Banded structure in binary matrices

Gemma Garriga (1) , Esa Junttila (2) , Heikki Mannila (3)

1 MOSTRARE - Modeling Tree Structures, Machine Learning, and Information Extraction
2 Department of Computer Science [Helsinki]
3 Department of Information and Computer Science

A binary matrix has a banded structure if both rows and columns can be permuted so that the non-zero entries exhibit a staircase pattern of overlapping rows. The concept of banded matrices has its origins in numerical analysis, where entries can be viewed as descriptions between the problem variables; the bandedness corresponds to variables that are coupled over short distances. Banded data occurs also in other applications, for example in the physical mapping problem of the human genome, in paleontological data, in network data and in the discovery of overlapping communities without cycles. We study the banded structure of binary matrices, give a formal definition of the concept and discuss its theoretical properties. We consider the algorithmic problems of computing how far a matrix is from being banded, and of finding a good submatrix of the original data that exhibits approximate bandedness. Finally, we show by experiments on real data from ecology and other applications the usefulness of the concept. Our results reveal that bands exist in real datasets and that the final obtained orderings of rows and columns have natural interpretations.

Domaines: Autres [stat.ML]

Gemma Garriga, Esa Junttila, Heikki Mannila. Banded structure in binary matrices. Knowledge and Information Systems (KAIS), 2011, 28 (1), pp.197-226. ⟨10.1007/s10115-010-0319-7⟩. ⟨hal-00658836⟩