An Efficient Approach to Higher Dimensional Data Clustering

Fusion of evolvable genome structure and multi-objective optimization for subspace clustering

Subspace clustering is an emerging area which explores clusters of objects in various subspaces. The existing subspace clustering algorithms are computationally expensive as they generate a large number of possibly redundant subspace clusters limiting the interpretability of the results. The problem gets even worse with the increase in data's dimensionality. So, this demands for efficient summarisation framework that generates limited number of interesting subspace clusters. A novel algorithm, Similarity Connectedness Based clustering on subspace clusters (SCoC) is proposed to form natural grouping of lower-dimensional subspace clusters. The concept of similarity connectedness is introduced to group and merge the subspace clusters formed in different lower-dimensional subspaces leaping through the lattice of dimensions. The resulted compact and summarised high-dimensional subspace clusters would easily be interpreted for making sound decisions. The SCoC algorithm is thoroughly tested on various benchmark datasets and found that it outperforms PCoC and SUBCLU both in cluster quality and execution time.

Block-diagonal structure learning for subspace clustering

Video co-segmentation refers to the task of jointly segmenting common objects appearing in a given group of videos. In practice, high-dimensional data such as videos can be conceptually thought as being drawn from a union of subspaces corresponding to categories rather than from a smooth manifold. Therefore, segmenting data into respective subspaces --- subspace clustering --- finds widespread applications in computer vision, including co-segmentation. State-of-the-art methods via subspace clustering seek to solve the problem in two steps: First, an affinity matrix is built from data, with appearance features or motion patterns. Second, the data are segmented by applying spectral clustering to the affinity matrix. However, this process is insufficient to obtain an optimal solution since it does not take into account the {\em interdependence} of the affinity matrix with the segmentation. In this work, we present a novel unified video co-segmentation framework inspired by the recent Structured Sparse Subspace Clustering ($\mathrm{S^{3}C}$) based on the {\em self-expressiveness} model. Our method yields more consistent segmentation results. In order to improve the detectability of motion features with missing trajectories due to occlusion or tracked points moving out of frames, we add an extra-dimensional signature to the motion trajectories. Moreover, we reformulate the $\mathrm{S^{3}C}$ algorithm by adding the affine subspace constraint in order to make it more suitable to segment rigid motions lying in affine subspaces of dimension at most $3$. Our experiments on MOViCS dataset show that our framework achieves the highest overall performance among baseline algorithms and demonstrate its robustness to heavy noise.

This paper presents an agent-based algorithm for discovering subspace clusters in high dimensional data. Each data object is represented by an agent, and the agents move from one local environment to another to find optimal clusters in subspaces. Heuristic rules and objective functions are defined to guide the movements of agents, so that similar agents(data objects) go to one group. The experimental results show that our proposed agent-based subspace clustering algorithm performs better than existing subspace clustering methods on both F1 measure and Entropy. The running time of our algorithm is scalable with the size and dimensionality of data. Furthermore, an application in stock market surveillance demonstrates its effectiveness in real world applications.

Self-organizing subspace clustering for high-dimensional and multi-view data

High dimensional data is a phenomenon in real-world data mining applications. Text data is a typical example. In text mining, a text document is viewed as a vector of terms whose dimension is equal to the total number of unique terms in a data set, which is usually in thousands. High dimensional data occurs in business as well. In retails, for example, to effectively manage supplier relationship, suppliers are often categorized according to their business behaviors (Zhang, Huang, Qian, Xu, & Jing, 2006). The supplier’s behavior data is high dimensional, which contains thousands of attributes to describe the supplier’s behaviors, including product items, ordered amounts, order frequencies, product quality and so forth. One more example is DNA microarray data. Clustering high-dimensional data requires special treatment (Swanson, 1990; Jain, Murty, & Flynn, 1999; Cai, He, & Han, 2005; Kontaki, Papadopoulos & Manolopoulos., 2007), although various methods for clustering are available (Jain & Dubes, 1988). One type of clustering methods for high dimensional data is referred to as subspace clustering, aiming at finding clusters from subspaces instead of the entire data space. In a subspace clustering, each cluster is a set of objects identified by a subset of dimensions and different clusters are represented in different subsets of dimensions. Soft subspace clustering considers that different dimensions make different contributions to the identification of objects in a cluster. It represents the importance of a dimension as a weight that can be treated as the degree of the dimension in contribution to the cluster. Soft subspace clustering can find the cluster memberships of objects and identify the subspace of each cluster in the same clustering process.

Subspace clustering has important and wide applications in computer vision and pattern recognition. It is a challenging task to learn low-dimensional subspace structures due to the possible errors (e.g., noise and corruptions) existing in high-dimensional data. Recent subspace clustering methods usually assume a sparse representation of corrupted errors and correct the errors iteratively. However large corruptions in real-world applications can not be well addressed by these methods. A novel optimization model for robust subspace clustering is proposed in this paper. The objective function of our model mainly includes two parts. The first part aims to achieve a sparse representation of each high-dimensional data point with other data points. The second part aims to maximize the correntropy between a given data point and its low-dimensional representation with other points. Correntropy is a robust measure so that the influence of large corruptions on subspace clustering can be greatly suppressed. An extension of our method with explicit introduction of representation error terms into the model is also proposed. Half-quadratic minimization is provided as an efficient solution to the proposed robust subspace clustering formulations. Experimental results on Hopkins 155 dataset and Extended Yale Database B demonstrate that our method outperforms state-of-the-art subspace clustering methods.

Gene expression data is a kind of high dimension and small sample size data. The clustering accuracy of conventional clustering techniques is lower on gene expression data due to its high dimension. Because some subspace segmentation approaches can be better applied in the high-dimensional space, three new subspace clustering models for gene expression data sets are proposed in this work. The proposed projection subspace clustering models have projection sparse subspace clustering, projection low-rank representation subspace clustering and projection least-squares regression subspace clustering which combine projection technique with sparse subspace clustering, low-rank representation and least-square regression, respectively. In order to compute the inner product in the high-dimensional space, the kernel function is used to the projection subspace clustering models. The experimental results on six gene expression data sets show these models are effective.

Nonlinear subspace clustering by functional link neural networks

We propose the Multi-resolution Correlation Cluster detection (MrCC), a novel, scalable method to detect correlation clusters able to analyze dimensional data in the range of around 5 to 30 axes. Existing methods typically exhibit super-linear behavior in terms of space or execution time. MrCC employs a novel data structure based on multi-resolution and gains over previous approaches in: (a) it finds clusters that stand out in the data in a statistical sense; (b) it is linear on running time and memory usage regarding number of data points and dimensionality of subspaces where clusters exist; (c) it is linear in memory usage and quasi-linear in running time regarding space dimensionality; and (d) it is accurate, deterministic, robust to noise, does not require stating the number of clusters as input parameter, does not perform distance calculation and is able to detect clusters in subspaces generated by original axes or linear combinations of original axes, including space rotation. We performed experiments on synthetic data ranging from 5 to 30 axes and from 12 k to 250 k points, and MrCC outperformed in time five of the recent and related work, being in average 10 times faster than the competitors that also presented high accuracy results for every tested dataset. Regarding real data, MrCC found clusters at least 9 times faster than the competitors, increasing their accuracy in up to 34 percent.

Dimensionality reduction is essential in multidimensional data mining since the dimensionality of real time data could easily reach higher dimensions. Most recent efforts on dimensionality reduction, however, are not adequate to multidimensional data due to lack of scalability. In this paper, we improve the performance of the fractional cuckoo search algorithm by utilizing the FCM function as objective function and we utilize the FCM operator which helps to improve the fitness value of worse solution. The fractional cuckoo search (FCS) algorithm is modified to reduce the dimension and select the best dimension. Once the high dimensional data is reduced in to low dimensional data, then the data is supplied to the clustering algorithm to make the partition easily. Finally, the experimentation is made with synthetic and real datasets and has been we have proved that efficiency of the FCM-FCS algorithm is 1.21% better than FPSO algorithm on iris dataset. The proposed algorithm is 0.66% better than FPSO 4.91% better than FCS on wine dataset and 3.4% better than FPSO 0.1% better than FCS on synthetic dataset.

This paper presents a novel clustering method, named Laplacian regularized deep subspace clustering (LRDSC), for unsupervised hyperspectral image (HSI) classification. We introduce the Laplacian regularization into the subspace clustering to consider the manifold structure reflecting geometric information. To enable the subspace clustering, which works in linear space, to deal with the complicated HSI data with nonlinear characteristics, we combine the subspace clustering as a self-expressive layer with deep convolutional auto-encoder. Furthermore, the 3-D convolutions and deconvolutions with skip connections are utilized to make full extraction of the spectral-spatial information and full use of the historical feature maps produced by the network. We compare the results of the proposed method with six existing cluster methods on four real hyperspectral data sets, showing that the proposed method is able to achieve state-of-the-art performance.

In high dimensional data, many dimensions are irrelevant to each other and clusters are usually hidden under noise. As an important extension of the traditional clustering, subspace clustering can be utilized to simultaneously cluster the high dimensional data into several subspaces and associate the low-dimensional subspaces with the corresponding points. In subspace clustering, it is a crucial step to construct an affinity matrix with block-diagonal form, in which the blocks correspond to different clusters. The distance-based methods and the representation-based methods are two major types of approaches for building an informative affinity matrix. In general, it is the difference between the density inside and outside the blocks that determines the efficiency and accuracy of the clustering. In this work, we introduce a well-known approach in statistic physics method, namely link prediction, to enhance subspace clustering by reinforcing the affinity matrix.More importantly,we introduce the idea to combine complex network theory with machine learning. By revealing the hidden links inside each block, we maximize the density of each block along the diagonal, while restrain the remaining non-blocks in the affinity matrix as sparse as possible. Our method has been shown to have a remarkably improved clustering accuracy comparing with the existing methods on well-known datasets.

Clustering in high dimensional data is an important task. Subspace clustering has emerged as a possible solution to the challenges associated with high dimensional clustering. A subspace cluster is a subset of points together with a subset of attributes, such that some category of value of cluster points has great aggregation in these attributes. This paper proposes a subspace clustering algorithm which follows the bottom-up strategy, evaluating each dimension separately and then using only those dimensions with great aggregation in further steps. Experimental results on synthetic data show that presented algorithm scales linearly with the number of the attributes and has good scalability as the size of the data objects is increased.