Triadic Formal Concept Analysis and triclustering: searching for optimal patterns

Abstract

This paper presents several definitions of "optimal patterns" in triadic data and results of experimental comparison of five triclustering algorithms on real-world and synthetic datasets. The evaluation is carried over such criteria as resource efficiency, noise tolerance and quality scores involving cardinality, density, coverage, and diversity of the patterns. An ideal triadic pattern is a totally dense maximal cuboid (formal triconcept). Relaxations of this notion under consideration are: OAC-triclusters; triclusters optimal with respect to the leastsquare criterion; and graph partitions obtained by using spectral clustering. We show that searching for an optimal tricluster cover is an NP-complete problem, whereas determining the number of such covers is #P-complete. Our extensive computational experiments lead us to a clear strategy for choosing a solution at a given dataset guided by the principle of Pareto-optimality according to the proposed criteria.

Mixed Integer Programming for Searching Maximum Quasi-Bicliques

“…Let us look at different maximizing criteria for related Mixed Integer Programming models. In papers (Ignatov et al, 2015;Mirkin and Kramarenko, 2011) dedicated to triclustering generation, K = (G, M, B, I) is a triadic context with G, the set of objects, M, the set of attributes, B, the set of conditions, and I ⊆ G × M × B, the ternary relation. The proposed triclustering algorithm searches for clusters that maximize the following criteria:…”

Watset: Local-Global Graph Clustering with Applications in Sense and Frame Induction

“…Such roles can be implicit, i.e., unexpressed in a given context (Schenk and Chiarcos 2016), so additional syntactic relationships between frame elements could be taken into account (Kallmeyer, QasemiZadeh, and Cheung 2018). We cast the frame induction problem as a triclustering task (Zhao and Zaki 2005;Ignatov et al 2015). Triclustering is a generalization of traditional clustering and biclustering problems (Mirkin 1996, p. 144), aiming at simultaneously clustering objects along three dimensions, i.e., subject, verb and object in our case (cf .…”

A comprehensive survey on formal concept analysis, its research trends and applications

“…, where X represents a set of objects, Y represents a set of attributes and Z represents a set of conditions, i.e, if (x, y, z) ∈Ĩ means object x has attribute y under condition z, whereas in the case of fuzzy attributes I represents the relationship among them using fuzzy membership-value. From a triadic context, the number of dyadic contexts can be derived as follows (Ignatov et al, 2015):…”

“…Finally, optimal factor F includes ({c 1 , c 2 , c 3 , c 4 }, which decompose the matrices (7×6) shown in Table 13 into two 7×4 and 4×7 matrices to be analyzed in a 4-dimensional space of factors instead of describing the galaxies in a 7-dimensional space. Recently, some applications of factor concepts have been shown in FCA with a fuzzy setting also (Belohlavek et al, 2013a;Bartl et al, 2011;Ignatov et al, 2015;Yao et al, 2012) as well as in graph theory (Helmi et al, 2014).…”