Rank K Binary Matrix Factorization (BMF) approximates a binary matrix by the product of two binary matrices of lower rank, K. Several researchers have addressed this problem, focusing on either approximations of rank 1 or higher, using either the L_1 or L_2-norms for measuring the quality of the ap proximation. The rank 1 problem (for which the L_1 and L_2-norms are equivalent) has been shown to be related to the Integer Linear Programming (ILP) problem. We first show here that the alternating strategy with the L_2-norm, at the core of several methods used to solve BMF, can be reformulated as an Unconstrained Binary Quadratic Programming (UBQP) problem. This reformulation allows us to use local search procedures designed for UBQP in order to improve the solutions of BMF. We then introduce a new local search dedicated to the BMF problem. We show in particular that this solution is in average faster than the previously proposed ones. We then assess its behavior on several collections and methods and show that it significantly improves methods targeting the L_2-norms on all the datasets considered; for the L_1-norm, the improvement is also significant for real, structured datasets and for the BMF problem without the binary reconstruction constraint.

Rank K Binary Matrix Factorization (BMF) approximates a binary matrix by the product of two binary matrices of lower rank, K, using either L1 or L2 norm. In this paper, we first show that the BMF with L2 norm can be reformulated as an Unconstrained Binary Quadratic Programming (UBQP) problem. We then review several local search strategies that can be used to improve the BMF solutions obtained by previously proposed methods, before introducing a new local search dedicated to the BMF problem. We show in particular that the proposed solution is in general faster than the previously proposed ones. We then assess its behavior on several collections and methods and show that it significantly improves methods targeting the L2 norms on all the datasets considered; for the L1 norm, the improvement is also significant for real, structured datasets and for the BMF problem without the binary reconstruction constraint.

The problem of summarizing a large collection of homogeneous items has been addressed extensively in particular in the case of geo-tagged datasets (e.g. Flickr photos and tags). In our work, we study the problem of summarizing large collections of heterogeneous items. For example, a user planning to spend extended periods of time in a given city would be interested in seeing a map of that city with item summaries in different geographic areas, each containing a theater, a gym, a bakery, a few restaurants and a subway station. We propose to solve that problem by building representative Composite Items (CIs).

We propose in this paper a new, alternative approach for the problem of finding a set of representative objects in large datasets. To do so, we first formulate the general Instance Selection Problem (ISP) and then study three variants of that in order to select instances from different regions of the data. These variants aim at finding the objects located in three very different locations of the data: the inner frontier, the central area and the outer frontier. Solutions to these problems have been discussed and their complexities have been studied. To illustrate the effectiveness of the proposed techniques, we first use a small, synthetic dataset for visualization purpose. We then study them on the Reuters dataset and show that the integration of instances selected by the ISP techniques is able to provide a good representation of the data and can be considered as a complementary approach for the state-of-the-art methods. Finally, we examine the quality of the selected objects by applying a topic-based analysis in order to show how well the selected documents cover the topics in the Reuters dataset.

We address in this paper the problem of efficiently finding a few number of representative frequent itemsets in transaction matrices. To do so, we propose to rely on matrix decomposition techniques, and more precisely on Constrained Binary Matrix Factorization (CBMF) which decomposes a given binary matrix into the product of two lower dimensional binary matrices, called factors. We first show, under binary constraints, that one can interpret the first factor as a transaction matrix operating on packets of items, whereas the second factor indicates which item belongs to which packet. We then formally prove that one can directly mine the CBMF factors in order to find (approximate) itemsets of a given size and support in the original transaction matrix. Then through a detailed experimental study, we show that the frequent itemsets produced by our method represent a significant portion of the set of all frequent itemsets according to existing metrics, while being up to several orders of magnitude less numerous.