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.
In @cite_23 , the authors also use tolerance classes over the set of numbers @math , and create one formal context for each class of tolerance. But they proposed a new algorithm called TriMax to mine the CTV biclusters from these formal contexts. TriMax is able to perform a complete, correct and non-redundat enumeration of all maximal CTV biclusters in a numerical dataset. But due to the scaling process, TriMax may be not feasible in many real-world scenarios too.
@cite_23
2088347683
Biclustering numerical data became a popular data-mining task at the beginning of 2000’s, especially for gene expression data analysis and recommender systems. A bicluster reflects a strong association between a subset of objects and a subset of attributes in a numerical object attribute data-table. So-called biclusters of similar values can be thought as maximal sub-tables with close values. Only few methods address a complete, correct and non-redundant enumeration of such patterns, a well-known intractable problem, while no formal framework exists. We introduce important links between biclustering and Formal Concept Analysis (FCA). Indeed, FCA is known to be, among others, a methodology for biclustering binary data. Handling numerical data is not direct, and we argue that Triadic Concept Analysis (TCA), the extension of FCA to ternary relations, provides a powerful mathematical and algorithmic framework for biclustering numerical data. We discuss hence both theoretical and computational aspects on biclustering numerical data with triadic concept analysis. These results also scale to n-dimensional numerical datasets.