Biclustering has proved to be a powerful data analysis technique due to its wide success in various application domains. However, the existing literature presents efficient solutions only for enumerating maximal biclusters with constant values, or heuristic-based approaches which can not find all biclusters or even support the maximality of the obtained biclusters. Here, we present a general family of biclustering algorithms for enumerating all maximal biclusters with (i) constant values on rows, (ii) constant values on columns, or (iii) coherent values. Versions for perfect and for perturbed biclusters are provided. Our algorithms have four key properties (just the algorithm for perturbed biclusters with coherent values fails to exhibit the first property): they are (1) efficient (take polynomial time per pattern), (2) complete (find all maximal biclusters), (3) correct (all biclusters attend the user-defined measure of similarity), and (4) non-redundant (all the obtained biclusters are maximal and the same bicluster is not enumerated twice). They are based on a generalization of an efficient formal concept analysis algorithm called In-Close2. Experimental results point to the necessity of having efficient enumerative biclustering algorithms and provide a valuable insight into the scalability of our family of algorithms and its sensitivity to user-defined parameters.
Maple @cite_4 is an improved version of pCluster and it is closer to an actual enumerative algorithm. It returns only non-redundant biclusters, but it does not have an efficient approach to do this: for each possible new bicluster, Maple must look at all previously generated biclusters to avoid redundancy. Besides, there are two scenarios where Maple fails in performing a complete and correct enumeration of all maximal biclusters. If two biclusters have the same set of objects and share some attributes, Maple would return only one bicluster containing both of them (thus violating the user-defined measure of similarity). Maple also may miss biclusters due to its routine of pruning unpromising biclusters: Maple keeps an attribute-list ordered by some criterion. If a bicluster has a subset of objects and a superset of attributes of another bicluster, and its extra attributes are subsequently considering Maple's attribute-list, Maple would prune it incorrectly. The worst-case time of Maple's search is also exponential w.r.t. the number of attributes.
@cite_4
1894726941 2011058337 2145799156 2195615921
Biclustering has proved to be a powerful data analysis technique due to its wide success in various application domains. However, the existing literature presents efficient solutions only for enumerating maximal biclusters with constant values, or heuristic-based approaches which can not find all biclusters or even support the maximality of the obtained biclusters. Here, we present a general family of biclustering algorithms for enumerating all maximal biclusters with (i) constant values on rows, (ii) constant values on columns, or (iii) coherent values. Versions for perfect and for perturbed biclusters are provided. Our algorithms have four key properties (just the algorithm for perturbed biclusters with coherent values fails to exhibit the first property): they are (1) efficient (take polynomial time per pattern), (2) complete (find all maximal biclusters), (3) correct (all biclusters attend the user-defined measure of similarity), and (4) non-redundant (all the obtained biclusters are maximal and the same bicluster is not enumerated twice). They are based on a generalization of an efficient formal concept analysis algorithm called In-Close2. Experimental results point to the necessity of having efficient enumerative biclustering algorithms and provide a valuable insight into the scalability of our family of algorithms and its sensitivity to user-defined parameters. We present a novel, general, optimally fast, incremental way of searching for a universal algorithm that solves each task in a sequence of tasks. The Optimal Ordered Problem Solver (OOPS) continually organizes and exploits previously found solutions to earlier tasks, efficiently searching not only the space of domain-specific algorithms, but also the space of search algorithms. Essentially we extend the principles of optimal nonincremental universal search to build an incremental universal learner that is able to improve itself through experience. The initial bias is embodied by a task-dependent probability distribution on possible program prefixes. Prefixes are self-delimiting and executed in online fashion while being generated. They compute the probabilities of their own possible continuations. Let p^n denote a found prefix solving the first n tasks. It may exploit previously stored solutions p^i, i >n, by calling them as subprograms, or by copying them and editing the copies before applying them. We provide equal resources for two searches that run in parallel until p^ n+1 is discovered and stored. The first search is exhaustive; it systematically tests all possible prefixes on all tasks up to n+1. The second search is much more focused; it only searches for prefixes that start with p^n, and only tests them on task n+1, which is safe, because we already know that such prefixes solve all tasks up to n. Both searches are depth-first and bias-optimal: the branches of the search trees are program prefixes, and backtracking is triggered once the sum of the runtimes of the current prefix on all current tasks exceeds the prefix probability multiplied by the total search time so far. In illustrative experiments, our self-improver becomes the first general system that learns to solve all n disk Towers of Hanoi tasks (solution size 2^n-1) for n up to 30, profiting from previously solved, simpler tasks involving samples of a simple context free language. We consider the problem of identifying a sparse set of relevant columns and rows in a large data matrix with highly corrupted entries. This problem of identifying groups from a collection of bipartite variables such as proteins and drugs, biological species and gene sequences, malware and signatures, etc is commonly referred to as biclustering or co-clustering. Despite its great practical relevance, and although several ad-hoc methods are available for biclustering, theoretical analysis of the problem is largely non-existent. The problem we consider is also closely related to structured multiple hypothesis testing, an area of statistics that has recently witnessed a flurry of activity. We make the following contributions 1. We prove lower bounds on the minimum signal strength needed for successful recovery of a bicluster as a function of the noise variance, size of the matrix and bicluster of interest. 2. We show that a combinatorial procedure based on the scan statistic achieves this optimal limit. 3. We characterize the SNR required by several computationally tractable procedures for biclustering including element-wise thresholding, column row average thresholding and a convex relaxation approach to sparse singular vector decomposition. Mining matrices to find relevant biclusters, subsets of rows exhibiting a coherent pattern over a subset of columns, is a critical task for a wide-set of biomedical and social applications. Since biclustering is a challenging combinatorial optimization task, existing approaches place restrictions on the allowed structure, coherence and quality of biclusters. Biclustering approaches relying on pattern mining (PM) allow an exhaustive yet efficient space exploration together with the possibility to discover flexible structures of biclusters with parameterizable coherency and noise-tolerance. Still, state-of-the-art contributions are dispersed and the potential of their integration remains unclear.This work proposes a structured and integrated view of the contributions of state-of-the-art PM-based biclustering approaches, makes available a set of principles for a guided definition of new PM-based biclustering approaches, and discusses their relevance for applications in pattern recognition. Empirical evidence shows that these principles guarantee the robustness, efficiency and flexibility of PM-based biclustering. HighlightsPattern mining (PM) searches enable flexible, exhaustive and efficient biclusteringIntegration of existing dispersed PM-inspired contributions for biclustering.Principles for guided design and evaluation of new PM-based biclustering approachesPM-based biclustering solutions have parameterizable coherency and quality.