Factorization of matrices with grades

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61989592%3A15310%2F16%3A33160375" target="_blank" >RIV/61989592:15310/16:33160375 - isvavai.cz</a>
Result on the web
<a href="http://www.sciencedirect.com/science/article/pii/S0165011415001694" target="_blank" >http://www.sciencedirect.com/science/article/pii/S0165011415001694</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.fss.2015.03.020" target="_blank" >10.1016/j.fss.2015.03.020</a>

Result language
angličtina
Original language name
Factorization of matrices with grades
Original language description
We present an approach to decomposition and factor analysis of matrices with ordinal data. The matrix entries are grades to which objects represented by rows satisfy attributes represented by columns, e.g. grades to which an image is red, a product has a given feature, or a person performs well in a test. We assume that the grades are taken from bounded scales equipped with certain aggregation operators that are involved in the decompositions. Particular cases of the decompositions include the well-known Boolean matrix decomposition, and the sup-t-norm and inf-residuum decompositions. We consider the problem of decomposition of a given matrix into a product of two matrices with grades such that the number of factors, i.e. the inner dimension, be as small as possible. We observe that computing such decompositions is NP-hard and present a greedy approximation algorithm. Our algorithm is based on a geometric insight provided by a theorem identifying particular rectangular-shaped submatrices as optimal factors for the decompositions. These factors correspond to fixpoints of certain Galois connections associated with the input matrix, which are called formal concepts, and allow an easy interpretation of the decomposition. We present illustrative examples and experimental evaluation of the algorithm.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
IN - Informatics
OECD FORD branch
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Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2016
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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