A novel pruning approach for robust data clustering

In this paper, we make an effort to overcome the sensitivity of traditional clustering algorithms to noisy data points (noise and outliers). A novel pruning method, in terms of information theory, is therefore proposed to phase out noisy points for robust data clustering. This approach identifies and prunes the noisy points based on the maximization of mutual information against input data distributions such that the resulting clusters are least affected by noise and outliers, where the degree of robustness is controlled through a separate parameter to make a trade-off between rejection of noisy points and optimal clustered data. The pruning approach is general, and it can improve the robustness of many existing traditional clustering methods. In particular, we apply the pruning approach to improve the robustness of fuzzy c-means clustering and its extensions, e.g., fuzzy c-spherical shells clustering and kernel-based fuzzy c-means clustering. As a result, we obtain three clustering algorithms that are the robust versions of the existing ones. The effectiveness of the proposed pruning approach is supported by experimental results.     

Mining Projected Clusters in High-Dimensional Spaces

Clustering high-dimensional data has been a major challenge due to the inherent sparsity of the points. Most existing clustering algorithms become substantially inefficient if the required similarity measure is computed between data points in the full-dimensional space. To address this problem, a number of projected clustering algorithms have been proposed. However, most of them encounter difficulties when clusters hide in subspaces with very low dimensionality. These challenges motivate our effort to propose a robust partitional distance-based projected clustering algorithm. The algorithm consists of three phases. The first phase performs attribute relevance analysis by detecting dense and sparse regions and their location in each attribute. Starting from the results of the first phase, the goal of the second phase is to eliminate outliers, while the third phase aims to discover clusters in different subspaces. The clustering process is based on the K-means algorithm, with the computation of distance restricted to subsets of attributes where object values are dense. Our algorithm is capable of detecting projected clusters of low dimensionality embedded in a high-dimensional space and avoids the computation of the distance in the full-dimensional space. The suitability of our proposal has been demonstrated through an empirical study using synthetic and real datasets.     

Iterative projected clustering by subspace mining

Irrelevant attributes add noise to high-dimensional clusters and render traditional clustering techniques inappropriate. Recently, several algorithms that discover projected clusters and their associated subspaces have been proposed. We realize the analogy between mining frequent itemsets and discovering dense projected clusters around random points. Based on this, we propose a technique that improves the efficiency of a projected clustering algorithm (DOC). Our method is an optimized adaptation of the frequent pattern tree growth method used for mining frequent itemsets. We propose several techniques that employ the branch and bound paradigm to efficiently discover the projected clusters. An experimental study with synthetic and real data demonstrates that our technique significantly improves on the accuracy and speed of previous techniques.     

基于互信息和散度改进K-Means在交通数据聚类中的应用

K-means算法是一种常用的聚类算法,已应用于交通热点提取中.但是,由于聚类数目和初始聚类中心的主观设置,已有的聚类方法提取的交通热点往往难以满足要求.利用互信息和相对熵,提出SK-means算法,并应用于交通热点提取中.在所提方法中,基于不同点之间的互信息寻找初始聚类中心;此外,基于互信息和散度的比值,确定聚类数目.将所提方法应用于成都某段时间交通热点提取中,并与传统的K-means比较,实验结果表明,所提方法具有更高的聚类精度,提取的热点更符合实际.     

Robust projected clustering

Projected clustering partitions a data set into several disjoint clusters, plus outliers, so that each cluster exists in a subspace. Subspace clustering enumerates clusters of objects in all subspaces of a data set, and it tends to produce many overlapping clusters. Such algorithms have been extensively studied for numerical data, but only a few have been proposed for categorical data. Typical drawbacks of existing projected and subspace clustering algorithms for numerical or categorical data are that they rely on parameters whose appropriate values are difficult to set appropriately or that they are unable to identify projected clusters with few relevant attributes. We present P3C, a robust algorithm for projected clustering that can effectively discover projected clusters in the data while minimizing the number of required parameters. P3C does not need the number of projected clusters as input, and can discover, under very general conditions, the true number of projected clusters. P3C is effective in detecting very low-dimensional projected clusters embedded in high dimensional spaces. P3C positions itself between projected and subspace clustering in that it can compute both disjoint or overlapping clusters. P3C is the first projected clustering algorithm for both numerical and categorical data.     

Recovering Multiple View Geometry from Mutual Projections of Multiple Cameras

In this paper, we analyze the computation of epipolar geometry in some special cases where multiple cameras are projected each other in their images. In such cases, epipoles can be obtained directly from images as the projection of cameras. As the result, the epipolar geometry can be computed from less image correspondences with higher stability. In this paper, we analyze the number of corresponding points required for computing bifocal, trifocal and quadrifocal tensors linearly in the case where cameras are projected mutually. We next show a practical linear method for computing multifocal tensors by using the mutual projection of cameras. The degenerate configurations of points and cameras is also studied, and it is shown that some degenerate configurations in general cases are no longer degenerate under the mutual projection of cameras.     

Uncovering Representative Groups in Multidimensional Projections

Multidimensional projection‐based visualization methods typically rely on clustering and attribute selection mechanisms to enable visual analysis of multidimensional data. Clustering is often employed to group similar instances according to their distance in the visual space. However, considering only distances in the visual space may be misleading due to projection errors as well as the lack of guarantees to ensure that distinct clusters contain instances with different content. Identifying clusters made up of a few elements is also an issue for most clustering methods. In this work we propose a novel multidimensional projection‐based visualization technique that relies on representative instances to define clusters in the visual space. Representative instances are selected by a deterministic sampling scheme derived from matrix decomposition, which is sensitive to the variability of data while still been able to handle classes with a small number of instances. Moreover, the sampling mechanism can easily be adapted to select relevant attributes from each cluster. Therefore, our methodology unifies sampling, clustering, and feature selection in a simple framework. A comprehensive set of experiments validate our methodology, showing it outperforms most existing sampling and feature selection techniques. A case study shows the effectiveness of the proposed methodology as a visual data analysis tool.     

Redefining clustering for high-dimensional applications

Clustering problems are well-known in the database literature for their use in numerous applications, such as customer segmentation, classification, and trend analysis. High-dimensional data has always been a challenge for clustering algorithms because of the inherent sparsity of the points. Recent research results indicate that, in high-dimensional data, even the concept of proximity or clustering may not be meaningful. We introduce a very general concept of projected clustering which is able to construct clusters in arbitrarily aligned subspaces of lower dimensionality. The subspaces are specific to the clusters themselves. This definition is substantially more general and realistic than the currently available techniques which limit the method to only projections from the original set of attributes. The generalized projected clustering technique may also be viewed as a way of trying to redefine clustering for high-dimensional applications by searching for hidden subspaces with clusters which are created by interattribute correlations. We provide a new concept of using extended cluster feature vectors in order to make the algorithm scalable for very large databases. The running time and space requirements of the algorithm are adjustable and are likely to trade-off with better accuracy     

Ranking inter-relationships between clusters

The evaluation of the relationships between clusters is important to identify vital unknown information in many real-life applications, such as in the fields of crime detection, evolution trees, metallurgical industry and biology engraftment. This article proposes a method called ‘mode pattern?+?mutual information’ to rank the inter-relationship between clusters. The idea of the mode pattern is used to find outstanding objects from each cluster, and the mutual information criterion measures the close proximity of a pair of clusters. Our approach is different from the conventional algorithms of classifying and clustering, because our focus is not to classify objects into different clusters, but instead, we aim to rank the inter-relationship between clusters when the clusters are given. We conducted experiments on a wide range of real-life datasets, including image data and cancer diagnosis data. The experimental results show that our algorithm is effective and promising.     

Information-theoretic clustering: A representative and evolutionary approach

This paper proposes a new perspective on non-parametric entropy-based clustering. We developed a new cost evaluation function for clustering that measures the cross information potential (CIP) between clusters on a dataset using representative points, which we called representative CIP (rCIP). We did this based on the idea that optimizing the cross information potential is equivalent to minimizing cross entropy between clusters. Our measure is different because, instead of using all points in a dataset, it uses only representative points to quantify the interaction between distributions without any loss of the original properties of cross information potential. This brings a double advantage: decreases the computational cost of computing the cross information potential, thus drastically reducing the running time, and uses the underlying statistics of the space region where representative points are in order to measure interaction. With this, created a useful non-parametric estimator of entropy and makes possible using cross information potential in applications where it was not. Due to the nature of clustering problems, we proposed a genetic algorithm in order to use rCIP as cost function. We ran several tests and compared the results with single linkage hierarchical algorithm, finite mixture of Gaussians and spectral clustering in both synthetic and real image segmentation datasets. Experiments showed that our approach achieved better results compared to the other algorithms and it was capable of capture the real structure of the data in most cases regardless of its complexity. It also produced good image segmentation with the advantage of a tuning parameter that provides a way of refining segmentation.     

A Novel Approach to Noise Clustering for Outlier Detection

Noise clustering, as a robust clustering method, performs partitioning of data sets reducing errors caused by outliers. Noise clustering defines outliers in terms of a certain distance, which is called noise distance. The probability or membership degree of data points belonging to the noise cluster increases with their distance to regular clusters. The main purpose of noise clustering is to reduce the influence of outliers on the regular clusters. The emphasis is not put on exactly identifying outliers. However, in many applications outliers contain important information and their correct identification is crucial. In this paper we present a method to estimate the noise distance in noise clustering based on the preservation of the hypervolume of the feature space. Our examples will demonstrate the efficiency of this approach.     

A linear discriminant analysis method based on mutual information maximization

We present a new linear discriminant analysis method based on information theory, where the mutual information between linearly transformed input data and the class labels is maximized. First, we introduce a kernel-based estimate of mutual information with a variable kernel size. Furthermore, we devise a learning algorithm that maximizes the mutual information w.r.t. the linear transformation. Two experiments are conducted: the first one uses a toy problem to visualize and compare the transformation vectors in the original input space; the second one evaluates the performance of the method for classification by employing cross-validation tests on four datasets from the UCI repository. Various classifiers are investigated. Our results show that this method can significantly boost class separability over conventional methods, especially for nonlinear classification.     

HARP: a practical projected clustering algorithm

In high-dimensional data, clusters can exist in subspaces that hide themselves from traditional clustering methods. A number of algorithms have been proposed to identify such projected clusters, but most of them rely on some user parameters to guide the clustering process. The clustering accuracy can be seriously degraded if incorrect values are used. Unfortunately, in real situations, it is rarely possible for users to supply the parameter values accurately, which causes practical difficulties in applying these algorithms to real data. In this paper, we analyze the major challenges of projected clustering and suggest why these algorithms need to depend heavily on user parameters. Based on the analysis, we propose a new algorithm that exploits the clustering status to adjust the internal thresholds dynamically without the assistance of user parameters. According to the results of extensive experiments on real and synthetic data, the new method has excellent accuracy and usability. It outperformed the other algorithms even when correct parameter values were artificially supplied to them. The encouraging results suggest that projected clustering can be a practical tool for various kinds of real applications.     

类属型数据核子空间聚类算法

现有的类属型数据子空间聚类方法大多基于特征间相互独立假设,未考虑属性间存在的线性或非线性相关性.提出一种类属型数据核子空间聚类方法.首先引入原作用于连续型数据的核函数将类属型数据投影到核空间,定义了核空间中特征加权的类属型数据相似性度量.其次,基于该度量推导了类属型数据核子空间聚类目标函数,并提出一种高效求解该目标函数的优化方法.最后,定义了一种类属型数据核子空间聚类算法.该算法不仅在非线性空间中考虑了属性间的关系,而且在聚类过程中赋予每个属性衡量其与簇类相关程度的特征权重,实现了类属型属性的嵌入式特征选择.还定义了一个聚类有效性指标,以评价类属型数据聚类结果的质量.在合成数据和实际数据集上的实验结果表明,与现有子空间聚类算法相比,核子空间聚类算法可以发掘类属型属性间的非线性关系,并有效提高了聚类结果的质量.     

Colour segmentation and highlight detection on a 1-D feature space

Specularly reflecting surfaces often corrupt a real scene and degrade system performance. In this paper, we present an algorithm for colour segmentation as well as highlight detection. This algorithm models the human colour vision perception with the physical properties of sensors, illumination and surface reflectances. For image to be dependent only on the body reflectance, the reflection due to illumination, shading and highlights should be discounted. The dichromatic reflection model which describes the colour of the reflected light as a linear combination of the colour of the light due to surface reflection (highlights) and body reflection (object colour) is used. The input image data is first mapped from device coordinates onto the CIE 1931 xy chromaticity diagram where colour clustering occurs. Since the feature space is 3-D, colour clustering is a computationally expensive process. A more efficient method is the dimensionality reduction of the feature space. Using information from the xy chromaticity diagram, the estimated colour clusters are then projected onto a line for 1-D thresholding. The error of projection is minimized by the Fisher's linear discriminant method. A program recognizing nine colours on a 1024 × 1024 image is presented. Colour segmentation of 99% accuracy within 25 s is also achieved.     

一种基于MST的自适应优化相异性度量的半监督聚类方法

针对混合属性空间中具有同一(或相近)分布特性的带类别标记的小样本集和无类别标记的大样本数据集,提出了一种基于MST的自适应优化相异性度量的半监督聚类方法。该方法首先采用决策树方法来获取小样本集的"规则聚类区域",然后根据"同一聚类的数据点更为接近"的原则自适应优化建构在该混合属性空间中的相异性度量,最后将优化后的相异性度量应用于基于MST的聚类算法中,以获得更为有效的聚类结果。仿真实验结果表明,该方法对有些数据集是有改进效果的。为进一步推广并在实际中发掘出该方法的应用价值,本文在最后给出了一个较有价值的研究展望。     

Enhancing principal direction divisive clustering

While data clustering has a long history and a large amount of research has been devoted to the development of numerous clustering techniques, significant challenges still remain. One of the most important of them is associated with high data dimensionality. A particular class of clustering algorithms has been very successful in dealing with such datasets, utilising information driven by the principal component analysis. In this work, we try to deepen our understanding on what can be achieved by this kind of approaches. We attempt to theoretically discover the relationship between true clusters in the data and the distribution of their projection onto the principal components. Based on such findings, we propose appropriate criteria for the various steps involved in hierarchical divisive clustering and develop compilations of them into new algorithms. The proposed algorithms require minimal user-defined parameters and have the desirable feature of being able to provide approximations for the number of clusters present in the data. The experimental results indicate that the proposed techniques are effective in simulated as well as real data scenarios.     

Linear Discriminative Star Coordinates for Exploring Class and Cluster Separation of High Dimensional Data

One main task for domain experts in analysing their nD data is to detect and interpret class/cluster separations and outliers. In fact, an important question is, which features/dimensions separate classes best or allow a cluster‐based data classification. Common approaches rely on projections from nD to 2D, which comes with some challenges, such as: The space of projection contains an infinite number of items. How to find the right one? The projection approaches suffers from distortions and misleading effects. How to rely to the projected class/cluster separation? The projections involve the complete set of dimensions/features. How to identify irrelevant dimensions? Thus, to address these challenges, we introduce a visual analytics concept for the feature selection based on linear discriminative star coordinates (DSC), which generate optimal cluster separating views in a linear sense for both labeled and unlabeled data. This way the user is able to explore how each dimension contributes to clustering. To support to explore relations between clusters and data dimensions, we provide a set of cluster‐aware interactions allowing to smartly iterate through subspaces of both records and features in a guided manner. We demonstrate our features selection approach for optimal cluster/class separation analysis with a couple of experiments on real‐life benchmark high‐dimensional data sets.     

Reducing Redundancy in Subspace Clustering

In this paper, we first study an important but unsolved dilemma in the literature of subspace clustering, which is referred to as “information overlapping-data coverage” challenge. Current solutions of subspace clustering usually invoke a grid-based Apriori-like procedure to identify dense regions and construct subspace clusters afterward. Due to the nature of monotonicity property in Apriori-like procedures, it is inherent that if a region is identified as dense, all its projected regions are also identified as dense, causing overlapping/redundant clustering information to be inevitably reported to users when generating clusters from such highly correlated regions. However, naive methods to filter redundant clusters will incur a challenging problem in the other side of the dilemma, called the “data coverage” issue. Note that two clusters may have highly correlated dense regions but their data members could be highly different to each other. Arbitrarily removing one of them may lose the coverage of data with clustering information, thus likely reporting an incomplete and biased clustering result. In this paper, therefore, we further propose an innovative algorithm, called "NOnRedundant Subspace Cluster mining” (NORSC), to efficiently discover a succinct collection of subspace clusters while also maintaining the required degree of data coverage. NORSC not only avoids generating the redundant clusters with most of the contained data covered by higher dimensional clusters to resolve the information overlapping problem but also limits the information loss to cope with the data coverage problem. As shown by our experimental results, NORSC is very effective in identifying a concise and small set of subspace clusters, while incurring time complexity in orders of magnitude better than that of previous works.     

Wasserstein discriminant analysis

Wasserstein discriminant analysis (WDA) is a new supervised linear dimensionality reduction algorithm. Following the blueprint of classical Fisher Discriminant Analysis, WDA selects the projection matrix that maximizes the ratio of the dispersion of projected points pertaining to different classes and the dispersion of projected points belonging to a same class. To quantify dispersion, WDA uses regularized Wasserstein distances. Thanks to the underlying principles of optimal transport, WDA is able to capture both global (at distribution scale) and local (at samples’ scale) interactions between classes. In addition, we show that WDA leverages a mechanism that induces neighborhood preservation. Regularized Wasserstein distances can be computed using the Sinkhorn matrix scaling algorithm; the optimization problem of WDA can be tackled using automatic differentiation of Sinkhorn’s fixed-point iterations. Numerical experiments show promising results both in terms of prediction and visualization on toy examples and real datasets such as MNIST and on deep features obtained from a subset of the Caltech dataset.