Elsevier

Fuzzy Sets and Systems

Volume 360, 1 April 2019, Pages 97-116
Fuzzy Sets and Systems

Factorization of matrices with grades via essential entries

https://doi.org/10.1016/j.fss.2018.07.004Get rights and content

Abstract

We present new results regarding the problem of factorization of matrices with grades, or, equivalently, decomposition of fuzzy relations. In particular, we examine geometry of factorizations and the role of fuzzy concept lattices in factorizations of matrices with grades. The results make it possible to reduce input data and enable a more focused search for factors in the search space, and are intended to guide the design of greedy and other approximation algorithms for the decomposition problem, which itself is NP-hard. To demonstrate usefulness of these results, we propose a new factorization algorithm based on these results. Our experiments demonstrate improvements in the quality of factorizations due to the new approach. We conclude by presenting further research topics implied by our findings.

Section snippets

Problem description

The problem we consider may be formulated in terms of matrices or, equivalently, in terms of relations. We proceed for matrices, which framework is commonly used for this problem. Let L denote a partially ordered set of grades bounded by 0 and 1. We primarily interpret the grades as truth degrees and denote by Ln×m the set of all matrices I with n rows and m columns. That is, Iij is a truth degree in L and we interpret it as the degree to which the object represented by i has the attribute

Hardness of the approximate factorization problem

We start by the following observation which is crucially important for algorithmic considerations. In view of the fact that the problem of exact decomposition is NP-hard ([13], see also [7], [12]), this theorem is actually not surprising.

Theorem 1

The approximate factorization problem is an NP-hard optimization problem for any L.

Proof

The proof proceeds by adaptation of the proofs of NP-hardness of the exact decomposition problem in the Boolean case (see e.g. [7]). We need to take into account that instead of

A new factorization algorithm

The

algorithm (see Algorithm 1, Algorithm 2), which we now present, is based on the results presented in the previous section and some other properties mentioned below in this section. Due to algorithmic considerations, we assume that the set L of grades is finite throughout this section.
computes, for a given matrix ILn×m and a prescribed precision ε>0 a set F of formal concepts of I, i.e. FB(I), such that the corresponding matrices AF and BF provide an approximate

Experiments

In our experimental evaluation, we performed two kinds of experiments. The first consists in performing factor analysis of real data. Since it has been repeatedly demonstrated that factor analysis of matrices with grades using formal concepts yields well-interpretable, informative factors (see e.g. [5] for numerous examples), our aim is to show that our new algorithm retains this convenient property. We performed analyses of several datasets and, for space reasons, present in Section 4.1 a

Conclusions

The contributions of our paper concern two aspects of the problem of factorization of matrices with grades. The first concerns theoretical insight regarding the problem. Such insight is largely needed because the existing factorization algorithms including those for Boolean matrices are mostly based on various greedy strategies and utilize only a very limited insight. Our results concern geometry of factorizations and make it possible to identify entries in the input matrix to which one may

Acknowledgement

Supported by grant No. 15-17899S of the Czech Science Foundation. Preliminary results in this paper were presented in Belohlavek, R., Krmelova M., Proc. ICDM 2013, pp. 961–966. R. Belohlavek acknowledges support by the ECOP (Education for Competitiveness Operational Programme) project no. CZ.1.07/2.3.00/20.0059, which is co-financed by the European Social Fund and the state budget of the Czech Republic. The present work was done during the sustainability period of this project.

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