Biclustering, the discovery of sets of objects with a coherent pattern across a subset of conditions, is a critical task to study a wide-set of biomedical problems, where molecular units or patients are meaningfully related with a set of properties. The challenging combinatorial nature of this task led to the development of approaches with restrictions on the allowed type, number and quality of biclusters. Contrasting, recent biclustering approaches relying on pattern mining methods can exhaustively discover flexible structures of robust biclusters. However, these approaches are only prepared to discover constant biclusters and their underlying contributions remain dispersed. The proposed BicPAM biclustering approach integrates existing principles made available by state-of-the-art pattern-based approaches with two new contributions. First, BicPAM is the first efficient attempt to exhaustively mine non-constant types of biclusters, including additive and multiplicative coherencies in the presence or absence of symmetries. Second, BicPAM provides strategies to effectively compose different biclustering structures and to handle arbitrary levels of noise inherent to data and with discretization procedures. Results show BicPAM’s superiority against its peers and its ability to retrieve unique types of biclusters of interest, to efficiently deliver exhaustive solutions and to successfully recover planted biclusters in datasets with varying levels of missing values and noise. Its application over gene expression data leads to unique solutions with heightened biological relevance. BicPAM approaches integrate existing disperse efforts towards pattern-based biclustering and provides the first critical strategies to efficiently discover exhaustive solutions of biclusters with shifting, scaling and symmetric assumptions with varying quality and underlying structures. Additionally, BicPAM dynamically adapts its behavior to mine data with different levels of missing values and noise.

Three-dimensional data are increasingly prevalent across biomedical and social domains. Notable examples are gene-sample-time, individual-feature-time, or node-node-time data, generally referred to as observation-attribute-context data. The unsupervised analysis of three-dimensional data can be pursued to discover putative biological modules, disease progression profiles, and communities of individuals with coherent behavior, among other patterns of interest. It is thus key to enhance the understanding of complex biological, individual, and societal systems. In this context, although clustering can be applied to group observations, its relevance is limited since observations in three-dimensional data domains are typically only meaningfully correlated on subspaces of the overall space. Biclustering tackles this challenge but disregards the third dimension. In this scenario, triclustering—the discovery of coherent subspaces within three-dimensional data—has been largely researched to tackle these problems. Despite the diversity of contributions in this field, there still lacks a structured view on the major requirements of triclustering, desirable forms of homogeneity (including coherency, structure, quality, locality, and orthonormality criteria), and algorithmic approaches. This work formalizes the triclustering task and its scope, introduces a taxonomy to categorize the contributions in the field, provides a comprehensive comparison of state-of-the-art triclustering algorithms according to their behavior and output, and lists relevant real-world applications. Finally, it highlights challenges and opportunities to advance the field of triclustering and its applicability to complex three-dimensional data analysis.

Biclustering has been largely applied for the unsupervised analysis of biological data, being recognised today as a key technique to discover putative modules in both expression data (subsets of genes correlated in subsets of conditions) and network data (groups of coherently interconnected biological entities). However, given its computational complexity, only recent breakthroughs on pattern-based biclustering enabled efficient searches without the restrictions that state-of-the-art biclustering algorithms place on the structure and homogeneity of biclusters. As a result, pattern-based biclustering provides the unprecedented opportunity to discover non-trivial yet meaningful biological modules with putative functions, whose coherency and tolerance to noise can be tuned and made problem-specific. To enable the effective use of pattern-based biclustering by the scientific community, we developed BicPAMS (Biclustering based on PAttern Mining Software), a software that: 1) makes available state-of-the-art pattern-based biclustering algorithms (BicPAM (Henriques and Madeira, Alg Mol Biol 9:27, 2014), BicNET (Henriques and Madeira, Alg Mol Biol 11:23, 2016), BicSPAM (Henriques and Madeira, BMC Bioinforma 15:130, 2014), BiC2PAM (Henriques and Madeira, Alg Mol Biol 11:1–30, 2016), BiP (Henriques and Madeira, IEEE/ACM Trans Comput Biol Bioinforma, 2015), DeBi (Serin and Vingron, AMB 6:1–12, 2011) and BiModule (Okada et al., IPSJ Trans Bioinf 48(SIG5):39–48, 2007)); 2) consistently integrates their dispersed contributions; 3) further explores additional accuracy and efficiency gains; and 4) makes available graphical and application programming interfaces. Results on both synthetic and real data confirm the relevance of BicPAMS for biological data analysis, highlighting its essential role for the discovery of putative modules with non-trivial yet biologically significant functions from expression and network data. BicPAMS is the first biclustering tool offering the possibility to: 1) parametrically customize the structure, coherency and quality of biclusters; 2) analyze large-scale biological networks; and 3) tackle the restrictive assumptions placed by state-of-the-art biclustering algorithms. These contributions are shown to be key for an adequate, complete and user-assisted unsupervised analysis of biological data. BicPAMS and its tutorial available in http://www.bicpams.com .

Biclustering is a critical task for biomedical applications. Order-preserving biclusters, submatrices where the values of rows induce the same linear ordering across columns, capture local regularities with constant, shifting, scaling and sequential assumptions. Additionally, biclustering approaches relying on pattern mining output deliver exhaustive solutions with an arbitrary number and positioning of biclusters. However, existing order-preserving approaches suffer from robustness, scalability and/or flexibility issues. Additionally, they are not able to discover biclusters with symmetries and parameterizable levels of noise.