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

Conventional clustering algorithms are not so efficient on higher dimensional data due to the problem of dimensionality curse. To address this issue searching for clusters in appropriate subspaces is performed. But searching all possible subspaces is exhaustive. In this paper we propose an efficient approach to effectively find the relevant subspaces in high dimensional data and apply clustering in those subspaces. Experiments are conducted on real and synthetic data sets and compared with other approaches and our approach is able to return good clustering results.

In the field of pattern recognition, the study of the gene expression profiles for different tissue samples over different experimental conditions has became feasible with the arrival of micro-array based technology. In cancer research, classification of tissue samples is necessary for cancer diagnosis, which can be done with the help of micro-array technology. In this article we have presented a multi-objective optimization ( MOO ) based clustering technique utilizing AMOSA ( Archived Multi-Objective Simulated Annealing ) as the underlying optimization strategy for classification of tissue samples from cancer data sets. As objective functions three cluster validity indices namely, XB, PBM, and FCM indices are optimized simultaneously to form more accurate clusters of tissue samples. The presented clustering technique is evaluated for two open source benchmark cancer data sets, which are Brain tumor data set and Adult Malignancy data set. In order to evaluate the quality or goodness of produced clusters two cluster quality measures viz, Adjusted Rand Index ( ARI ) and Classification Accuracy ( %CoA ) are calculated for each data set. Comparative results of the presented clustering algorithm with 10 state-of-the-art existing single-objective, multi-objective based clustering algorithms are shown for two benchmark data sets.

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.

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.

We propose Ordered Subspace Clustering (OSC) to segment data drawn from a sequentially ordered union of subspaces. Current subspace clustering techniques learn the relationships within a set of data and then use a separate clustering algorithm such as NCut for final segmentation. In contrast our technique, under certain conditions, is capable of segmenting clusters intrinsically without providing the number of clusters as a parameter. Similar to Sparse Subspace Clustering (SSC) we formulate the problem as one of finding a sparse representation but include a new penalty term to take care of sequential data. We test our method on data drawn from infrared hyper spectral data, video sequences and face images. Our experiments show that our method, OSC, outperforms the state of the art methods: Spatial Subspace Clustering (SpatSC), Low-Rank Representation (LRR) and SSC.

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.

<p>Differential Evolution (DE) is an algorithm for evolutionary optimization. Clustering problems have been solved by using DE based clustering methods but these methods may fail to find clusters hidden in subspaces of high dimensional datasets. Subspace and projected clustering methods have been proposed in literature to find subspace clusters that are present in subspaces of dataset. In this paper we propose VINAYAKA, a semi-supervised projected clustering method based on DE. In this method DE optimizes a hybrid cluster validation index. Subspace Clustering Quality Estimate index (SCQE index) is used for internal cluster validation and Gini index gain is used for external cluster validation in the proposed hybrid cluster validation index. Proposed method is applied on Wisconsin breast cancer dataset.</p>

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

In an effort to achieve improved classifier accuracy, extensive research has been conducted in classifier ensembles. Very recently, cluster ensembles have emerged. It is well known that off-the-shelf clustering methods may discover different structures in a given set of data. This is because each clustering algorithm has its own bias resulting from the optimization of different criteria. Furthermore, there is no ground truth against which the clustering result can be validated. Thus, no cross-validation technique can be carried out to tune input parameters involved in the clustering process. As a consequence, the user is not equipped with any guidelines for choosing the proper clustering method for a given dataset. Cluster ensembles offer a solution to challenges inherent to clustering arising from its ill-posed nature. Cluster ensembles can provide more robust and stable solutions by leveraging the consensus across multiple clustering results, while averaging out emergent spurious structures that arise due to the various biases to which each participating algorithm is tuned. In this chapter, we discuss the problem of combining multiple weighted clusters, discovered by a locally adaptive algorithm (Domeniconi, Papadopoulos, Gunopulos, & Ma, 2004) which detects clusters in different subspaces of the input space. We believe that our approach is the first attempt to design a cluster ensemble for subspace clustering (Al-Razgan & Domeniconi, 2006). Recently, several subspace clustering methods have been proposed (Parsons, Haque, & Liu, 2004). They all attempt to dodge the curse of dimensionality which affects any algorithm in high dimensional spaces. In high dimensional spaces, it is highly likely that, for any given pair of points within the same cluster, there exist at least a few dimensions on which the points are far apart from each other. As a consequence, distance functions that equally use all input features may not be effective. Furthermore, several clusters may exist in different subspaces comprised of different combinations of features. In many real-world problems, some points are correlated with respect to a given set of dimensions, while others are correlated with respect to different dimensions. Each dimension could be relevant to at least one of the clusters. Global dimensionality reduction techniques are unable to capture local correlations of data. Thus, a proper feature selection procedure should operate locally in input space. Local feature selection allows one to embed different distance measures in different regions of the input space; such distance metrics reflect local correlations of data. In (Domeniconi, Papadopoulos, Gunopulos, & Ma, 2004) we proposed a soft feature selection procedure (called LAC) that assigns weights to features according to the local correlations of data along each dimension. Dimensions along which data are loosely correlated receive a small weight, which has the effect of elongating distances along that dimension. Features along which data are strongly correlated receive a large weight, which has the effect of constricting distances along that dimension. Thus the learned weights perform a directional local reshaping of distances which allows a better separation of clusters, and therefore the discovery of different patterns in different subspaces of the original input space.

Subspace clustering by directly solving Discriminative K-means

Subspace clustering is the task of grouping objects based on mutual similarity in subspaces of the full-dimensional space.<br/>The INSCY algorithm extends the well-known density-based clustering algorithm DBSCAN. It finds dimensionality-unbiased non-redundant subspace clusters using a tree structure to speed up the processing of subspaces. Still, finding density-based clusters in all subspaces implies an exponential search space in the number of dimensions. Thus, the running time of INSCY is still measured in hours on even small datasets of 2000 points. For larger datasets, it becomes prohibitively expensive.<br/><br/>To benefit from INSCY for real-world sized datasets, we propose a novel GPU-parallel approach that runs on standard graphics cards. To utilize the many cores of the GPU, we need new algorithmic strategies that fit the computational model of the GPU. While the GPU provides a large number of threads, traditional algorithms incur diverging threads and poor memory alignment, both of which lead to idle time and poor runtime performance. In INSCY, extracting subspace regions from the SCY-tree structure and the density-based clustering of regions itself are thus unfit for the GPU. <br/><br/>Our novel GPU-friendly algorithm GPU-INSCY computes the same subspace clustering as INSCY at dramatically reduced runtimes. <br/>To achieve this, we devise a restructured SCY-tree index-structure and associated operations for the GPU, as well as a GPU-parallel density-based subspace clustering.<br/><br/>We experimentally show that GPU-INSCY scales well with the size of the dataset and the number of dimensions, and improves the running time of INSCY by a factor of several thousand for large datasets of high dimensionality.

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.

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.