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A Robust Density-Based Clustering Algorithm for Multi-Manifold Structure

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A Robust Density-Based Clustering Algorithm for Multi-Manifold Structure A Robust Density-based Clustering Algorithm for Multi-Manifold Structure Jianpeng Zhang, Mykola Pechenizkiy, Yulong Pei, Julia Efremova Department of Mathematics and Computer Science Eindhoven University of Technology, 5600 MB Eindhoven, the Netherlands {j.zhang.4, m.pechenizkiy, y.pei.1, i.efremova}@tue.nl ABSTRACT ing methods is K-means [8] algorithm. It attempts to cluster In real-world pattern recognition tasks, the data with mul- data by minimizing the distance between cluster centers and tiple manifolds structure is ubiquitous and unpredictable. other data points. However, K-means is known to be sensi- Performing an effective clustering on such data is a challeng- tive to the selection of the initial cluster centers and is easy ing problem. In particular, it is not obvious how to design to fall into the local optimum. K-means is also not good a similarity measure for multiple manifolds. In this paper, at detecting non-spherical clusters which has poor ability to we address this problem proposing a new manifold distance represent the manifold structure. Another clustering algo- measure, which can better capture both local and global s- rithm is Affinity propagation (AP) [6]. It does not require to patial manifold information. We define a new way of local specify the initial cluster centers and the number of clusters density estimation accounting for the density characteristic. but the AP algorithm is not suitable for the datasets with ar- It represents local density more accurately. Meanwhile, it is bitrary shape or multiple scales. Other density-based meth- less sensitive to the parameter settings. Besides, in order to ods, such as DBSCAN [5], are able to detect non-spherical select the cluster centers automatically, a two-phase exem- clusters using a predefined density threshold. They can eas- plar determination method is proposed. The experiments ily find clusters with different sizes and shapes. One of the on several synthetic and real-world datasets show that the most significant disadvantages of these algorithms is their proposed algorithm has higher clustering effectiveness and weakness to recognize clusters with variant densities. In better robustness for data with varying density, multi-scale general, these algorithms use a global neighborhood radius and noise overlap characteristics. to determine whether two points are in the same cluster or not. Therefore, they might merge two adjacency clusters with different densities. If a smaller value is chosen, then CCS Concepts sparser clusters will be over clustered. Besides, this method •Information systems ! Clustering; is also limited in its ability to represent the data accurately because choosing an appropriate threshold is nontrivial. Keywords Recently, a new clustering algorithm (in this paper, we name it DensityClust for simplicity) has been introduced [13] and cluster analysis; density peaks; manifold distance; power-law it clusters data by fast search and then finds density peak- distribution, PauTa criterion s. DensityClust algorithm assumes that the cluster centers are characterized by a higher density than their neighbors 1. INTRODUCTION and by a relatively large distance from points with higher densities. It recognizes the clusters based on the similarity The data with multi-manifold structure [7] is ubiquitous in distance, and detects distinct non-spherical clusters in the real-world pattern recognition tasks, and designing an effec- datasets. Furthermore, it identifies the number of clusters in tive clustering algorithm for this type of data is a challenging the decision graph intuitively through an interactive process. problem. In many practical problems, e.g., handwritten dig- The algorithm has two effective criteria: the local density ρ it recognition, image segmentation, and web analysis etc., and the minimum distance δ. Based on the data distribution clustering analysis is aimed at finding correlations within assumption, only density peaks have large δ and relatively subsets of the dataset and assessing similarity among ele- high ρ, so the density peaks can pop out and be separable ments within these subsets. One of the most famous cluster- from the remaining points in the decision graph. Permission to make digital or hard copies of all or part of this work for Although DensityClust algorithm is simple and can solve personal or classroom use is granted without fee provided that copies the problems in previous methods to some extend, the char- are not made or distributed for profit or commercial advantage and that acteristics of real-world data are more unpredictable and copies bear this notice and the full citation on the first page. To copy complicated, and the Euclidean distance measure may not otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. fully reflect the complex spatial distribution of the datasets. SAC 2016,April 04-08, 2016, Pisa, Italy The local density selection is still quite sensitive to param- eter settings. In addition, facing the less-than-ideal deci- Copyright 2016 ACM 978-1-4503-3739-7/16/04. $15.00 sion graphs, users have difficulty in making decision on the http://dx.doi.org/10.1145/2851613.2851644 Table 1: Main difference of clustering algorithms: A(Excellent), B(Good), C(Fair), D(Poor), F(Failure). Clustering algorithm non-predetermined clusters varying density arbitrary shapes multi-scale noise sensitivity high-dimensional algorithmic efficiency K-means [8] F F D D F D C Affinity propagation [6] A F D D F D C DBSCAN [5] A D A D A D C Spectral clustering [16] F D B C F A D DensityClust [13] F D C D B D A RDensityClust A A A A A B C number of clusters since the decision graphs do not pro- by a relatively large distance from points with higher den- vide any other useful information of the original datasets. sities. There are two leading criteria in this method: local To solve those problems, in this paper we propose a ro- density(ρi) of each point i and minimum distance (δi) from bust density-based clustering (RDensityClust) for multi- other points with higher density, ρi is defined as: manifold structure. First, we design a manifold distance as X − the similarity measure which reflects the inherent manifold ρi = χ (dij dc) ; (1) structure information effectively. Second, we define a new j local density calculation based on the topology structure of where ρi is the local density, in which, manifold. It is less sensitive to parameter settings and can ( represent local density better. Finally, in order to select the 1 x ≤0 χ(x) = (2) cluster centers automatically, we describe a two-phase ex- 0 otherwise emplar determination method. In order to demonstrate an advantage of our algorithm, we compare performance result- and dc is a cutoff distance. ρi is basically equal to the num- s to several state-of-the-art clustering techniques. Table 1 ber of points that are closer than dc to point i. It should be presents main characteristics of applied algorithms. It can noticed that this algorithm is sensitive to the relative mag- be observed that our approach outperforms in dealing with nitude of ρ in different points. For large datasets, this algo- the data with varying density, multi-scale and noise overlap rithm gives an empirical value selection of dc which makes characteristics. the average number of neighbors is around 1% to 2% of the To summarize, the main contributions of this work include: total number in the dataset, but empirical analysis shows that the results are less robust with respect to the choice of d . δ is measured by computing the minimum distance • We present a new manifold distance which connects t- c i between the point i and any other point with higher density: wo points in the same manifold by shorter edges, while min (d ) and for the point with highest density, the connect two points in different manifolds by longer j:ρj >ρi ij paper defines it as: max (d ). For more details about this edges. Our innovation is that by marking the small j ij algorithm, reader can refer to [13]. components in the graphs, it is able to eliminate the the effects of outliers in noisy case. As a result, the proposed manifold distance is very robust against the 3. PROPOSED ROBUST DENSITYCLUST noise and outliers. (RDENSITYCLUST) ALGORITHM • We propose a new local density definition based on Since the local density selection of DensityClust algorithm the proposed manifold distance. It represents the local relies on the cutoff threshold dc, it is quite sensitive to the density better as well as reflects the global consistency parameter settings. If dc is set too large, the overlapping of the datasets. This algorithm is more accurate than neighborhood can even include the data from other clusters, the original algorithm, and simultaneously it is very and it may cause that each point has similar heavy density stable to the selection of parameters. neighborhood; if dc is set too small, every point has similar • We propose a novel two-phase exemplar determination sparse density neighborhood. In both cases, the density has method in order to select the cluster centers automat- less significant discriminative power. DensityClust uses a ically. The approach can reject the real outliers effec- heuristic value which makes the average number of neighbors tively and determine the final cluster centers accurate- is around 1% to 2% of the total number in the dataset. ly and automatically. However, the local density (ρ) computation may be sensitive to the parameter dc and it is not clear how to choose that sensitive parameter without human intervention. Besides, The rest of the paper is organized as follows. In Section 2, we the selection of d only depends on the Euclidean distance present a brief overview of the basic DensityClust algorith- c which only reflects the local information of the datasets, and m.
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