that uses convex programming in the grouping stage and study how well it works. This algorithm is designed based on the following observation. If a graph is well-clustered, then the nodes with the largest degree in each cluster can be found by computing an enclosing ellipsoid of the nodes embedded in real space, and the clusters can be identified by using those nodes. We show that, for well-clustered graphs, the algorithm can find clusters of nodes with minimal conductance. We also give an experimental assessment of the algorithm's performance.

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Spectral concentration and greedy k-clustering

Cited by 6 publications

References 21 publications

Clustering on Laplacian-embedded latent manifolds when clusters overlap

Clustering on Laplacian-embedded latent manifolds when clusters overlap

Cluster Editing for Multi-Layer and Temporal Graphs

Convex programming based spectral clustering

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