text: Subspace clustering aims to find groups of similar objects (clusters) that exist in lower dimensional subspaces from a high dimensional dataset. It has a wide range of applications, such as analysing high dimensional sensor data or DNA sequences. However, existing algorithms have limitations in finding clusters in non-disjoint subspaces and scaling to large data, which impinge their applicability in areas such as bioinformatics and the Internet of Things. We aim to address such limitations by proposing a subspace clustering algorithm using a bottom-up strategy. Our algorithm first searches for base clusters in low dimensional subspaces. It then forms clusters in higher-dimensional subspaces using these base clusters, which we formulate as a frequent pattern mining problem. This formulation enables efficient search for clusters in higher-dimensional subspaces, which is done using FP-trees. The proposed algorithm is evaluated against traditional bottom-up clustering algorithms and state-of-the-art subspace clustering algorithms. The experimental results show that the proposed algorithm produces clusters with high accuracy, and scales well to large volumes of data. We also demonstrate the algorithm’s performance using real-life ten genomic datasets.
  @cite_35: 
  @cite_18: 
  @cite_36: Two-dimensional contingency tables or co-occurrence matrices arise frequently in various important applications such as text analysis and web-log mining. As a fundamental research topic, co-clustering aims to generate a meaningful partition of the contingency table to reveal hidden relationships between rows and columns. Traditional co-clustering algorithms usually produce a predefined number of flat partition of both rows and columns, which do not reveal relationship among clusters. To address this limitation, hierarchical co-clustering algorithms have attracted a lot of research interests recently. Although successful in various applications, the existing hierarchical co-clustering algorithms are usually based on certain heuristics and do not have solid theoretical background. In this paper, we present a new co-clustering algorithm, HICC, with solid theoretical background. It simultaneously constructs a hierarchical structure of both row and column clusters, which retains sufficient mutual information between rows and columns of the contingency table. An efficient and effective greedy algorithm is developed, which grows a co-cluster hierarchy by successively performing row-wise or column-wise splits that lead to the maximal mutual information gain. Extensive experiments on both synthetic and real datasets demonstrate that our algorithm can reveal essential relationships of row (and column) clusters and has better clustering precision than existing algorithms. Moreover, the experiments on real dataset show that HICC can effectively reveal hidden relationships between rows and columns in the contingency table.
  @cite_9: As a prolific research area in data mining, subspace clustering and related problems induced a vast quantity of proposed solutions. However, many publications compare a new proposition—if at all—with one or two competitors, or even with a so-called “naive” ad hoc solution, but fail to clarify the exact problem definition. As a consequence, even if two solutions are thoroughly compared experimentally, it will often remain unclear whether both solutions tackle the same problem or, if they do, whether they agree in certain tacit assumptions and how such assumptions may influence the outcome of an algorithm. In this survey, we try to clarify: (i) the different problem definitions related to subspace clustering in general; (ii) the specific difficulties encountered in this field of research; (iii) the varying assumptions, heuristics, and intuitions forming the basis of different approaches; and (iv) how several prominent solutions tackle different problems.
  @cite_23: Microarray technology enables the collection of vast amounts of gene expression data from biological experiments. Clustering algorithms have been successfully applied to exploring the gene expression data. Since a set of genes may be only correlated to a subset of samples, it is useful to use co-clustering to recover co-clusters in the gene expression data. In this paper, we propose a novel algorithm, called Subspace Weighting Co-Clustering (SWCC), for high dimensional gene expression data. In SWCC, a gene subspace weight matrix is introduced to identify the contribution of gene objects in distinguishing different sample clusters. We design a new co-clustering objective function to recover the co-clusters in the gene expression data, in which the subspace weight matrix is introduced. An iterative algorithm is developed to solve the objective function, in which the subspace weight matrix is automatically computed during the iterative co-clustering process. Our empirical study shows encouraging results of the proposed algorithm in comparison with six state-of-the-art clustering algorithms on ten gene expression data sets. We also propose to use SWCC for gene clustering and selection. The experimental results show that the selected genes can improve the classification performance of Random Forests.
  @cite_5: Co-clustering, or simultaneous clustering of rows and columns of a two-dimensional data matrix, is rapidly becoming a powerful data analysis technique. Co-clustering has enjoyed wide success in varied application domains such as text clustering, gene-microarray analysis, natural language processing and image, speech and video analysis. In this paper, we introduce a partitional co-clustering formulation that is driven by the search for a good matrix approximation---every co-clustering is associated with an approximation of the original data matrix and the quality of co-clustering is determined by the approximation error. We allow the approximation error to be measured using a large class of loss functions called Bregman divergences that include squared Euclidean distance and KL-divergence as special cases. In addition, we permit multiple structurally different co-clustering schemes that preserve various linear statistics of the original data matrix. To accomplish the above tasks, we introduce a new minimum Bregman information (MBI) principle that simultaneously generalizes the maximum entropy and standard least squares principles, and leads to a matrix approximation that is optimal among all generalized additive models in a certain natural parameter space. Analysis based on this principle yields an elegant meta algorithm, special cases of which include most previously known alternate minimization based clustering algorithms such as kmeans and co-clustering algorithms such as information theoretic (, 2003b) and minimum sum-squared residue co-clustering (, 2004). To demonstrate the generality and flexibility of our co-clustering framework, we provide examples and empirical evidence on a variety of problem domains and also describe novel co-clustering applications such as missing value prediction and compression of categorical data matrices.
  @cite_34: Fuzzy co-clustering is an unsupervised technique that performs simultaneous fuzzy clustering of objects and features. In this paper, we propose a new flexible fuzzy co-clustering algorithm which incorporates feature-cluster weighting in the formulation. We call it Flexible Fuzzy Co-clustering with Feature-cluster Weighting (FFCFW). By flexible we mean the algorithm allows the number of object clusters to be different from the number of feature clusters. There are two motivations behind this work. First, in the fuzzy framework, many co-clustering algorithms still require the number of object clusters to be the same as the number of feature clusters [1][2][3][4]. This is despite the fact that such rigid structure is hardly found in real-world applications. The second motivation is that while there have been numerous attempts for flexible co-clustering, it is common that in such scheme the relationships between object and feature clusters are not clearly represented. For this reason we incorporate a feature-cluster weighting scheme for each object cluster generated by FFCFW so that the relationships between the two types of clusters are manifested in the feature-cluster weights. This enables the new algorithm to generate more accurate representation of fuzzy co-clusters. FFCFW is formulated by fusing together the core components of two existing algorithms [2][5]. Like its predecessors, FFCFW adopts an iterative optimization procedure. We discuss in details the derivation of the proposed algorithm and the advantages it has over other existing works. Experiments on several large benchmark document datasets reveal the feasibility of our proposed algorithm.
summary: Another relevant topic is co-clustering (a.k.a bi-clustering or pattern-based clustering) @cite_35 . Co-clustering can be considered as a more general class of clustering high dimensional data by simultaneously clustering rows (points) and columns (dimensions). The main point that differentiates co-clustering from subspace clustering lies in the approach to the problem, and the homogeneous methodology to find clusters in both axis-parallel and arbitrarily oriented subspaces @cite_9 . In this paper, we also compare the performance of our algorithm on gene expression data with a range of co-clustering algorithms, including SWCC @cite_23 , BBAC-S @cite_5 , ITCC @cite_18 , FFCFW @cite_34 , and HICC @cite_36 .