Generalized Matrix Factorizations as a Unifying Framework for Pattern Set Mining: Complexity Beyond Blocks

Abstract. Matrix factorizations are a popular tool to mine regularities from data. There are many ways to interpret the factorizations, but one particularly suited for data mining utilizes the fact that a matrix product can be interpreted as a sum of rank-1 matrices. Then the factorization of a matrix becomes the task of finding a small number of rank-1 matrices, sum of which is a good representation of the original matrix. Seen this way, it becomes obvious that many problems in data mining can be expressed as matrix factorizations with correct definitions of what a rank-1 matrix and a sum of rank-1 matrices mean. This paper develops a unified theory, based on generalized outer product operators, that encompasses many pattern set mining tasks. The focus is on the computational aspects of the theory and studying the computational complexity and approximability of many problems related to generalized matrix factorizations. The results immediately apply to a large number of data mining problems, and hopefully allow generalizing future results and algorithms, as well.

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Fully Dynamic Quasi-Biclique Edge Covers via Boolean Matrix Factorizations

An important way of summarizing a bipartite graph is to give a set of (quasi-) bicliques that contain (almost) all of its edges. These quasi-bicliques are somewhat similar to clustering of the nodes, giving sets of similar nodes. Unlike clustering, however, the quasi-bicliques are not required to partition the nodes, allowing greater flexibility when creating them. When we identify the bipartite graph with its bi-adjacency matrix, the problem of finding these quasi-bicliques turns into the problem of finding the Boolean matrix factorization of the bi-adjacency matrix – a problem that has received increasing research interest in data mining in recent years. But many real-world graphs are dynamic and evolve over time. How can we update our bicliques without having to re-compute them from the scratch? An algorithm was recently proposed for this task (Miettinen, ICMD 2012). The algorithm, however, is only able to handle the case where the new 1s are added to the matrix – it cannot handle the removal of existing 1s. Furthermore, the algorithm cannot adjust the rank of the factorization. This paper extends said algorithm with the capability of working in fully dynamic setting (with both additions and deletions) and with capability of adjusting its rank dynamically, as well. The behaviour and performance of the algorithm is studied in experiments conducted with both real-world and synthetic data

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum